Geometric Measure of Entanglement and Schmidt Decomposition of Multipartite Systems
aa r X i v : . [ qu a n t - ph ] M a r A. ALIKHANYAN NATIONAL SCIENCE LABORATORYYEREVAN PHYSICS INSTITUTE
DEPARTMENT OF THEORETICAL PHYSICS
Geometric Measure of Entanglement andSchmidt Decomposition of Multipartite Systems
Levon Tamaryan
Yerevan Physics InstituteAlikhanian Br. Str. 2, 0036 Yerevan, Armenia
PhD Thesis
Advisor: Dr. Lekdar Gevorgyan
Abstract
The thesis includes the original results of our articles [30, 37, 40, 42, 51, 53, 75].These results are described in a concise form below.A method is developed to compute analytically entanglement measures ofthree-qubit pure states. The methods leans on the theorem stating that entan-glement measures of the n-party pure state can be expressed by the (n-1)-partyreduced state density operator directly. Owing to this theorem algebraic equa-tions are derived for the geometric measure of entanglement and solved explicitlyin the cases of most interest. The solutions give analytic expressions for the ge-ometric entanglement measure in a wide range of three-qubit systems, includingthe general class of W-type states and states which are symmetric under thepermutation of two qubits [37, 40].The same method is used to find the geometric measure of entanglement ofgeneric three-qubit pure states. Closed-form expressions are presented for thegeometric measure of entanglement for three-qubit states that are linear com-binations of four orthogonal product states. It turns out that the geometricmeasure for these states has three different expressions depending on the rangeof definition in parameter space. Each expression of the measure has its own ge-ometrically meaningful interpretation and thus the Hilbert space of three-qubitsconsists of three different entangled regions. The states that lie on joint sur-faces separating different entangled regions, designated as shared states, haveparticularly interesting features and are dual quantum channels for the perfectteleportation and superdense coding [42].A powerful method is developed to compute analytically multipartite entan-glement measures. The method uses the duality concept and creates a bijectionbetween highly entangled quantum states and their nearest separable states. Thebijection gives explicitly the geometric entanglement measure of arbitrary gener-alized W states of n qubits and singles out two critical points of entanglementin quantum state parameter space. The first critical value separates symmetricand asymmetric entangled regions of highly entangled states, while the secondone separates highly and slightly entangled states [30, 75].The behavior of the geometric entanglement measure of many-qubit W statesis analyzed and an interpolating formula is derived. The importance of the inter-polating formula in quantum information is threefold. First, it connects quantitiesthat can be easily estimated in experiments. Second, it is an example of how wecompute entanglement of a quantum state with many unknowns. Third, one canprepare the W state with a given entanglement bringing into the position a singlequantity [51].Generalized Schmidt decomposition of pure three-qubit states has four posi-tive and one complex coefficients. In contrast to the bipartite case, they are notarbitrary and the largest Schmidt coefficient restricts severely other coefficients.It is derived a non-strict inequality between three-qubit Schmidt coefficients,where the largest coefficient defines the least upper bound for the three nondi-agonal coefficients or, equivalently, the three nondiagonal coefficients togetherdefine the greatest lower bound for the largest coefficient. Besides, it is shownthe existence of another inequality which should establish an upper bound forthe remaining Schmidt coefficient [53]. ontents
Introduction 71 Three Qubit Geometric Measure 15 P max . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.1 General Feature . . . . . . . . . . . . . . . . . . . . . . . . 302.3.2 Type 1 (Product States): J = J = J = J = J = 0 . . . 312.3.3 Type2a (biseparable states) . . . . . . . . . . . . . . . . . 312.3.4 Type2b (generalized GHZ states): J = 0, J = J = J = J = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.5 Type3a (tri-Bell states) . . . . . . . . . . . . . . . . . . . . 332.3.6 Type3b (extended GHZ states) . . . . . . . . . . . . . . . 342.3.7 Type4a ( λ = 0) . . . . . . . . . . . . . . . . . . . . . . . 362.3.8 Type4b . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.3.9 Type4c ( λ = 0) . . . . . . . . . . . . . . . . . . . . . . . . 372.3.10 Type5 (real states): ϕ = 0, π . . . . . . . . . . . . . . . . 382.4 New Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.4.1 standard form . . . . . . . . . . . . . . . . . . . . . . . . . 392.4.2 LU-invariants . . . . . . . . . . . . . . . . . . . . . . . . . 402.4.3 calculation of P max . . . . . . . . . . . . . . . . . . . . . . 412.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ONTENTS O ONTENTS Appendix C Geometrical interpretation 102References 104 ntroduction
Building quantum information processing devices is a great challenge for scien-tists and engineers of the third millennium [1, 2]. Compound quantum systemshave potential for many quantum processes, including the following applications:factoring of large composite numbers [3, 4], quantum cryptography [5, 6], su-perdense coding [7, 8], quantum teleportation [9, 10] and exponential speedup ofquantum computers [11, 12, 13]. These remarkable phenomena have provided abasis for the development of modern quantum information science.The superior performance of quantum systems in computation and commu-nication applications is rooted in a property of quantum mechanical states calledentanglement [14, 15, 16]. Quantum entanglement is a physical resource associ-ated with the peculiar nonclassical correlations that are possible between sepa-rated quantum systems. It is a fundamental property of quantum systems and abasic physical resource for quantum information science [17]. In general, any taskinvolving distant parties and using up entangled states as a resource benefits froma better understanding of entanglement. It is increasingly realized that quantumentanglement is at the heart of quantum physics and as such it may be of verybroad importance for modern science and future technologies.Entanglement is usually created by direct interactions between subatomicparticles. If two particles are entangled, then there is a correlation between theresults of measurements performed on entangled pairs, and this correlation isobserved even though the entangled pair may have been separated by arbitrarilylarge distances. In the multipartite case the entanglement is more complicatedconcept and to distinguish entangled and unentangled quantum states in thiscase it is necessary to define product states and separable states.Consider multipartite systems. The Hilbert space of a such system is thetensor product of the Hilbert spaces of single particles. There is a simple definitionof unentangled states in the case of pure states. Indeed, the vector(pure state)belonging to the Hilbert space of the multipartite system is called a productstate if it is a tensor product of vectors(pure states) belonging to Hilbert spacesof single particles. In other words, a pure state of a multi-particle system isa product state if and only if all subsystems are pure states. Clearly, there isno correlation between subsystems of product states and they are unentangledstates. 7
ONTENTS • Separable states contain no entanglement. • All nonseparable states are entangled. • The entanglement of states does not increase under LOCC operations. • Entanglement does not change under LU-transformations.Many entanglement measures have been proposed for the two-particle as wellas for the multi-particle case [18]. They are very difficult to compute as theirdefinition contains optimizations over certain quantum states or quantum infor-mation protocols. In bipartite case entanglement is relatively well understood,while in multipartite case quantifying entanglement of pure states is a questionof vital importance.The geometric measure of entanglement(GM) is one of the most reliable quan-tifiers of multipartite entanglement [22, 23, 24, 25]. It measures the distance of agiven quantum state from the set of product states and is a decreasing function ofthe maximal product overlap of the quantum state. The maximal product over-lap (MPO) of a quantum pure state is the absolute value of the inner product ofthe quantum state and its nearest separable state. It(or its square) has severalnames and we list all of them for the completeness: entanglement eigenvalue [25],injective tensor norm [26], maximal probability of success [28], maximum singularvalue [29] and maximal product overlap [30].The geometric measure of entanglement (GM) has the following remarkableproperties and applications:
ONTENTS
91. It has identified irregularity in channel capacity additivity. Using this mea-sure, one can show that a family of quantities, which were thought to beadditive in earlier papers, actually are not [26].2. It has an operational treatment and quantifies how well a given state servesas an input state to Grover’s search algorithm [27, 28].3. It has useful connections to other entanglement measures and gives rise toa lower bound on the relative entropy of entanglement [31] and generalizedrobustness [32].4. It quantifies the difficulty to distinguish multipartite quantum states bylocal means [33].5. It exhibits interesting connections with entanglement witnesses and can beefficiently estimated in experiments [34].6. It has been used to prove that one dimensional quantum systems tendto be globally separable along renormalization group flows by following auniversal scaling law in the correlation length of the system. Owing to thisone can understand the physical implication of Zamolodchikov’s c-theoremmore deeply [35].7. It has been used to study quantum phase transitions in spin models [36].8. It singles out states that can be used as a quantum channel for the perfectteleportation and superdence coding [37].9. It gives the largest coefficient of the generalized Schmidt decomposition andthe corresponding nearest product state uniquely defines the factorisablebasis of the decomposition [38].10. It has been used to derive a single-parameter family of the maximallyentangled three-qubit states, where the paradigmatic Greenberger-Horne-Zeilinger and W states emerge as the extreme members in this family ofmaximally entangled states [39].Owing to these features, GM can play an important role in the investigation ofdifferent problems related to entanglement. In spite of its usefulness one obsta-cle to use GM fully in quantum information theories is the that it is difficult tocompute it analytically for generic states. The usual maximization method gen-erates a system of nonlinear equations which are unsolvable in general. Thus, it isimportant to develop a technique for the computation of GM [40, 41, 42, 43, 44].For bipartite systems maim problems problems related to entanglement havebeen solved with the help of the Schmidt decomposition [45, 46]. Therefore its
ONTENTS etal [47], where it is shown that an arbitrary pure state can be written as a linearcombination of five product states. Independently, Carteret et al developed amethod for such a generalization for pure states of arbitrary multipartite sys-tem, where the dimensions of the individual state spaces are finite but otherwisearbitrary [38].However, for a given quantum state the canonical form is not unique and thesame state can have different canonical forms and therefore different sets of suchamplitudes. The reason is that the stationarity equations defining stationaritypoints are nonlinear equations and in general have several solutions of differenttypes. Then the question is which of amplitude sets should be treated as Schmidtcoefficients and which ones should be treated as insignificant mathematical so-lutions. A criterion should exist that can distinguish right Schmidt coefficientsfrom false ones and we need such a criterion. It is unlikely that we can solveproblems of three-qubit entanglement without knowledge of what quantities arethe relevant entanglement parameters.The main goals of the thesis are:a) to develop methods that allow us to compute analytically multipartite en-tanglement measures,b) to derive analytic expressions for the geometric measure of entanglementof multi-particle systems,c) to analyze basic phenomena in quantum information theory using closedform solutions for geometric measure.d) to find inequalities which define a unique Schmidt decomposition for genericmultipartite systemsWe have developed two powerful methods to compute analytically multipar-tite entanglement measures.The first method, hereafter referred to as reduced density method , allows usto compute analytically entanglement measures of three-qubit pure states. Thethree-qubit system is important in the sense that it is the simplest system whichgives a nontrivial effect in the entanglement. Thus, we should understand thegeneral properties of the entanglement in this system as much as possible to gofurther to more complicated higher-qubit systems. The three-qubit system can beentangled in two inequivalent ways – Greenberger-Horne-Zeilinger (GHZ) [48] andW – and neither form can be transformed into the other with any probability ofsuccess [49]. This picture is complete: any fully entangled three-qubit pure statecan be obtained from either the GHZ or W state via stochastic local operationsand classical communication (SLOCC).
ONTENTS duality method , allows us tocompute analytically the entanglement measures of highly entangled n-qubit purestates. The main point of the method is the theorem stating that the nearestproduct state is essentially unique if the quantum state is highly entangled [30].This makes it possible to map highly entangled state to its nearest product stateand quickly obtain its geometric measure of entanglement. More precisely, weconstruct two bijections. The first one creates a map between highly entangledn-qubit quantum states and n-dimensional unit vectors. The second one does thesame between n-dimensional unit vectors and n-part product states. Thus weobtain a double map, or duality , as followsn-qubit pure states ↔ n-dimensional spatial vectors ↔ n-part product states.The main advantage of the map is that if one knows any of the three vectors,then one instantly finds the other two. Hence we find the geometric measure ofentanglement of general multiqubit W states.The derived answer shows that highly entangled W states have two exceptionalpoints in the parameter space. At the second exceptional point the reduceddensity operator of a some qubit is a constant multiple of the unit operator andthen the maximal product overlap of these states is a constant regardless howmany qubits are involved and what are the values of the remaining entanglementparameters. These states are known as shared quantum states and can be usedas quantum channels for the perfect teleportation and dense coding.Next it is shown that W-states have two different entangled regions: thesymmetric and asymmetric entangled regions. In the computational basis theseregions can be defined as follows. If a W state is in the symmetric region, thenthe entanglement is a fully symmetric function on the state parameters. Con-versely, if a W state is in the asymmetric region, then there is an exceptionalparameter such that the entanglement dependence on the exceptional parameterdiffers dramatically from the dependencies of the remaining parameters. Hencethe point of intersection of the symmetric and asymmetric regions is the firstexceptional point.The first exceptional point is important for large-scale W states [51]. Itapproaches to a fixed point when number of qubits n increases and becomesstate-independent(up to 1/n corrections) when n ≫
1. As a consequence the en-
ONTENTS
Chapter
Chapter
ONTENTS
Chapter
Chapter
Chapter n ≫ z com-ponent. Hence one can prepare a W state with the required maximal productoverlap by altering the Bloch vector of a single qubit. Next we compute analyti-cally the geometric measure of large-scale W states by describing these systemsin terms of very few parameters. The final formula relates two quantities, namelythe maximal product overlap and the Bloch vector, that can be easily estimatedin experiments.In Chapter
ONTENTS
Summary we give the main points of our results and conclusions.In
Bibliography we list our references in order of appearance. hapter 1Analytic Expressions forGeometric Measure of ThreeQubit States
In this chapter we compute analytically the geometric measure of entanglementof three-qubit pure states [37].The entanglement of bipartite systems is well-understood [19, 20, 21, 55], whilethe entanglement of multipartite systems offers a real challenge to physicists. Themain point which makes difficult to understand the entanglement for the multi-qubit systems is mainly due to the fact that the analytic expressions for thevarious entanglement measures is extremely hard to derive.We consider pure three qubit systems [47, 56, 57, 58, 59], although the en-tanglement of mixed states attracts a considerable attention. Only very fewanalytical results for tripartite entanglement have been obtained so far and weneed more light on the subject.Recently the idea was suggested that nonlinear eigenproblem can be reducedto the linear eigenproblem for the case of three qubit pure states [50]. The idea isbased on theorem stating that any reduced ( n − n -qubit pure state. This means that twoqubit mixed states can be used to calculate the geometric measure of three qubitpure states and this will be fully addressed in this work.The method gives two algebraic equations of degree six defining the geomet-ric measure of entanglement. Thus the difficult problem of geometric measurecalculation is reduced to the algebraic equation root finding. Equations containvaluable information, are good bases for the numerical calculations and may testnumerical calculations based on other numerical techniques [60].Furthermore, the method allows to find the nearest separable states for threequbit states of most interest and get analytic expressions for their geometricmeasures. It turn out that highly entangled states have their own feature. Eachhighly entangled state has a vicinity with no product state and all nearest product15 HAPTER 1. THREE QUBIT GEOMETRIC MEASURE
We start by developing a general formulation, appropriate for multipartite sys-tems comprising n parts, in which each part has its distinct Hilbert space. Let | ψ i be a pure state of an n -party system H = H ⊗ H ⊗ · · · ⊗ H n , where thedimensions of the individual state spaces H k are finite but otherwise arbitrary.Denote by | q q ...q n i product states which are defined as the tensor products | q q ...q n i ≡ | q i ⊗ | q i ⊗ · · · ⊗ | q n i , where | q k i ∈ H k , k = 1 , , ..., n .The geometric measure of entanglement E g for an n -part pure state ψ isdefined as E g ( ψ ) = − ln Λ ( ψ ), where the maximal product overlap Λ max ( ψ ) isgiven by [25] Λ max = max q ,q ,...,q n | h ψ | q q ...q n i | , (1.1)where the normalization condition h q k | q k i = 1( k = 1 , , ..., n ) is understood andthe maximization is performed over all product states.The nearest product state is a stationary point for the overlap with | ψ i , sothe states | q k i satisfy the nonlinear eigenvalue equations h q q · · · b q k · · · q n | ψ i = Λ k | q k i ; k = 1 , , · · · , n, (1.2)where the caret means exclusion and eigenvalues Λ k are associated with the La-grange multipliers enforcing constraints h q k | q k i = 1( k = 1 , , ..., n ).Since phases of local states | q k i are irrelevant one can choose them such thatΛ k ’s are all positive. On the other hand | Λ k | = Λ max and therefore the stationarityequations can be rewritten as h q q · · · b q k · · · q n | ψ i = Λ max | q k i ; k = 1 , , · · · , n. (1.3)This is a system of nonlinear equations and its maximal eigenvalue and corre-sponding eigenvector are the maximal product overlap and the nearest productstate of a given pure states | ψ i , respectively. HAPTER 1. THREE QUBIT GEOMETRIC MEASURE
Consider now three qubits A,B,C with state function | ψ i . The entanglementeigenvalue Λ max ( ψ ) is given byΛ max = max q q q |h q q q | ψ i| (1.4)and the maximization runs over all normalized complete product states | q i ⊗| q i ⊗ | q i . Superscripts label single qubit states and spin indices are omitted forsimplicity. Since in the following we will use density matrices rather than statefunctions, our first aim is to rewrite Eq.(1.4) in terms of density matrices. Letus denote by ρ ABC = | ψ ih ψ | the density matrix of the three-qubit state and by ̺ k = | q k ih q k | the density matrices of the single qubit states. The equation for thesquare of the entanglement eigenvalue takes the formΛ ( ψ ) = max ̺ ̺ ̺ tr (cid:0) ρ ABC ̺ ⊗ ̺ ⊗ ̺ (cid:1) . (1.5)An important equalitymax ̺ tr( ρ ABC ̺ ⊗ ̺ ⊗ ̺ ) = tr( ρ ABC ̺ ⊗ ̺ ⊗ ) (1.6)was derived in [50] where is a unit matrix. It has a clear meaning. Thematrix tr( ρ ABC ̺ ⊗ ̺ ) is 2 ⊗ ρ AB of qubits A and B in a formΛ ( ψ ) = max ̺ ̺ tr (cid:0) ρ AB ̺ ⊗ ̺ (cid:1) . (1.7)We denote by s and s the unit Bloch vectors of the density matrices ̺ and ̺ respectively and adopt the usual summation convention on repeated indices i and j . Then Λ = 14 max s = s =1 (1 + s · r + s · r + g ij s i s j ) , (1.8)where HAPTER 1. THREE QUBIT GEOMETRIC MEASURE r = tr( ρ A σ ) , r = tr( ρ B σ ) , g ij = tr( ρ AB σ i ⊗ σ j ) (1.9)and σ i ’s are Pauli matrices. The matrix g ij is not necessarily to be symmetricbut must has only real entries. The maximization gives a pair of equations r + g s = λ s , r + g T s = λ s , (1.10)where Lagrange multipliers λ and λ are enforcing unit nature of the Blochvectors. The solution of Eq.(1.10) is s = (cid:0) λ λ − g g T (cid:1) − ( λ r + g r ) , (1.11a) s = (cid:0) λ λ − g T g (cid:1) − (cid:0) λ r + g T r (cid:1) . (1.11b)Now, the only unknowns are Lagrange multipliers, which should be determinedby equations | s | = 1 , | s | = 1 . (1.12)In general, Eq.(1.12) give two algebraic equations of degree six. However, thesolution (1.11) is valid if Eq.(1.10) supports a unique solution and this is by nomeans always the case. If the solution of Eq.(1.10) contains a free parameter,then det( λ λ − gg T ) = 0 and, as a result, Eq.(1.11) cannot not applicable. Theexample presented in Section III will demonstrate this situation.In order to test Eq.(1.11) let us consider an arbitrary superposition of W | W i = 1 √ | i + | i + | i ) (1.13)and flipped W | f W i = 1 √ | i + | i + | i ) (1.14)states, i.e. the state | ψ i = cos θ | W i + sin θ | f W i . (1.15)Straightforward calculation yields r = r = 13 (2 sin 2 θ i + cos 2 θ n ) , (1.16a) g = 13 − , (1.16b)where unit vectors i and n are aligned with the axes x and z , respectively. Bothvectors i and n are eigenvectors of matrices g and g T . Therefore s and s are HAPTER 1. THREE QUBIT GEOMETRIC MEASURE i and n . Also from r = r and g = g T it follows that s = s and λ = λ . Then Eq.(1.11) for general solution give s = s = sin 2 ϕ i + cos 2 ϕ n (1.17)where sin 2 ϕ = 2 sin 2 θ λ − , cos 2 ϕ = cos 2 θ λ + 1 . (1.18)The elimination of the Lagrange multiplier λ from Eq.(1.18) gives3 sin 2 ϕ cos 2 ϕ = cos 2 θ sin 2 ϕ − θ cos 2 ϕ. (1.19)Let us denote by t = tan ϕ . After the separation of the irrelevant root t = − tan θ , Eq.(1.19) takes the formsin θ t + 2 cos θ t − θ t − cos θ = 0 . (1.20)This equation exactly coincides with that derived in [25]. Since a detailed analysiswas given in Ref.[25], we do not want to repeat the same calculation here. Insteadwe would like to consider the three-qubit states that allow the analytic expressionsfor the geometric entanglement measure by making use of Eq.(1.10). Consider W-type state | ψ i = a | i + b | i + c | i , a + b + c = 1 . (1.21)Without loss of generality we consider only the case of positive parameters a, b, c .Direct calculation yields r = r n , r = r n , g = ω ω
00 0 − r , (1.22)where r = b + c − a , r = a + c − b , r = a + b − c (1.23)and ω = 2 ab . The unit vector n is aligned with the axis z . Any vector perpendic-ular to n is an eigenvector of g with eigenvalue ω . Then from Eq.(1.10) it followsthat the components of vectors s and s perpendicular to n are collinear. Wedenote by m the unit vector along that direction and parameterize vectors s and s as follows s = cos α n + sin α m , s = cos β n + sin β m . (1.24) HAPTER 1. THREE QUBIT GEOMETRIC MEASURE r − r cos β = λ cos α, r − r cos α = λ cos β, (1.25a) ω sin β = λ sin α, ω sin α = λ sin β, (1.25b)which are used to solve the four unknown constants λ , λ , α and β . Eq.(1.25b)impose either λ λ − ω = 0 (1.26)or sin α sin β = 0 . (1.27)First consider the case r > , r > , r > a, b, c forman acute triangle. Eq.(1.27) does not give a true maximum and this can beunderstood as follows. If both vectors s and s are aligned with the axis z ,then the last term in Eq.(1.8) is negative. If vectors s and s are antiparallel,then one of scalar products in Eq.(1.8) is negative. In this reason Λ cannotbe maximal. Then Eq.(1.26) gives true maximum and we have to choose positivevalues for λ and λ to get maximum.First we use Eq.(1.25a) to connect the angles α and β with the Lagrangemultipliers λ and λ cos α = λ r − r r ω − r , cos β = λ r − r r ω − r . (1.28)Then Eq.(1.25b) and (1.26) give the following expressions for Lagrange mul-tipliers λ and λ λ = ω (cid:18) ω + r − r ω + r − r (cid:19) / , (1.29a) λ = ω (cid:18) ω + r − r ω + r − r (cid:19) / . (1.29b)Eq.(1.10) allows to write a shorter expression for the entanglement eigenvalueΛ = 14 (1 + λ + r cos α ) . (1.30)Now we insert the values of λ and cos α into Eq.(1.30) and obtain4Λ = 1 + ω p ( ω + r − r )( ω + r − r ) − r r r ω − r . (1.31)The denominator in above expression is multiple of the area S of the triangle a, b, c HAPTER 1. THREE QUBIT GEOMETRIC MEASURE ω − r = 16 S . (1.32)A little algebra yields for the numerator ω q ( ω + r − r ) + ( ω + r − r ) − r r r (1.33)= 16 a b c − ω + r . Combining together the numerator and denominator, we obtain the final ex-pression for the entanglement eigenvalueΛ = 4 R , (1.34)where R is the circumradius of the triangle a, b, c . Entanglement value is minimalwhen triangle is regular, i.e. for W-state and Λ ( W ) = 4 / r <
0. Since r + r = 2 b ≥
0, we have r > r >
0. Eq.(1.27) gives true maximum in this case and both vectors arealigned with the axis z s = s = n (1.35)resulting in Λ = c . In view of symmetryΛ = max( a , b , c ) , max( a , b , c ) > . (1.36)Since the matrix g and vectors r and r are invariant under rotations aroundaxis z the same properties must have Bloch vectors s and s . There are twopossibilities:i)Bloch vectors are unique and aligned with the axis z . The solution given byEq.(1.35) corresponds to this situation and the resulting entanglement eigenvalueEq.(1.36) satisfies the inequality 12 < Λ ≤ . (1.37)ii)Bloch vectors have nonzero components in xy plane and the solution is notunique. Eq.(1.24) corresponds to this situation and contains a free parameter.The free parameter is the angle defining the direction of the vector m in the xy plane. Then Eq.(1.34) gives the entanglement eigenvalue in highly entangledregion 49 ≤ Λ < . (1.38)Eq.(1.34) and (1.36) have joint curves when parameters a, b, c form a righttriangle and give Λ = 1 /
2. The GHZ states have same entanglement value and
HAPTER 1. THREE QUBIT GEOMETRIC MEASURE = 1 / Now let us consider the state which is symmetric under permutation of qubits Aand B and contains three real independent parameters | ψ i = a | i + b | i + c | i + d | i , (1.39)where a + b + c + d = 1. According to Generalized Schmidt Decomposition [47]the states with different sets of parameters are local-unitary(LU) inequivalent.The relevant quantities are r = r = r n , g = ω − ω
00 0 1 , (1.40)where r = a + c − b − d , ω = 2 ad + 2 bc (1.41)and the unit vector n again is aligned with the axis z .All three terms in the l.h.s. of Eq.(1.8) are bounded above: • s · r ≤ | r | , • s · r ≤ | r | , • and owing to inequality | ω | ≤ , g ij s i s j ≤ s = s = Sign( r ) n , (1.42)which results in Λ = 12 (1 + | r | ) . (1.43)This expression has a clear meaning. To understand it we parameterize thestate as | ψ i = k | q i + k | q i , (1.44) HAPTER 1. THREE QUBIT GEOMETRIC MEASURE q and q are arbitrary single normalized qubit states and positive param-eters k and k satisfy k + k = 1. ThenΛ = max( k , k ) , (1.45)i.e. the maximization takes a larger coefficient in Eq.(1.44). In bipartite casethe maximization takes the largest coefficient in Schmidt decomposition [28, 63]and in this sense Eq.(1.44) effectively takes the place of Schmidt decomposition.When | q i = | i and | q i = | i , Eq.(1.45) gives the known answer for generalizedGHZ state [25, 61].The entanglement eigenvalue is minimal Λ = 1 / k = k . These states can be described as follows | ψ i = | q i + | q i (1.46)where q and q are arbitrary single qubit normalized states. The entanglementeigenvalue is constant Λ = 1 / = 1 /
2. On the other hand W-state isunique up to LU transformations and the low bound Λ = 4 / a = b = c . However, one cannot make such conclusions in general.Five real parameters are necessary to parameterize the set of inequivalent threequbit pure states [47]. And there is no explicit argument that W-state is not justone of LU inequivalent states that have Λ = 4 / We have derived algebraic equations defining geometric measure of three qubitpure states. These equations have a degree higher than four and explicit solutionsfor general cases cannot be derived analytically. However, the explicit expressionsare not important. Remember that explicit expressions for the algebraic equationsof degree three and four have a limited practical significance but the equationsitself are more important. This is especially true for equations of higher degree;main results can be derived from the equations rather than from the expressionsof their roots.Eq.(1.10) give the nearest separable state directly and this separable stateshave useful applications. In order to construct an entanglement witness, forexample, the crucial point lies in finding the nearest separable state [64]. Thiswill be especially interesting for highly entangled states that have a whole set ofnearest separable states and allow to construct a set of entanglement witnesses.
HAPTER 1. THREE QUBIT GEOMETRIC MEASURE hapter 2Three-Qubit Groverian Measure
In this chapter we connect the geometric entanglement measure with polynomialinvariants in the case of three-quibt pure states [40].About decade ago the axioms which entanglement measures should satisfywere studied [23]. The most important property for measure is monotonic-ity under local operation and classical communication(LOCC) [65]. Follow-ing the axioms, many entanglement measures were constructed such as relativeentropy[66], entanglement of distillation[21] and formation[19, 20, 67, 68], geomet-ric measure[22, 24, 25, 69], Schmidt measure[70] and Groverian measure[28]. En-tanglement measures are used in various branches of quantum mechanics. Espe-cially, recently, they are used to try to understand Zamolodchikov’s c-theorem[71]more profoundly. It may be an important application of the quantum informationtechniques to understand the effect of renormalization group in field theories[35].The purpose of this paper is to compute the Groverian measure for variousthree-qubit quantum states. The Groverian measure G ( ψ ) for three-qubit state | ψ i is defined by G ( ψ ) ≡ √ − P max where P max = Λ (2.1)Thus P max can be interpreted as a maximal overlap between the given state | ψ i and product states. Groverian measure is an operational treatment of ageometric measure. Thus, if one can compute G ( ψ ), one can also compute thegeometric measure of pure state by G ( ψ ). Sometimes it is more convenient tore-express Eq.(2.1) in terms of the density matrix ρ = | ψ ih ψ | . This can be easilyaccomplished by an expression P max = max R ,R ,R Tr (cid:2) ρR ⊗ R ⊗ R (cid:3) (2.2)where R i ≡ | q i ih q i | density matrix for the product state. Eq.(2.1) and Eq.(2.2)manifestly show that P max and G ( ψ ) are local-unitary(LU) invariant quanti-ties. Since it is well-known that three-qubit system has five independent LU-invariants[47, 54, 57, 72], say J i ( i = 1 , · · · , J i ’s in this paper.25 HAPTER 2. THREE-QUBIT GROVERIAN MEASURE i.e. two-qubit system. Using Bloch formof the density matrix it is shown in this section that two-qubit system has onlyone independent LU-invariant quantity, say J . It is also shown that Groverianmeasure and P max for arbitrary two-qubit states can be expressed solely in termsof J .In Section 2.2 we have discussed how to derive LU-invariants in higher-qubitsystems. In fact, we have derived many LU-invariant quantities using Bloch formof the density matrix in three-qubit system. It is shown that all LU-invariantsderived can be expressed in terms of J i ’s discussed in Ref.[47]. Recently, it wasshown in Ref.[50] that P max for n -qubit state can be computed from ( n − J i ’s. For type 4 and type 5 the analytical computation seems to behighly nontrivial and may need separate publications. Thus the analytical calcu-lation for these types is not presented in this paper. The results of this sectionare summarized in Table I.In Section 2.4 we have discussed the modified W-like state, which has three-independent real parameters. In fact, this state cannot be categorized in the fivetypes discussed in Section 2.3. The analytic expressions of the Groverian measurefor this state was computed recently in Ref.[42]. It was shown that the measurehas three different expressions depending on the domains of the parameter space.It turned out that each expression has its own geometrical meaning. In thissection we have re-expressed all expressions of the Groverian measure in terms ofLU-invariants.In Section 2.5 brief conclusion is given. In this section we consider P max for the two-qubit system. The Groverian measurefor two-qubit system is already well-known[61]. However, we revisit this issue hereto explore how the measure is expressed in terms of the LU-invariant quantities.The Schmidt decomposition[45, 46] makes the most general expression of thetwo-qubit state vector to be simple form | ψ i = λ | i + λ | i (2.3) HAPTER 2. THREE-QUBIT GROVERIAN MEASURE λ , λ ≥ λ + λ = 1. The density matrix for | ψ i can be expressed inthe Bloch form as following: ρ = | ψ ih ψ | = 14 [ ⊗ + v α σ α ⊗ + v α ⊗ σ α + g αβ σ α ⊗ σ β ] , (2.4)where ~v = ~v = λ − λ , g αβ = λ λ − λ λ
00 0 1 . (2.5)In order to discuss the LU transformation we consider first the quantity U σ α U † where U is 2 × U σ α U † = O αβ σ β , (2.6)where the explicit expression of O αβ is given in appendix A. Since O αβ is a realmatrix satisfying OO T = O T O = , it is an element of the rotation group O(3).Therefore, Eq.(2.6) implies that the LU-invariants in the density matrix (2.4) are | ~v | , | ~v | , Tr[ gg T ] etc.All LU-invariant quantities can be written in terms of one quantity, say J ≡ λ λ . In fact, J can be expressed in terms of two-qubit concurrence[20] C by C /
4. Then it is easy to show | ~v | = | ~v | = 1 − J, (2.7) g αβ g αβ = 1 + 8 J. It is well-known that P max is simply square of larger Schmidt number in two-qubit case P max = max (cid:0) λ , λ (cid:1) . (2.8)It can be re-expressed in terms of reduced density operators P max = 12 h p − ρ A i , (2.9)where ρ A = Tr B ρ = (1 + v α σ α ) /
2. Since P max is invariant under LU-transfor-mation, it should be expressed in terms of LU-invariant quantities. In fact, P max in Eq.(2.9) can be re-written as P max = 12 h √ − J i . (2.10)Eq.(2.10) implies that P max is manifestly LU-invariant. HAPTER 2. THREE-QUBIT GROVERIAN MEASURE The Bloch representation of the 3-qubit density matrix can be written in the form ρ = 18 " ⊗ ⊗ + v α σ α ⊗ ⊗ + v α ⊗ σ α ⊗ + v α ⊗ ⊗ σ α + h (1) αβ ⊗ σ α ⊗ σ β + h (2) αβ σ α ⊗ ⊗ σ β + h (3) αβ σ α ⊗ σ β ⊗ + g αβγ σ α ⊗ σ β ⊗ σ γ , (2.11)where σ α is Pauli matrix. According to Eq.(2.6) and appendix A it is easy to showthat the LU-invariants in the density matrix (2.11) are | ~v | , | ~v | , | ~v | , Tr[ h (1) h (1) T ],Tr[ h (2) h (2) T ], Tr[ h (3) h (3) T ], g αβγ g αβγ etc.Few years ago Ac´ın et al[47] represented the three-qubit arbitrary states in asimple form using a generalized Schmidt decomposition[45] as following: | ψ i = λ | i + λ e iϕ | i + λ | i + λ | i + λ | i (2.12)with λ i ≥
0, 0 ≤ ϕ ≤ π , and P i λ i = 1. The five algebraically independentpolynomial LU-invariants were also constructed in Ref.[47]: J = λ λ + λ λ − λ λ λ λ cos ϕ, (2.13) J = λ λ , J = λ λ , J = λ λ ,J = λ ( J + λ λ − λ λ ) . In order to determine how many states have the same values of the invariants J , J , ...J , and therefore how many further discrete-valued invariants are neededto specify uniquely a pure state of three qubits up to local transformations, onewould need to find the number of different sets of parameters ϕ and λ i ( i =0 , , ... λ is found, other parameters aredetermined uniquely and therefore we derive an equation defining λ in terms ofpolynomial invariants.( J + J ) λ − ( J + J ) λ + J J + J J + J J + J = 0 . (2.14)This equation has at most two positive roots and consequently an additionaldiscrete-valued invariant is required to specify uniquely a pure three qubit state.Generally 18 LU-invariants, nine of which may be taken to have only discretevalues, are needed to determine a mixed 2-qubit state [73].If one represents the density matrix | ψ ih ψ | as a Bloch form like Eq.(2.11), itis possible to construct v α , v α , v α , h (1) αβ , h (2) αβ , h (3) αβ , and g αβγ explicitly, whichare summarized in appendix B. Using these explicit expressions one can show HAPTER 2. THREE-QUBIT GROVERIAN MEASURE J i as following: | ~v | = 1 − J + J + J ) , | ~v | = 1 − J + J + J ) (2.15) | ~v | = 1 − J + J + J ) , Tr[ h (1) h (1) T ] = 1 + 4(2 J − J − J )Tr[ h (2) h (2) T ] = 1 − J − J + J ) , Tr[ h (3) h (3) T ] = 1 − J + J − J ) g αβγ g αβγ = 1 + 4(2 J + 2 J + 2 J + 3 J ) h (3) αβ v (1) α v (2) β = 1 − J + J + J + J − J ) . Recently, Ref.[50] has shown that P max for n -qubit pure state can be computedfrom ( n − R ,R , ··· ,R n Tr (cid:2) ρR ⊗ R ⊗ · · · ⊗ R n (cid:3) = (2.16)max R ,R , ··· ,R n − Tr (cid:2) ρR ⊗ R ⊗ · · · ⊗ R n − ⊗ (cid:3) which is Theorem I of Ref.[50]. Here, we would like to discuss the physicalmeaning of Eq.(2.16) from the aspect of LU-invariance. Eq.(2.16) in 3-qubitsystem reduces to P max = max R ,R Tr (cid:2) ρ AB R ⊗ R (cid:3) (2.17)where ρ AB = Tr C ρ . From Eq.(2.11) ρ AB simply reduces to ρ = 14 h ⊗ + v α σ α ⊗ + v α ⊗ σ α + h (3) αβ σ α ⊗ σ β i (2.18)where v α , v α and h (3) αβ are explicitly given in appendix B. Of course, the LU-invariant quantities of ρ AB are | ~v | , | ~v | , Tr[ h (3) h (3) T ], h (3) αβ v α v β etc, all of which,of course, can be re-expressed in terms of J , J , J , J and J . It is worthwhilenoting that we need all J i ’s to express the LU-invariant quantities of ρ AB . Thismeans that the reduced state ρ AB does have full information on the LU-invarianceof the original pure state ρ .Indeed, any reduced state resulting from a partial trace over a single qubituniquely determines any entanglement measure of original system, given that theinitial state is pure. Consider an ( n − | ψ ′ i be any joint pure state.All other purifications can be obtained from the state | ψ ′ i by LU-transformations U ⊗ ⊗ ( n − , where U is a local unitary matrix acting on single qubit. Since anyentanglement measure must be invariant under LU-transformations, it must besame for all purifications independently of U . Hence the reduced density matrixdetermines any entanglement measure on the initial pure state. That is why wecan compute P max of n -qubit pure state from the ( n − HAPTER 2. THREE-QUBIT GROVERIAN MEASURE n -qubit stateis partly lost if we take partial trace twice. In order to show this explicitly let usconsider ρ A ≡ Tr B ρ AB and ρ B ≡ Tr A ρ AB : ρ A = 12 [ + v α σ α ] (2.19) ρ B = 12 [ + v α σ α ] . Eq.(2.6) and appendix A imply that their LU-invariant quantities are only | ~v | and | ~v | respectively. Thus, we do not need J to express the LU-invariant quantitiesof ρ A and ρ B . This fact indicates that the mixed states ρ A and ρ B partly loosethe information of the LU-invariance of the original pure state ρ . This is why( n − P max of n -qubit pure state. P max If we insert the Bloch representation R = + ~s · ~σ R = + ~s · ~σ | ~s | = | ~s | = 1 into Eq.(2.17), P max for 3-qubit state becomes P max = 14 max | ~s | = | ~s | =1 [1 + ~r · ~s + ~r · ~s + g ij s i s j ] (2.21)where ~r = Tr (cid:2) ρ A ~σ (cid:3) (2.22) ~r = Tr (cid:2) ρ B ~σ (cid:3) g ij = Tr (cid:2) ρ AB σ i ⊗ σ j (cid:3) . Since in Eq.(2.21) P max is maximization with constraint | ~s | = | ~s | = 1, we shoulduse the Lagrange multiplier method, which yields a pair of equations ~r + g~s = Λ ~s (2.23) ~r + g T ~s = Λ ~s , where the symbol g represents the matrix g ij in Eq.(2.22). Thus we should solve ~s , ~s , Λ and Λ by eq.(2.23) and the constraint | ~s | = | ~s | = 1. Although itis highly nontrivial to solve Eq.(2.23), sometimes it is not difficult if the given3-qubit state | ψ i has rich symmetries. Now, we would like to compute P max forvarious types of 3-qubit system. HAPTER 2. THREE-QUBIT GROVERIAN MEASURE J = J = J = J = J = 0 In order for all J i ’s to be zero we have two cases λ = J = 0 or λ = λ = λ = 0. λ = J = 0If λ = 0, | ψ i in Eq.(2.12) becomes | ψ i = | i ⊗ | BC i where | BC i = λ e iϕ | i + λ | i + λ | i + λ | i . (2.24)Thus P max for | ψ i equals to that for | BC i . Since | BC i is two-qubit state, onecan easily compute P max using Eq.(2.9), which is P max = 12 h p − B | BC ih BC | ) i = 12 h p − J i . (2.25)If, therefore, λ = J = 0, we have P max = 1, which gives a vanishing Groverianmeasure. λ = λ = λ = 0In this case | ψ i in Eq.(2.12) becomes | ψ i = (cid:0) λ | i + λ e iϕ | i (cid:1) ⊗ | i ⊗ | i . (2.26)Since | ψ i is completely product state, P max becomes one. In this type we have following three cases. J = 0 and J = J = J = J = 0In this case we have λ = 0. Thus P max for this case is exactly same withEq.(2.25). J = 0 and J = J = J = J = 0In this case we have λ = λ = 0. Thus P max for | ψ i equals to that for | AC i ,where | AC i = λ | i + λ e iϕ | i + λ | i . (2.27)Using Eq.(2.9), therefore, one can easily compute P max , which is P max = 12 h p − J i . (2.28) HAPTER 2. THREE-QUBIT GROVERIAN MEASURE J = 0 and J = J = J = J = 0In this case P max for | ψ i equals to that for | AB i , where | AB i = λ | i + λ e iϕ | i + λ | i . (2.29)Thus P max for | ψ i is P max = 12 h p − J i . (2.30) J = 0 , J = J = J = J = 0 In this case we have λ = λ = λ = 0 and | ψ i becomes | ψ i = λ | i + λ | i (2.31)with λ + λ = 1. Then it is easy to show ~r = Tr (cid:2) ρ A ~σ (cid:3) = (0 , , λ − λ ) (2.32) ~r = Tr (cid:2) ρ B ~σ (cid:3) = (0 , , λ − λ ) g ij = Tr (cid:2) ρ AB σ i ⊗ σ j (cid:3) = . Thus P max reduces to P max = 14 max | ~s | = | ~s | =1 (cid:2) λ − λ )( s z + s z ) + s z s z (cid:3) . (2.33)Since Eq.(2.33) is simple, we do not need to solve Eq.(2.23) for the maximization.If λ > λ , the maximization can be achieved by simply choosing ~s = ~s =(0 , , λ < λ , we choose ~s = ~s = (0 , , − P max = max( λ , λ ) . (2.34)In order to express P max in Eq.(2.34) in terms of LU-invariants we follow thefollowing procedure. First we note P max = 12 (cid:2) ( λ + λ ) + | λ − λ | (cid:3) . (2.35)Since | λ − λ | = p ( λ + λ ) − λ λ = √ − J , we get finally P max = 12 h p − J i . (2.36) HAPTER 2. THREE-QUBIT GROVERIAN MEASURE In this case we have λ = λ = 0 and | ψ i becomes | ψ i = λ | i + λ | i + λ | i (2.37)with λ + λ + λ = 1. If we take LU-transformation σ x in the first-qubit, | ψ i ischanged into | ψ ′ i which is usual W-type state[49] as follows: | ψ ′ i = λ | i + λ | i + λ | i . (2.38)The LU-invariants in this type are J = λ λ J = λ λ (2.39) J = λ λ J = 2 λ λ λ . Then it is easy to derive a relation J J + J J + J J = p J J J = 12 J . (2.40)Recently, P max for | ψ ′ i is computed analytically in Ref.[37] by solving theLagrange multiplier equations (2.23) explicitly. In order to express P max explicitlywe first define r = λ + λ − λ (2.41) r = λ + λ − λ r = λ + λ − λ ω = 2 λ λ . Also we define a = max( λ , λ , λ ) (2.42) b = mid( λ , λ , λ ) c = min( λ , λ , λ ) . Then P max is expressed differently in two different regions as follows. If a ≥ b + c , P max becomes P >max = a = max( λ , λ , λ ) . (2.43)In order to express P max in terms of LU-invariants we express Eq.(2.43) differentlyas follows P >max = (2.44)14 (cid:2) ( λ + λ + λ ) + | λ + λ − λ | + | λ − λ + λ | + | λ − λ − λ | (cid:3) . HAPTER 2. THREE-QUBIT GROVERIAN MEASURE | λ + λ − λ | = q − λ λ − λ λ = p − J + J ) (2.45) | λ − λ + λ | = q − λ λ − λ λ = p − J + J ) | λ − λ − λ | = q − λ λ − λ λ = p − J + J ) , we can express P max in Eq.(2.43) as follows: P >max = (2.46)14 h p − J + J ) + p − J + J ) + p − J + J ) i . If a ≤ b + c , P max becomes P
1) (2.64) ~r = Tr[ ρ B ~σ ] = (2 λ λ cos ϕ, − λ λ sin ϕ, − λ ) g ij = Tr[ ρ AB σ i ⊗ σ j ] = λ λ λ λ cos ϕ − λ λ λ λ sin ϕ − λ λ cos ϕ λ λ sin ϕ λ − λ − λ + λ . Although we have freedom to choose the phase factor ϕ , it is impossible to findsingular values of the matrix g , which makes it formidable task to solve Eq.(2.23).Based on Ref.[37] and Ref.[42], furthermore, we can conjecture that P max for thistype may have several different expressions depending on the domains in param-eter space. Therefore, it may need long calculation to compute P max analytically.We would like to leave this issue for our future research work and the explicitexpressions of P max are not presented in this paper. HAPTER 2. THREE-QUBIT GROVERIAN MEASURE This type consists of the 2 cases, i.e. λ = 0 and λ = 0. λ = 0In this case the state vector | ψ i in Eq.(2.12) reduces to | ψ i = λ | i + λ e iϕ | i + λ | i + λ | i (2.65)with λ + λ + λ + λ = 1. The LU-invariants are J = λ λ J = λ λ J = λ λ . (2.66)Eq.(2.66) implies that the Groverian measure for Eq.(2.65) is independent ofthe phase factor ϕ like type 4a. This fact may drastically reduce the calculationprocedure for solving the Lagrange multiplier equation (2.23). In spite of this fact,however, solving Eq.(2.23) is highly non-trivial as we commented in the previoustype. The explicit expressions of the Groverian measure are not presented in thispaper and we hope to present them elsewhere in the near future. λ = 0In this case the state vector | ψ i in Eq.(2.12) reduces to | ψ i = λ | i + λ e iϕ | i + λ | i + λ | i (2.67)with λ + λ + λ + λ = 1. The LU-invariants are J = λ λ J = λ λ J = λ λ . (2.68)Eq.(2.68) implies that the Groverian measure for Eq.(2.67) is independent of thephase factor ϕ like type 4a. λ = 0 ) In this case the state vector | ψ i in Eq.(2.12) reduces to | ψ i = λ | i + λ | i + λ | i + λ | i (2.69)with λ + λ + λ + λ = 1. The LU-invariants in this type are J = λ λ J = λ λ J = λ λ (2.70) J = λ λ J = 2 λ λ λ . From Eq.(2.70) it is easy to show J ( J + J + J ) + J J = p J J J = 12 J . (2.71) HAPTER 2. THREE-QUBIT GROVERIAN MEASURE ~r , ~r and g ij defined in Eq.(2.22) are ~r = (0 , , λ −
1) (2.72) ~r = (2 λ λ , , λ + λ − λ − λ ) g ij = λ λ − λ λ − λ λ − λ . Like type 4a and type 4b solving Eq.(2.23) is highly non-trivial mainly due tonon-diagonalization of g ij . Of course, the fact that the first component of ~r is non-zero makes hard to solve Eq.(2.23) too. The explicit expressions of theGroverian measure in this type are not given in this paper. ϕ = 0 , π ϕ = 0In this case the state vector | ψ i in Eq.(2.12) reduces to | ψ i = λ | i + λ | i + λ | i + λ | i + λ | i (2.73)with λ + λ + λ + λ + λ = 1. The LU-invariants in this case are J = ( λ λ − λ λ ) J = λ λ J = λ λ (2.74) J = λ λ J = 2 λ λ λ ( λ λ − λ λ ) . It is easy to show √ J J J = J / ϕ = π In this case the state vector | ψ i in Eq.(2.12) reduces to | ψ i = λ | i − λ | i + λ | i + λ | i + λ | i (2.75)with λ + λ + λ + λ + λ = 1. The LU-invariants in this case are J = ( λ λ + λ λ ) J = λ λ J = λ λ (2.76) J = λ λ J = 2 λ λ λ ( λ λ + λ λ ) . It is easy to show √ J J J = J / P max in type 5 is most difficult problem. In addi-tion, we don’t know whether it is mathematically possible or not. However, thegeometric interpretation of P max presented in Ref.[37] and Ref.[42] may provideus valuable insight. We hope to leave this issue for our future research work too.The results in this section is summarized in Table I. HAPTER 2. THREE-QUBIT GROVERIAN MEASURE P max Type I J i = 0 1 J i = 0 except J (cid:0) √ − J (cid:1) Type II a J i = 0 except J (cid:0) √ − J (cid:1) J i = 0 except J (cid:0) √ − J (cid:1) b J i = 0 except J (cid:0) √ − J (cid:1) (cid:16) √ − J + J )+ √ − J + J )+ √ − J + J ) (cid:17) / a λ = λ = 0 if a ≥ b + c √ J J J / (4( J + J + J ) − a ≤ b + c Type III λ = λ = 0 (cid:16) p − J + J ) (cid:17) b λ = λ = 0 (cid:16) p − J + J ) (cid:17) λ = λ = 0 (cid:16) p − J + J ) (cid:17) a λ = 0 independent of ϕ : not presentedType IV b λ = 0 independent of ϕ : not presented λ = 0 independent of ϕ : not presentedc λ = 0 not presentedType V ϕ = 0 not presented ϕ = π not presentedTable I: Summary of P max in various types. In this section we consider new type in 3-qubit states. The type we consider is | Φ i = a | i + b | i + c | i + q | i , a + b + c + q = 1 . (2.77)First, we would like to derive the standard form like Eq.(2.12) from | Φ i . Thiscan be achieved as following. First, we consider LU-transformation of | Φ i , i.e. ( U ⊗ ⊗ ) | Φ i , where U = 1 √ aq + bc (cid:18) √ aqe iθ √ bce iθ −√ bc √ aq (cid:19) . (2.78)After LU-transformation, we perform Schmidt decomposition following Ref.[47].Finally we choose θ to make all λ i to be positive. Then we can derive the standard HAPTER 2. THREE-QUBIT GROVERIAN MEASURE | Φ i with ϕ = 0 or π , and λ = s ( ac + bq )( ab + cq ) aq + bc (2.79) λ = √ abcq p ( ab + cq )( ac + bq )( aq + bc ) | a + q − b − c | λ = 1 λ | ac − bq | λ = 1 λ | ab − cq | λ = 2 √ abcqλ . It is easy to prove that the normalization condition a + b + c + q = 1 guaranteesthe normalization λ + λ + λ + λ + λ = 1 . (2.80)Since | Φ i has three free parameters, we need one more constraint between λ i ’s.This additional constraint can be derived by trial and error. The explicit expres-sion for this additional relation is λ ( λ + λ + λ ) = 14 − λ λ ( λ + λ )( λ + λ ) . (2.81)Since all λ i ’s are not vanishing but there are only three free parameters, | Φ i isnot involved in the types discussed in the previous section. Using Eq.(2.79) it is easy to derive LU-invariants which are J = ( λ λ − λ λ ) = 1( ab + cq ) ( ac + bq ) (2.82) × (cid:2) abcq | a + q − b − c | − ( aq + bc ) | ( ab − cq )( ac − bq ) | (cid:3) J = λ λ = ( ac − bq ) J = λ λ = ( ab − cq ) J = λ λ = 4 abcqJ = λ (cid:0) J + λ λ − λ λ (cid:1) . One can show directly that J = 2 √ J J J . Since | Φ i has three free parame-ters, there should exist additional relation between J i ’s. However, the explicitexpression may be hardly derived. In principle, this constraint can be derived asfollowing. First, we express the coefficients a , b , c , and q in terms of J , J , J HAPTER 2. THREE-QUBIT GROVERIAN MEASURE J using first four equations of Eq.(2.82). Then the normalization condition a + b + c + q = 1 gives explicit expression of this additional constraint. Since,however, this procedure requires the solutions of quartic equation, it seems to behard to derive it explicitly.Since J contains absolute value, it is dependent on the regions in the param-eter space. Direct calculation shows that J is J = ( aq − bc ) , (2.83)when ( a + q − b − c )( ab − cq )( ac − bq ) ≥ J = ( aq − bc ) (cid:20) ab − cq )( ac − bq )( aq + bc )( ab + cq )( ac + bq )( aq − bc ) (cid:21) , (2.84)when ( a + q − b − c )( ab − cq )( ac − bq ) < P max is manifestly LU-invariant quantity, it is obvious that it also de-pends on the regions on the parameter space. P max P max for state | Φ i in Eq.(2.77) has been analytically computed recently in Ref.[42].It turns out that P max is differently expressed in three distinct ranges of defini-tion in parameter space. The final expressions can be interpreted geometricallyas discussed in Ref.[42]. To express P max explicitly we define r ≡ b + c − a − q r ≡ a + c − b − q (2.85) r ≡ a + b − c − q ω ≡ ab + qc µ ≡ ab − qc. The first expression of P max , which can be expressed in terms of circumradiusof convex quadrangle is P ( Q ) max = 4( ab + qc )( ac + qb )( aq + bc )4 ω − r . (2.86)The second expression of P max , which can be expressed in terms of circumradiusof crossed-quadrangle is P ( CQ ) max = ( ab − cq )( ac − bq )( bc − aq )4 S x (2.87)where S x = 116 ( a + b + c + q )( a + b − c − q )( a − b + c − q )( − a + b + c − q ) . (2.88)The final expression of P max corresponds to the largest coefficient: P ( L ) max = max( a , b , c , q ) = 14 (1 + | r | + | r | + | r | ) . (2.89) HAPTER 2. THREE-QUBIT GROVERIAN MEASURE P max is fully discussed in Ref.[42].Now we would like to express all expressions of P max in terms of LU-invariants.For the simplicity we choose a simplified case, that is ( a + q − b − c )( ab − cq )( ac − bq ) ≥
0. Then it is easy to derive r = 1 − J + J + J ) r = 1 − J + J + J ) (2.90) r = 1 − J + J + J ) ω = J + J . Then it is simple to express P ( Q ) max and P ( CQ ) max as following: P ( Q ) max = 4 p ( J + J )( J + J )( J + J )4( J + J + J + 2 J ) − P ( CQ ) max = 4 √ J J J J + J + J + J ) − . If we take q = 0 limit, we have λ = J = 0. Thus P ( Q ) max and P ( CQ ) max reduce to4 √ J J J / (4( J + J + J ) − P
In this section we briefly review the derivation of the stationarity equations andtheir general solutions [37]. Denote by ρ ABC the density matrix of the three-qubitpure state and define the entanglement eigenvalue Λ [25]Λ = max ̺ ̺ ̺ tr (cid:0) ρ ABC ̺ ⊗ ̺ ⊗ ̺ (cid:1) , (3.1)where the maximization runs over all normalized complete product states. The-orem 1 of Ref.[50] states that the maximization of a pure state over a singlequbit state can be completely derived by using a particle traced over densitymatrix. Hence the theorem allows us to re-express the entanglement eigenvalueby reduced density matrix ρ AB of qubits A and BΛ = max ̺ ̺ tr (cid:0) ρ AB ̺ ⊗ ̺ (cid:1) . (3.2)Now we introduce four Bloch vectors:1) r A for the reduced density matrix ρ A of the qubit A,2) r B for the reduced density matrix ρ B of the qubit B,3) u for the single qubit state ̺ ,4) v for the single qubit state ̺ .Then the expression for entanglement eigenvalue (3.2) takes the formΛ = 14 max u = v =1 (1 + u · r A + v · r B + g ij u i v j ) , (3.3)where(summation on repeated indices i and j is understood) g ij = tr( ρ AB σ i ⊗ σ j ) (3.4) HAPTER 3. SHARED QUANTUM STATES σ i ’s are Pauli matrices. The closest product state satisfies the stationarityconditions r A + g v = λ u , r B + g T u = λ v , (3.5)where Lagrange multipliers λ and λ enforce the unit Bloch vectors u and v .The solutions of Eq.(3.5) are u = (cid:0) λ λ − g g T (cid:1) − ( λ r A + g r B ) , v = (cid:0) λ λ − g T g (cid:1) − (cid:0) λ r B + g T r A (cid:1) . (3.6)Unknown Lagrange multipliers are defined by equations u = 1 , v = 1 . (3.7)In general, Eq.(3.7) gives algebraic equations of degree six. The reason for thisis that stationarity equations define all extremes of the reduced density matrix ρ AB over product states, regardless of them being global or local. And the degreeof the algebraic equations is the number of possible extremes.Eq.(3.6) contains valuable information. It provides solid bases for a newnumerical approach. This can be compared with the numerical calculations basedon other technique [61]. We consider a four-parameter state | ψ i = a | i + b | i + c | i + d | i , (3.8)where free parameters a, b, c, d satisfy the normalization condition a + b + c + d = 1. Without loss of generality we consider only the case of positive parameters a, b, c, d . At first sight, it is not obvious whether the state allows analytic solutionsor not. However, it does and our first task is to confirm the existence of theanalytic solutions.In fact, entanglement of the state Eq.(3.8) is invariant under the permutationsof four parameters a, b, c, d . The invariance under the permutations of threeparameters a, b, c is the consequence of the invariance under the permutationsof qubits A,B,C. Now we make a local unitary(LU) transformation that relabelsthe bases of qubits B and C, i.e. 0 B ↔ B , C ↔ C , and does not change thebasis of qubit A. This LU-transformation interchanges the coefficients as follows: a ↔ d, b ↔ c . Since any entanglement measure must be invariant under LU-transformations and the permutation b ↔ c , it must be also invariant underthe permutation a ↔ d . In view of this symmetry, any entanglement measuremust be invariant under the permutations of all the state parameters a, b, c, d . HAPTER 3. SHARED QUANTUM STATES (cid:0) λ λ − g g T (cid:1) = 0 . (3.9)Indeed, if the condition (3.9) is fulfilled, then the expressions (3.6) for thegeneral solutions are not applicable and Eq.(3.5) admits further simplification.Denote by i , j , k unit vectors along axes x, y, z respectively. Straightforwardcalculation yields r A = r k , r B = r k , g = ω µ
00 0 − r , (3.10)where r = b + c − a − d , r = a + c − b − d , (3.11) r = a + b − c − d , ω = ab + dc, µ = ab − dc. Vectors u and v can be written as linear combinations u = u i i + u j j + u k k , v = v i i + v j j + v k k (3.12)of vectors i , j , k . The substitution of the Eq.(3.12) into Eq.(3.5) gives a coupleof equations in each direction. The result is a system of six linear equations2 ω v i = λ u i , ω u i = λ v i , (3.13a)2 µ v j = λ u j , µ u j = λ v j , (3.13b) r − r v k = λ u k , r − r u k = λ v k . (3.13c)Above equations impose two conditions( λ λ − ω ) u i v i = 0 , (3.14a)( λ λ − µ ) u j v j = 0 . (3.14b)From these equations it can be deduced that the condition (3.9) is valid andthe system of equations (3.5) and (3.7) is solvable. Note that as a consequencesof Eq.(3.13) x and/or y components of vectors u and v vanish simultaneously.Hence, conditions (3.14) are satisfied in following three cases: • vectors u and v lie in xz plane λ λ − ω = 0 , u j v j = 0 , (3.15) HAPTER 3. SHARED QUANTUM STATES • vectors u and v lie in yz plane λ λ − µ = 0 , u i v i = 0 , (3.16) • vectors u and v are aligned with axis zu i v i = u j v j = 0 . (3.17)These cases are examined individually in next section. In this section we analyze all three cases and derive explicit expressions for en-tanglement eigenvalue. Each expression has its own range of definition in whichthey are deemed applicable. Three ranges of definition cover the four dimen-sional sphere given by normalization condition. It is necessary to separate thevalidity domains and to make clear which of expressions should be applied fora given state. It turns out that the separation of domains requires solving in-equalities that contain polynomials of degree six. This is a nontrivial task andwe investigate it in the next section.
Let us consider the first case. Our main task is to find Lagrange multipliers λ and λ . From equations (3.13c) and (3.15) we have u k = λ r − r r ω − r , v k = λ r − r r ω − r . (3.18)In its turn Eq.(3.13a) gives λ u i = λ v i . (3.19)Eq.(3.7) allows the substitution of expressions (3.18) into Eq.(3.19). Then wecan obtain the second equation for Lagrange multipliers λ (cid:0) ω + r − r (cid:1) = λ (cid:0) ω + r − r (cid:1) . (3.20)This equation has a simple form owing to condition (3.9). Thus we can fac-torize the equation of degree six into the quadratic equations. Equations (3.20)and (3.15) together yield λ = 2 ω bc + adac + bd , λ = 2 ω ac + bdbc + ad . (3.21) HAPTER 3. SHARED QUANTUM STATES . Now Eq.(3.3) takes the form4Λ = 1 + 8( ab + cd )( ac + bd )( ad + bc ) − r r r ω − r . (3.22)In fact, entanglement eigenvalue is the sum of two equal terms and this state-ment follows from the identity1 − r r r ω − r = 8 ( ab + cd )( ac + bd )( ad + bc )4 ω − r . (3.23)To derive this identity one has to use the normalization condition a + b + c + d =1. The identity allows to rewrite Eq.(3.22) as followsΛ = 4 R q , (3.24)where R q = ( ab + cd )( ac + bd )( ad + bc )4 ω − r . (3.25)Above formula has a geometric interpretation and now we demonstrate it.Let us define a quantity p ≡ ( a + b + c + d ) /
2. Then the denominator can berewritten as 4 ω − r = 16( p − a )( p − b )( p − c )( p − d ) . (3.26)Five independent parameters are necessary to construct a convex quadrangle.However, four independent parameters are necessary to construct a convex quad-rangle that has circumradius. For such quadrangles the area S q is given exactlyby Eq.(3.26) up to numerical factor, that is S q = ( p − a )( p − b )( p − c )( p − d ).Hence Eq.(3.25) can be rewritten as R q = ( ab + cd )( ac + bd )( ad + bc )16 S q . (3.27)Thus R q can be interpreted as a circumradius of the convex quadrangle. Eq.(3.27)is the generalization of the corresponding formula of Ref.[37] and reduces to thecircumradius of the triangle if one of parameters is zero.Eq.(3.24) is valid if vectors u and v are unit and have non-vanishing x com-ponents. These conditions have short formulations | u k | ≤ , | v k | ≤ . (3.28)Above inequalities are polynomials of degree six and algebraic solutions areunlikely. However, it is still possible do define the domain of validity of Eq.(3.27). HAPTER 3. SHARED QUANTUM STATES Here, we consider the second case given by Eq.(3.16). Derivations repeat stepsof the previous subsection and the only difference is the interchange ω ↔ µ .Therefore we skip some obvious steps and present only main results. Componentsof vectors u and v along axis z are u k = λ r − r r µ − r , v k = λ r − r r µ − r . (3.29)The second equation for Lagrange multipliers λ (cid:0) µ + r − r (cid:1) = λ (cid:0) µ + r − r (cid:1) (3.30)together with Eq.(3.16) yields λ = ± µ bc − adac − bd , λ = ± µ ac − bdbc − ad . (3.31)Using these expressions, one can derive the following expression for entanglementeigenvalue 4Λ = 1 + λ (4 µ + r − r ) − r r r µ − r . (3.32)Now the restrictions 1 / < Λ ≤ d by − d in the identity (3.23)and rewrite Eq.(3.32) as follows4Λ = 12 ( ac − bd )( bc − ad )( ab − cd ) p ( p − c − d )( p − b − d )( p − a − d ) ±
12 ( ac − bd )( bc − ad )( ab − cd ) p ( p − c − d )( p − b − d )( p − a − d ) . (3.33)Lower sign yields zero and is wrong. It shows that reduced density matrix ρ AB still has zero eigenvalue.Upper sign may yield a true answer. Entanglement eigenvalue isΛ = 4 R × , (3.34)where R × = ( ac − bd )( bc − ad )( ab − cd )16 S × , (3.35)and S × = p ( p − c − d )( p − b − d )( p − a − d ). The formula (3.35) may seemsuspicious because it is not clear whether right hand side is positive and lies in HAPTER 3. SHARED QUANTUM STATES
ABCD in Fig.1A is not a quadrangle and is not apolygon at all. The reason is that it has crossed sides AD and BC . We callfigure ABCD crossed-quadrangle in a figurative sense as it has four sides anda cross point. Another justification of this term is that we will compare figure
ABCD in Fig.1A with a convex quadrangle
ABCD containing the same sides.Consider a crossed-quadrangle
ABCD with sides AB = a, BC = b, CD = c, DA = d that has circumcircle. It is easy to find the length of the interval ACAC = ( ac − bd )( bc − ad ) ab − cd . (3.36)This relation is true unless triangles ABC and
ADC have the same heightand as a consequence equal areas. Note that S × is not an area of the crossed-quadrangle. It is the difference between the areas of the noted triangles.Using Eq.(3.36), one can derive exactly Eq.(3.35) for the circumradius of thecrossed-quadrangle.Eq.(3.34) is meaningful if vectors u and v are unit and have nonzero compo-nents along the axis y . In this subsection we consider the last case described by Eq.(3.17). Entanglementeigenvalue takes maximal value if all terms in r.h.s. of Eq.(3.3) are positive. Thenequations (3.17) and (3.10) together impose
HAPTER 3. SHARED QUANTUM STATES u = Sign( r ) k , v = Sign( r ) k , r r r < , (3.37)where Sign(x) gives -1, 0 or 1 depending on whether x is negative, zero, or positive.Substituting these values into Eq.(3.3), we obtainΛ = 14 (1 + | r | + | r | + | r | ) . (3.38)Owing to inequality, r r r <
0, above expression always gives a square of thelargest coefficient l l = max( a, b, c, d ) (3.39)in Eq.(3.8). Indeed, let us consider the case r > , r > , r <
0. Frominequalities r > , r > c > d + | a − b | and therefore c > d .Note, c > d is necessary but not sufficient condition. Now if d > b , then r > c > a and if d < b , then r < c > a . Thus inequality c > a is truein all cases. Similarly c > b and c is the largest coefficient. On the other handΛ = c and Eq.(3.38) really gives the largest coefficient in this case.Similarly, cases r > , r < , r > r < , r > , r > = b and Λ = a , respectively. And again entanglement eigenvalue takesthe value of the largest coefficient.The last possibility r < , r < , r < = d and d is the largest coefficient.Combining all cases mentioned earlier, we rewrite Eq.(3.38) as followsΛ = l . (3.40)This expression is valid if both vectors u and v are collinear with the axes z .We have derived three expressions for (3.24),(3.34) and (3.40) for entangle-ment eigenvalue. They are valid when vectors u and v lie in xz plane, lie in yz plane and are collinear with axis z , respectively. The following section goes onto specify these domains by parameters a, b, c, d . Mainly, two points are being analyzed. First, we probe into the meaningful ge-ometrical interpretations of quantities R q and R × . Second, we separate validitydomains of equations (3.24),(3.34) and (3.40). It is mentioned earlier that alge-braic methods for solving the inequalities of degree six are ineffective. Hence, weuse geometric tools that are elegant and concise in this case.We consider four parameters a, b, c, d as free parameters as the normalizationcondition is irrelevant here. Indeed, one can use the state | ψ i / √ a + b + c + d HAPTER 3. SHARED QUANTUM STATES is replaced by Λ / ( a + b + c + d ). Inother words, normalization condition re-scales the quadrangle, convex or crossed,so that the circumradius always lies in the required region. Consequently, inconstructing quadrangles we can neglect the normalization condition and considerfour free parameters a, b, c, d . It is known that four sides a, b, c, d of the convex quadrangle must obey theinequality p − l >
0. Any set of such parameters forms a cyclic quadrilateral. Notethat the quadrangle is not unique as the sides can be arranged in different orders.But all these quadrangles have the same circumcircle and the circumradius isunique.The sides of a crossed-quadrangle must obey the same condition. Indeed,from Fig.1A it follows that BC − AB < AC < AD + DC and DC − AD < AC
0. However, theexistence of the circumcircle requires an additional condition and it is explainedhere. The relation r = 2 µ cos ABC forces 4 µ ≥ r and, therefore S × ≥ . (3.41)Thus the denominator in Eq.(3.35) must be positive. On the other hand theinequality AC ≥ ac − bd )( bc − ad )( ab − cd ) ≥ . (3.42)These two inequalities impose conditions on parameters a, b, c, d . For thefuture considerations, we need to write explicitly the condition imposed by in-equality (3.42). The numerator is a symmetric function on parameters a, b, c, d and it suffices to analyze only the case a ≥ b ≥ c ≥ d . Obviously ( ac − bd ) ≥ , ( ab − cd ) ≥ bc ≥ ad . The last inequalitystates that the product of the largest and smallest coefficients must not exceedthe product of remaining coefficients. Denote by s the smallest coefficient s = min( a, b, c, d ) . (3.43)We can summarize all cases as follows l s ≤ abcd. (3.44)This is necessary but not sufficient condition for the existence of R × . The nextcondition S × > HAPTER 3. SHARED QUANTUM STATES In this section we define applicable domains of expressions (3.24),(3.34) and (3.40)step by step.
Circumradius of convex quadrangle.
First we separate the validity domainsbetween the convex quadrangle and the largest coefficient. In a highly entangledregion, where the center of circumcircle lies inside the quadrangle, the circum-radius is greater than any of sides and yield a correct answer. This situationis changed when the center lies on the largest side of the quadrangle and bothequations (3.24) and (3.40) give equal answers. Suppose that the side a is thelargest one and the center lies on the side a . A little geometrical speculationyields a = b + c + d + 2 bcda . (3.45)From this equation we deduce that if a is smaller than r.h.s., i.e. a ≤ b + c + d + 2 bcda , (3.46)then the circumradius-formula is valid. If a is greater than r.h.s in Eq.(3.45),then the largest coefficient formula is valid. The inequality (3.46) also guaranteesthe existence of the cyclic quadrilateral. Indeed, using the inequality bc + cd + bd ≥ bcda , (3.47)one derives ( b + c + d ) ≥ b + c + d + 6 bcda ≥ a . (3.48)Above inequality ensures the existence of a convex quadrangle with the givensides.To get a confidence, we can solve equation u k = ± ω − r )(1 + u k ) = 2 adbc + ad (cid:18) b + c + d + 2 bcda − a (cid:19) × (cid:18) a + b + c + 2 abcd − d (cid:19) (3.49a)(4 ω − r )(1 − u k ) = 2 bcbc + ad (cid:18) a + c + d + 2 acdb − b (cid:19) × (cid:18) a + b + d + 2 abdc − c (cid:19) . (3.49b) HAPTER 3. SHARED QUANTUM STATES ω − r )(1 + v k ) = 2 bdac + bd (cid:18) a + c + d + 2 acdb − b (cid:19) × (cid:18) a + b + c + 2 abcd − d (cid:19) (3.50a)(4 ω − r )(1 − v k ) = 2 acac + bd (cid:18) b + c + d + 2 bcda − a (cid:19) × (cid:18) a + b + d + 2 abdc − c (cid:19) . (3.50b)Thus, the circumradius of the convex quadrangle gives a correct answer if allbrackets in the above equations are positive. In general, Eq.(3.24) is valid if l ≤
12 + abcdl . (3.51)When one of parameters vanishes, i.e. abcd = 0, inequality (3.51) coincideswith the corresponding condition in Ref.[37]. Circumradius of crossed quadrangle.
Next we separate the validity do-mains between the convex and the crossed quadrangles. If S × <
0, then crossedone has no circumcircle and the only choice is the circumradius of the convexquadrangle. If S × >
0, then we use the equality4 R q − R × = r abcdS q S × (3.52)where r = r r r . It shows that r > R q > R × and vice-versa. Entangle-ment eigenvalue always takes the maximal value. Therefore, Λ = 4 R q if r > = 4 R × if r <
0. Thus r = 0 is the separating surface and it is necessaryto analyze the condition r < a ≥ b ≥ c ≥ d . Then r and r are positive. Therefore r is negativeif and only if r is negative, which implies a + d > b + c . (3.53)Now suppose a ≥ d ≥ b ≥ c . Then r is negative and r is positive. Therefore r must be positive, which implies a + c > b + d . (3.54) HAPTER 3. SHARED QUANTUM STATES r ≤ l ≥ − s . (3.55)It remains to separate the validity domains between the crossed-quadrangleand the largest coefficient. We can use three equivalent ways to make this sepa-ration:1)to use the geometric picture and to see when 4 R × and l coincide,2)directly factorize equation u k = ± d .All of these give the same result stating that Eq.(3.34) is valid if l ≤ − abcdl . (3.56)Inequalities (3.55) and (3.56) together yield l s ≥ abcd. (3.57)This inequality is contradicted by (3.44) unless l s = abcd . Special cases like l s = abcd are considered in the next section. Now we would like to commentthe fact that crossed quadrangle survives only in exceptional cases. Actuallycrossed case can be obtained from the convex cases by changing the sign of anyparameter. It crucially depends on signs of parameters or, in general, on phasesof parameters. On the other hand all phases in Eq.(3.8) can be eliminated by LU-transformations. For example, the phase of d can be eliminated by redefinitionof the phase of the state function | ψ i and the phases of remaining parameterscan be absorbed in the definitions of basis vectors | i of the qubits A, B andC. Owing to this entanglement eigenvalue being LU invariant quantity does notdepend on phases. However, crossed case is relevant if one considers states givenby Generalized Schmidt Decomposition(GSD) [47]. In this case phases can notbe gauged away and crossed case has its own range of definition. This range hasshrunk to the separating surface r = 0 in our case.Now we are ready to present a distinct separation of the validity domains:Λ = ( R q , if l ≤ / abcd/l l if l ≥ / abcd/l (3.58)As an illustration we present the plot of d -dependence of Λ in Fig.2 when a = b = c .We have distinguished three types of quantum states depending on which ex-pression takes entanglement eigenvalue. Also there are states that lie on surfaces HAPTER 3. SHARED QUANTUM STATES m a x d(4/7) (4/7) (4/9) Figure 3.2: Plot of d -dependence of Λ max when a = b = c . When d →
1, Λ goes to 1 as expected. When d = 0, Λ becomes 4 /
9, which coincides with theresult of Ref.[25]. When r = 0 which implies a = d = 1 /
2, Λ becomes 1 / d = 2 a , which implies d = p /
7, Λ goes to4 /
7, which is one of shared states (it is also shown as another dotted line).separating different applicable domains. They are shared by two types of quan-tum states and may have interesting features. We will call those shared states.Such shared states are considered in the next section.
Consider quantum states for which both convex and crossed quadrangles yieldthe same entanglement eigenvalue. Eq.(3.36) is not applicable and we rewriteequations (3.27) and (3.35) as follows4 R q = 12 (cid:18) − r S q (cid:19) , R × = 12 (cid:18) − r S × (cid:19) . (3.59)These equations show that if the state lies on the separating surface r = 0,then entanglement eigenvalue is a constantΛ = 12 (3.60)and does not depend on the state parameters. This fact has a simple inter-pretation. Consider the case r = 0. Then b + c = a + d = 1 / HAPTER 3. SHARED QUANTUM STATES b, c and a, d , respectively, regardless of the triangles beingin the same semicircle or in opposite semicircles. In both cases they yield samecircumradius. Decisive factor is that the center of the circumcircle lies on thediagonal. Thus the perimeter and diagonals of the quadrangle divide ranges ofdefinition of the convex quadrangle. When the center of circumcircle passes theperimeter, entanglement eigenvalue changes-over from convex circumradius tothe largest coefficient. And if the center lies on the diagonal, convex and crossedcircumradiuses become equal.We would like to bring plausible arguments that this picture is incompleteand there is a region that has been shrunk to the point. Consider three-qubitstate given by GSD | ψ i = a | i + b | i + b | i + d | i + e | i . (3.61)One of parameters must have non-vanishing phase[47] and we can treat thisphase as an angle. Then, we have five sides and an angle. This set defines asexangle that has circumcircle. One can guess that in a highly entangled regionentanglement eigenvalue is the circumradius of the sexangle. However, there is acrucial difference. Any convex sexangle contains a star type area and the sides ofthis area are the diagonals of the sexangle. The perimeter of the star separatesthe convex and the crossed cases. Unfortunately, we can not see this picture inour case because the diagonals of a quadrangle confine a single point. It is left forfuture to calculate the entanglement eigenvalues for arbitrary three qubit statesand justify this general picture.Shared states given by r = 0 acquire new properties. They can be used forperfect teleportation and superdense coding [37, 62]. This statement is not provenclearly, but also no exceptions are known.Now consider a case where the largest coefficient and circumradius of theconvex quadrangle coincide with each other. The separating surface is given by l = 12 + abcdl . (3.62)Entanglement eigenvalue ranges within the narrow interval12 ≤ Λ ≤ . (3.63)It separates slightly and highly entangled states. When one of coefficients islarge enough and satisfies the relation l > / abcd/l , entanglement eigenvaluetakes a larger coefficient. And the expression (3.8) for the state function effec-tively takes the place of Schmidt decomposition. In highly entangled region nosimilar picture exists and all coefficients participate in equal parts and yield thecircumradius. Thus, shared states given by Eq.(3.62) separate slightly entangledstates from highly entangled ones, and can be ascribed to both types. HAPTER 3. SHARED QUANTUM STATES r = 0 acquirenew and important features. One can expect that shared states dividing highlyand slightly entangled states also must acquire some new features. However,these features are yet to be discovered. We have considered four-parametric families of three qubit states and derivedexplicit expressions for entanglement eigenvalue. The final expressions have theirown geometrical interpretation. The result in this paper with the results ofRef.[37] show that the geometric measure has two visiting cards: the circum-radius and the largest coefficient. The geometric interpretation may enable usto predict the answer for the states given by GSD. If the center of circumcirclelies in star type area confined by diagonals of the sexangle, then entanglementeigenvalue is the circumradius of the crossed sexangle(s). If the center lies in theremaining part of sexangle, the entanglement eigenvalue is the circumradius of theconvex sexangle. And when the center passes the perimeter, then entanglementeigenvalue is the largest coefficient. Although we cannot justify our predictiondue to lack of computational technique, this picture surely enables us to take astep toward a deeper understanding of the entanglement measure.Shared states given by r = 0 play an important role in quantum informationtheory. The application of shared states given by Eq.(3.62) is somewhat ques-tionable, and should be analyzed further. It should be pointed out that one hasto understand the properties of these states and find the possible applications.We would like to investigate this issue elsewhere.Finally following our procedure, one can obtain the nearest product state of agiven three-parametric W-type state. These two states will always be separatedby a line of densities composed of the convex combination of W-type states andthe nearest product states [69]. There is a separable density matrix ̺ which splitsthe line into two parts as follows. One part consists of separable densities andanother part consists of non-separable densities. It was shown in Ref.[69] that anoperator W = ̺ − ρ ABC − tr[ ̺ ( ̺ − ρ ABC )] I has the properties tr( W ρ
ABC ) < W ̺ ) ≥ ̺ . The operator W is clearlyHermitian and thus is an entanglement witness for the state. Thus our resultsallow oneself to construct the entanglement witnesses for W-type three qubitstates. However, the explicit derivation of ̺ seems to be highly non-trivial. hapter 4Duality and the geometricmeasure of entanglement ofgeneral multiqubit W states In this chapter we find the nearest product states and geometric measure ofentanglement for arbitrary generalized W states of n qubits [30, 75].Quantifying entanglement of multipartite pure states presents a real challengeto physicists. Intensive studies are under way and different entanglement mea-sures have been proposed over the years [22, 23, 76, 77, 78, 79, 80]. However,it is generally impossible to calculate their value because the definition of anymultipartite entanglement measure usually includes a massive optimization overcertain quantum protocols or states [20, 37, 81].Inextricable difficulties of the optimization are rooted in a tangle of differentobstacles. First, the number of entanglement parameters grows exponentiallywith the number of particles involved [82]. Second, in the multipartite settingseveral inequivalent classes of entanglement exist [49, 83]. Third, the geometryof entangled regions of robust states is complicated [42]. All of these make theusual optimization methods ineffective [29, 42, 44]. Concise and elegant tools arerequired to overcome this problem.A widely used measure for multipartite systems is the maximal product over-lap Λ max . In what follows states with Λ > / < / = 1 / | GHZ i = a | . . . i + b | . . . i in some product basis. Suchstates are fragile under local decoherence, i.e. they become disentangled by theloss of any one party, and they are not highly entangled in the sense defined above.60 HAPTER 4. ENTANGLEMENT OF N-QUBIT W STATES | a | or | b | [60]. Accordingly, the nearest separable state is the product state with the largercoefficient. Thus many generalized GHZ states with different maximal overlapscan have the same nearest product state.Consider now generalized W-states [84], which can be written | W n i = c | ... i + c | ... i + · · · + c n | ... i . (4.1)Without loss of generality we consider only the case of positive parameters c k sincethe phases of the coefficients c k can be eliminated by redefinitions of local states | k i , k = 1 , , ..., n . The states (4.1) are robust against decoherence [85], i.e. lossof any n − n -qubit W states and n -dimensional unit vectors x . The second one does the samebetween n -dimensional unit vectors and n -part product states. Thus we obtaina double map, or duality , as follows | W n i ↔ x ↔ | u i ⊗ | u i ⊗ · · · ⊗ | u n i . (4.2)The main advantage of the map is that if one knows any of the three vectors,then one instantly finds the other two.This chapter is organized as follows. In Section 4.1 we construct a classifyingmap. In Section 4.2 we consider highly entangled multi-qubit W states. InSection 4.3 we derive a closed-form expression for the maximal overlap of n-qubitW states. In Section 4.4 we summarize our results. Now we prove a theorem that provides a basis for the map.
Theorem 1.
Let | W n i be an arbitrary W state (4.1) with non-negative real co-efficients c i , and let | u i ⊗ | u i ⊗ · · · ⊗ | u n i be its nearest product state. Then thephase of | u k i can be chosen so that | u k i = sin θ k | i + cos θ k | i , ≤ θ k ≤ π , k = 1 , , ..., n. HAPTER 4. ENTANGLEMENT OF N-QUBIT W STATES where cos θ + cos θ + · · · + cos θ n = 1 . (4.3) Proof.
The nearest product state is a stationary point for the overlap with | W n i ,so the states | u k i satisfy the nonlinear eigenvalue equations [25, 38, 37] h u u · · · b u k · · · u n | W n i = Λ max | u k i ; k = 1 , , · · · , n (4.4)where the caret means exclusion. We can choose the phase of | u k i so that | u k i =sin θ k | i + e iφ k cos θ k | i , and then (4.4) gives the pair of equations c k Y j = k sin θ j = Λ max e iφ k cos θ k , (4.5a) X l = k e − iφ l c l cos θ l Y j = k,l sin θ j = Λ max sin θ k . (4.5b)Eq. (4.5a) shows that Λ max e iφ k is real, so φ k = − arg(Λ max ) is independent of k .Then the modulus of the overlap |h u · · · u n | W n i| is independent of φ , so we canassume that φ = 0. Now multiplying eq.(4.5b) by sin θ k and using eq.(4.5a) givesEq.(4.3).Thus the angles cos θ k define a unit n -dimensional Euclidean vector x . Wecan also define a length r as follows. From Eq.(4.5a) it follows that the ratiosin 2 θ k /c k does not depend on k . If this ratio is non-zero we can define1 r ≡ sin 2 θ c = sin 2 θ c = · · · = sin 2 θ n c n . (4.6) Equations (5) admit a trivial solution sin 2 θ k = 0 , k = 1 , , · · · , n and a specialsolution with nonzero values of all sines. The trivial solution gives the largestcoefficient of | W n i for the maximal overlap and is valid for slightly entangledstates. We consider them later and now focus on the special solutions. FromEq.(4.6) it follows thatcos θ k = 12 ± r − c k r ! , k = 1 , , · · · , n. (4.7)The plus sign means that cos 2 θ k >
0. Then from Eq.(4.3) it follows that this ispossible for at most one angle; specifically, we prove that if cos 2 θ k > k ,then c k is the largest coefficient in Eq.(4.1). Suppose cos 2 θ k > c k is not thelargest coefficient and there exists a greater coefficient, say c l . Then from Eq.(4.6) HAPTER 4. ENTANGLEMENT OF N-QUBIT W STATES θ l > sin 2 θ k > | cos 2 θ l | < | cos 2 θ k | . Nowwe rewrite Eq.(4.3) as follows: − cos 2 θ − cos 2 θ − · · · − cos 2 θ n = n − . (4.8)From | cos 2 θ l | < | cos 2 θ k | and cos 2 θ k > − cos 2 θ k − cos 2 θ l < c k must be the largest coefficient.Without loss of generality we assume that 0 ≤ c ≤ · · · ≤ c n . Then in (4.7)we must take the − sign for k = 1 , . . . , n − r − c r + · · · + r − c n − r ± r − c n r = n − f ± ( r ). We also use f ( r ) todenote this expression without the last term. The function r ( c , c , ..., c n ) definedby f + ( r ) = n − c k .In contrast, the function defined by f − ( r ) = n − c n is different. Thus in equation (4.9)the upper and lower signs describe symmetric and asymmetric entangled regionsof highly entangled states, respectively.For highly entangled states, eqs. (4.9) ± uniquely define r as a function of thestate parameters c k . More precisely, Theorem 2.
There are two critical values r and r of the largest coefficient c n ,i.e. functions of c , . . . , c n − such that1. If c n ≤ r , there is a unique solution of (4.9 + ) and no solution of (4.9 − );2. If c n = r , both (4.9 + ) and (4.9 − ) have a unique solution, the same forboth;3. If r < c n ≤ r , there is no solution of (4.9) + and a unique solution of(4.9 − );4. If c n > r , neither (4.9 + ) nor (4.9 − ) has a solution. In this case the state | W n i is slightly entangled. The value r is the solution of f ( r ) = n −
2, which exists and is unique since f ( c n − ) < n − f ( r ) → n − r → ∞ ; and r is definedby r = c + · · · + c n − . (4.10)Then r ≥ r , for f ( r ) ≥ n − f ( r ) using √ x ≥ x for 0 ≤ x ≤
1. Since f is an increasing function of r, it follows that r ≥ r . Now the theorem followsfrom the following properties of the functions f ± ( r )( f ′− is the derivative of f − ):1. f and f + are monotonically increasing functions of r . HAPTER 4. ENTANGLEMENT OF N-QUBIT W STATES f + ( r ) → n as r → ∞ .3. If c n ≤ r , f + ( c n ) = f ( c n ) ≤ f ( r ) = n − c n ≥ r , then f + ( r ) ≥ n − r > r .5. If c n < r , then f − ( c n ) < n − c n > r , then f − ( c n ) > n − c n < r , then f − ( r ) < n − r .8. If c n > r then f − ( r ) > n − r .9. f ′− ( c n + ǫ ) < ǫ .10. If c n > r , then f ′− ( r ) < r ≥ c n .These properties are illustrated in Figure 4.1. r rr f nn-2 Figure 4.1: (Color online) The behaviour of the functions f ± for five-qubit Wstates. The function f + ( r ) (dotted line) and f − ( r ) (solid line) are plotted against r in the four cases c n < r , c n = r , r < c n < r and c n = r . We can now identify the nearest product state, and the largest product stateoverlap Λ max ( | W n i ), for any W-state | W n i , as follows. Theorem 3. If c n ≥ / , the state | W n i defined by (4.1) is slightly entangled.Its nearest product state is | . . . i , with overlap Λ max ( | W n i ) = c n .If c n ≤ / , the state | W n i is highly entangled and has nearest product state | u i . . . | u n i where | u k i = sin θ k | i + | e iφ cos θ k | i , (4.11) with which its overlap is Λ max = 2 r sin θ sin θ . . . sin θ n . (4.12) HAPTER 4. ENTANGLEMENT OF N-QUBIT W STATES Here r is the solution of (4.9) ± , whose existence and uniqueness are guaranteedby Theorem 2; the phase φ is arbitrary; and θ k is given by (4.7) with the − signfor k = 1 , . . . , n − , the − sign for k = n if r satisfies (4.9 + ), the + sign if r satisfies (4.9 − ).Proof. The nonlinear eigenvalue equations (4.4) always have n solutionsΛ max = c k , | u i i = ( | i if i = k, | i if i = k , k = 1 . . . n If c n ≥
2, i.e. in case (4) of Theorem 2, there are no other stationary values, sothe largest overlap Λ max ( | W n i ) equals the largest coefficient c n , the correspondingproduct state being | . . . i .If c n < / c k . We use thefollowing inequality: If y , . . . , y n are real numbers lying between 0 and 1, andsatisfying y + · · · + y n ≤
1, then(1 − y )(1 − y ) · · · (1 − y n ) ≥ − y − y − . . . − y n . (4.13)This is readily proved by induction. We can apply (4.13) to n − − cos θ ) · · · (1 − cos θ n − ) ≥ − cos θ − · · · − cos θ n − or sin θ sin θ sin θ n − ≥ cos θ n . (4.14)Now from Eq.(4.5a) it follows that Λ ≥ c n . Thus Λ max is the maximal productoverlap, and the nearest product state is | u i . . . u n i .Next we prove that if | W n i is normalised, then Λ < /
2. For this we needanother inequality: If y , . . . , y n are real numbers lying between 0 and 1, andsatisfying y + · · · + y n = n −
1, then y + · · · + y n ≥ y + · · · + y n + 2 y y . . . y n . (4.15)This can also be proved by induction.From (4.6), and using c + · · · + c n = 1, we find r = 1sin θ + · · · + sin θ n . (4.16)Hence (4.12) gives Λ = y y . . . y n y (1 − y ) + · · · y n (1 − y n ) (4.17)where y k = sin θ k . But y + · · · + y n = n −
1, so the inequality (4.15) applies,and gives Λ ≤ / HAPTER 4. ENTANGLEMENT OF N-QUBIT W STATES R n . Theorem 4.
There is a 1:1 correspondence between highly entangled states | W n i defined by (4.1) , their nearest product states with real non-negative coefficients,and unit vectors x ∈ R n with < x k < / √ ( k = 1 , . . . , n − ), < x n < .Proof. By Theorem 3, | W n i is highly entangled if and only if c n < /
2. If this isthe case, Theorem 1 and (4.7) show that its nearest product state is of the form(4.11) where x = (cos θ , . . . , cos θ n ) is a unit vector in R n in the region stated.The angles θ k are given in terms of the coefficients c k by (4.6), in which r is afunction of the coefficients which, by Theorem 2, is uniquely defined. The nearestproduct states | u i| u i . . . | u n i are determined by these angles, up to a phase φ , by | u k i = sin θ k | i + e iφ cos θ k | i , so there is only one nearest product state with realnon-negative coefficients, and only one unit vector x , for each highly entangledstate | W n i . Conversely, given a unit vector x = (cos θ , . . . , cos θ n ), the quantity r is determined by (4.16), and then the coefficients c , . . . , c n are determined by(4.6). Thus the correspondences (4.2) are bijections.The equations (4.9 ± ) cannot always be explicitly solved to give analytic ex-pressions for r in terms of the coefficients c k . However, in some cases, includ-ing all states for n = 3, explicit solutions can be obtained. Then the angles θ k can be calculated from (4.6) and eq.(4.12) gives a formula for the maximalproduct overlap Λ max ( | W n i ). This formula is valid unless any of the angles θ k vanishes, and restores all known results for the maximal overlap of highly entan-gled W states. When n = 3 it coincides with the formula (31) in Ref.[37]. When c = c = · · · = c n it coincides with the formula (52) in Ref.[61]. And when n = 4and c = c it coincides with the formula (37) derived in Ref.[75].When max( c , c , · · · c n ) = r = 1 / max ( | W n i ) givenin Theorem 3 coincide; these states are shared quantum states. The nearestproduct states and maximal overlaps of shared states are given by the first caseof Theorem 3, but also they appear as asymptotic limits of the second case.Indeed, at the limit θ n → θ n → r sin θ n → c n , lim θ n → r cos θ k → c k , k = n. (4.18)Thus the angle θ n vanishes and the length of the vector r goes to infinity, buttheir product has a finite limit. Substituting these limits into Eq.(4.3) one obtains c n → r . Therefore entangled regions of highly and slightly entangled states areseparated by the surface c n = 1 /
2; for states on the surface, r → ∞ . All ofthese states can be used as a quantum channel for the perfect teleportation andsuperdense coding [42]. HAPTER 4. ENTANGLEMENT OF N-QUBIT W STATES We have constructed correspondences between W states, n -dimensional unit vec-tors and separable pure states. The map reveals two critical values for quantumstate parameters. The first critical value separates symmetric and asymmetricentangled regions of highly entangled states, whiles the second one separateshighly and slightly entangled states. The method gives an explicit expressionsfor the geometric measure when the state allows analytical solutions, otherwiseit expresses the entanglement as an implicit function of state parameters.It should be noted that the bijection between W states and n -dimensional unitvectors is not related directly to the geometric measure of entanglement. There-fore it is possible to extend the method to other entanglement measures. To thisend one creates an appropriate bijection between unit vectors and optimizationpoints of an entanglement measure one wants to compute. hapter 5Universal behavior of thegeometric entanglement measureof many-qubit W states In this chapter we analyze geometric entanglement measure of many-qubit Wstates and derive an interpolating formula [51].The physics of many-particle systems differs fundamentally from the one ofa few particles and gives rise to new interesting phenomena, such as phase tran-sitions [35, 52] or quantum computing [3, 5, 9, 55]. Entanglement theory, inparticular, appears to have a much more complex and richer structure in theN-partite case than it has in the bipartite setting. This is reflected by the factthat multipartite entanglement is a very active field of research that has led toimportant insights into our understanding of many-particle physics [23, 25, 38,89, 90, 91, 92, 93]. In view of this, it seems worthy to investigate also the behav-ior of entanglement measures for large-scale systems. Despite the fact that thenumber of entanglement parameters scales exponentially in the number of parti-cles [56], it is sometimes possible to capture the most relevant physical propertiesby describing these systems in terms of very few parameters.Recently a duality between highly entangled W states and product states hasbeen established [30]. The important class of W states [49] represents a particularinteresting set of quantum states associated with high robustness against particleloss and nonlocal properties of genuine entangled multipartite states [84, 94, 95,96]. And different experimentally accessible schemes to generate multipartite Wstates have been proposed and put into practice over the years [85, 97, 98, 99]The duality specifies a single-valued function r of entanglement parameters.We shall refer to r as the entanglement diameter, as it will play a crucial rolethroughout this article. Another reason for the term entanglement diameter isthat r can be interpreted geometrically as a diameter of a circumscribing sphere.The geometrical interpretation and its illustration will be presented in the ap-pendix and now we focus on the physical significance of r .68 HAPTER 5. UNIVERSALITY OF MANY-QUBIT ENTANGLEMENT max of these states is a constant regardless howmany qubits are involved and what are the values of the remaining entanglementparameters. These states are known as shared quantum states and can be usedas quantum channels for the perfect teleportation and dense coding. Thus theshared quantum states are uniquely defined as the states whose entanglementdiameter is infinite.Furthermore, highly entangled W states have two different entangled regions:the symmetric and asymmetric entangled regions. In the computational basisthese regions can be defined as follows. If a W state is in the symmetric region,then the entanglement diameter is a fully symmetric function on the state pa-rameters. Conversely, if a W state is in the asymmetric region, then there is acoefficient c such that the c dependence of the entanglement diameter differs dra-matically from the dependencies of the remaining coefficients. Hence the pointof intersection of the symmetric and asymmetric regions is the first exceptionalpoint. It depends on state parameters and its role has not been revealed so far.One thing was clear that the first exceptional point does not play an importantrole for three- and four-qubit W states [37, 75].In this chapter we show that the first exceptional point is important for large-scale W states. It approaches to a fixed point when number of qubits N increasesand becomes state-independent(up to 1 /N corrections) when N ≫
1. As a con-sequence the entanglement diameter, as well as the maximal product overlap, be-comes state-independent too and therefore many-qubit W states have two state-independent exceptional points. The underlying concept is that states whoseentanglement parameters differ widely, may nevertheless have the same maximalproduct overlap and this phenomenon should occur at two fixed points. This isan analog of the universality of dynamical systems at critical points. It is anintriguing fact that systems with quite different microscopic parameters may be-have equivalently at criticality. Fortunately, the renormalization group providesan explanation for the emergence of universality in critical systems [35, 36, 52].The developed concept distinguishes three classes of W states. The first classconsists of highly entangled W states which are below both exceptional pointsand then r varies from r min = 1 / r ≈ / √ O (1 /N ). We will showthat these states are in the symmetric region and their entanglement diameteris a slowly oscillating function on entanglement parameters. Accordingly, themaximal product overlap is an almost everywhere constant close to its greatestlower bound. Similar results have been obtained in Ref.[88], where it is shownthat almost all multipartite pure states with sufficiently large number of partiesare nearly maximally entangled with respect to the geometric measure [25] and HAPTER 5. UNIVERSALITY OF MANY-QUBIT ENTANGLEMENT r varies from r to infinity.These states are in the asymmetric region and the behavior of the entanglementdiameter is curious. We will show that r is a one-variable function in this caseand depends only on the Bloch vector b of a single qubit. As a consequenceΛ max depends only on the same Bloch vector too and its behavior is universal.That is, regardless how many many qubits are involved and what are the re-maining N − max ( b ) is common. Wewill compute analytically Λ max ( b ) and thereby find the Groverian and geometricentanglement measures [25, 28] for the large-scale W states even if neither thenumber of particles nor the most of state parameters are known.The third class consists of slightly entangled W states which are above bothexceptional points. In this case the maximal product overlap takes the value ofthe largest coefficient and these states do not posses an entanglement diameter.We will not analyze this trivial case, but will combine the functions Λ max ( b )for slightly entangled and highly entangled asymmetric W states and obtain aninterpolating function Λ max ( b ) valid for both cases. It is in a perfect agreementwith numerical solutions and quantifies the many-qubit entanglement in highaccuracy(∆Λ max / Λ max ∼ − at N ∼ max ,bringing into the position the Bloch vector, say Λ max ( b ) = Λ max .This chapter is organized as follows. In Section 5.1, we review the main resultsof Ref.[30]. In Section 6.2, we consider two- and three-parameter W states inthe symmetric region and show that all of these states are almost maximallyentangled. In Section 5.3, we consider three- and four-parameter W states inthe asymmetric region and compute explicitly their maximal product overlap. InSection 5.4, we generalize the results of Sec.III and Sec.IV to arbitrary many-qubit W states. In Section 5.5, we discuss our results. In the Appendix C, weprovide a geometrical interpretation for the entanglement diameter. In the computational basis N-qubit W states can be written as | W n i = c | ... i + c | ... i + · · · + c N | ... i , (5.1) HAPTER 5. UNIVERSALITY OF MANY-QUBIT ENTANGLEMENT c k can be absorbed in the definitions of the local states | i i ( i =1 , , ..., N ) and without loss of generality we consider only the case of positiveparameters. For the simplicity we assume that c N is the maximal coefficient,that is, c N = max( c , c , · · · , c N ).The maximal product overlap Λ max ( ψ ) of a pure state | ψ i is given byΛ max ( ψ ) = max u ,u ,...,u N | h ψ | u u ...u N i | , (5.2)where the maximization runs over all product states. The larger Λ max is, theless entangled is | ψ i . Hence for a quantum multipartite system the geometricentanglement measure E Λ max is defined as E Λ max = − log Λ max ( ψ ) . The maximal product overlap demarcates three different entangled regions inthe parameter space of W states:1. The symmetric region of highly entangled W states. Here Λ max ( c , c , ..., c N )is a symmetric function on all coefficients c i .2. The asymmetric region of highly entangled W states. Here the invarianceof Λ max ( c , c , ..., c N ) under the permutations of coefficients c i ceases to betrue.3. The region of slightly entangled W states. Here the inequityΛ ( c , c , ..., c N ) > / c N . The first criticalvalue r ( c , c , ..., c N − ) is the solution of q r − c + q r − c + · · · + q r − c N − = ( N − r , (5.3)which always exists and is unique. Note that the first critical value r for thecoefficient c N depends on the remaining coefficients c i , i = 1 , , ..., N − c N . Nonetheless we will use the abbreviation r ( c N ) ≡ r ( c , c , ..., c N − ) whenever no confusion occurs.The second critical value r ( c , c , ..., c N − is given by r = c + c + · · · + c N − . (5.4) HAPTER 5. UNIVERSALITY OF MANY-QUBIT ENTANGLEMENT r ( c N ) ≡ r ( c , c , ..., c N − for thesimplicity.The second critical value is always greater than the first one and thus thereare three cases. The first case is c N < r and the maximal product overlap isexpressed via the fully symmetric entanglement diameter r ( c , c , ..., c N ), whichis the unique solution of q r − c + q r − c + · · · + q r − c N = ( N − r. (5.5)Then Λ max is given byΛ = r N − r − c r ! r − c r ! · · · r − c N r ! (5.6)and is a bounded function satisfying the inequalities c N < Λ ( c , c , ..., c N ) < / r < c N < r . In this case the entanglement diameter r ( c , c , ..., c N ) is the unique solution of q r − c + q r − c + · · · − q r − c N = ( N − r (5.7)where only the last radical has the − sign. Then Λ max takes the formΛ = r N − r − c r ! r − c r ! · · · − r − c N r ! , (5.8)where again the negative root is taken from the last radical. The expression (5.8)also has an upper and lower bounds and the inequalities c N < Λ ( c , c , ..., c N ) < / c N ≥ r and Λ max takes the value of the largest coefficientin this case Λ = c N . (5.9)Now Λ max is bounded below and satisfies the inequality Λ > / c N = r both Eqs. (5.5) and (5.7) have the same solution r = r = c N and expressions(5.6) and (5.8) for Λ max coincide. At c N → r the solution of (5.7) goes to infinity, r → ∞ , and (5.8) asymptotically comes to (5.9). At this limit Λ = c N = r =1 / ( c , c , ..., c N ) = 1 / HAPTER 5. UNIVERSALITY OF MANY-QUBIT ENTANGLEMENT In this section we analyze the maximal product overlap of two– and three–parameter W states that belong to the symmetric region of entanglement andshow that if all coefficients are small, then r is a slowly oscillating function closeto 1 / Equations (5.5) and (5.7) are solvable for N = 3 and the answer is [37]Λ max = ( R, if c ≤ c + c c , if c ≥ c + c (5.10)where R is the circumradius of the triangle c , c , c .When N ≥ r in terms of the coefficients c k unless the state posses a symmetry.For example, for N = 4 the equations are solvable if any two coefficients coincideand unsolvable if all coefficients are arbitrary [75].However, when N ≫ N = m + k qubits and coefficients c = c = · · · = c m = a, c m +1 = c m +2 = · · · = c m + k = b. (5.11)When m > n > m √ r − a + k √ r − b = ( N − r. (5.12)This equation is solvable by radicals. Setting a = cos θ/ √ m, b = sin θ/ √ k oneobtains r = 2 N mk − N − m cos θ + k sin θ ) + 2 mk ( N − √ D N − m − k − , (5.13)where D = 1 − N − mk sin θ. (5.14)At m = 1 or k = 1 the denominator and numerator vanish in Eq.(5.13), buttheir ratio gives the correct answer. We will not consider this case since it isanalyzed in detail in Ref.[75].If m, k ≫
1, then r is almost constant since r = 14 + O (cid:18) m (cid:19) + O (cid:18) k (cid:19) . (5.15) HAPTER 5. UNIVERSALITY OF MANY-QUBIT ENTANGLEMENT θ dependence of the exact solution (5.13) isplotted. The graphics show that ∆ r/r ∼ − at N ∼ Θ r Figure 5.1: (Color online) The plots of the θ dependence of the exact solution r ( θ ) for the state (5.11). The top, middle and bottom lines represent the cases( m = 10 , k = 10) , ( m = 12 , k = 18) and ( m = 30 , k = 30), respectively. Θ g Figure 5.2: (Color online) The maximal product overlap function Λ ( θ ) atdifferent values of m and k . The axes origin is put at the point (0 , /e ) to makeit easer the comparison of the exact ant approximate solutions. The top, middleand bottom lines correspond to the values ( m = 10 , k = 10) , ( m = 12 , k = 18)and ( m = 30 , k = 30), respectively.As a consequence of Eq.(5.15) Λ is also almost constant and close to itslower bound 1 /e [100]. Indeed, using approximations12 m r − a r ! m ≈ e − ma / r , k r − b r ! k ≈ e − kb / r (5.16)one obtains Λ = 1 e + O (cid:18) m (cid:19) + O (cid:18) k (cid:19) . (5.17) HAPTER 5. UNIVERSALITY OF MANY-QUBIT ENTANGLEMENT max ( θ ) given by Eqs. (5.6) and(5.13) is plotted in Fig.5.2, which shows that ∆Λ max / Λ max ∼ − at m, k ∼ N ≥ Consider now a three-parameter W state with N = m + k + l qubits and coefficients c = · · · = c m = a, c m +1 = · · · = c m + k = b, c m + k +1 = · · · = c m + k + l = c. (5.18)We will analyze the case m, k, l ≫
1. Then Eq. (5.5) can be rewritten as m √ r − a + k √ r − b + l √ r − c = ( N − r. (5.19)From the normalization condition ma + kb + lc = 1 it follows that a ≤ /m ≪ b , c ≪
1. On the other hand (5.19) shows that r ∼
1, and thereforewe can expand the radicals in powers of a /r , b /r and c /r . Then r = 14 + O (cid:18) m , k , l (cid:19) . (5.20)Again we got the same answer for r , which means that for partitions with largenumber of qubits r depends neither on m, k, l nor on a, b, c . More precisely, r depends only on the expression ma + kb + lc = | ψ | , which drops out owing tothe normalization condition.The equation (5.19) can be solved explicitly, but the resulting half-page answeris impractical and we will compare (5.20) with the numerical solution instead.For this purpose we use the parametrization a = sin θ cos ϕ/ √ m, b = sin θ sin ϕ/ √ k, c = cos θ. The behavior of the numerical solution r ( θ ) of Eq.(5.19) for various values m, k, l and ϕ is plotted in Fig.5.3. The graphics show that the approximate solution isin a perfect agreement with the numerical solution for N ≫ n , n , ..., n k product vectors in the com-putational basis have coefficients c , c , ..., c k , respectively. Then Λ max does notdepend on partition numbers n i or amplitudes c i and the approximate solution(5.15) with the maximal product overlap (5.17) quantifies the entanglement inhigh accuracy. For example, at N ∼
10 the accuracy is ∆Λ max / Λ max ∼ − .This approximation is true unless the condition n i ≫ i = 1 , , ..., k ) is violated.What is happening if this condition is violated, is analyzed in the next section. HAPTER 5. UNIVERSALITY OF MANY-QUBIT ENTANGLEMENT Figure 5.3: (Color online) The curves show the θ dependence of the function r ( θ ).The upper, middle and bottom curves represent the cases ( m = k = l = 10 , ϕ = π/ , ( m = k = l = 20 , ϕ = 5 π/
12) and ( m = 10 , k = 20 , l = 30 , ϕ = π/ In this section we consider three- and four-parameter W states in the asymmetricregion and show that if one of coefficients exceeds the first critical value r , then r is a rapidly increasing function and ranges from one-third to infinity when themaximal coefficient ranges from the first critical value to the second critical value. Consider now the case when l = 1 in (5.18) c = · · · = c m = a, c m +1 = · · · = c m + k = b, c m + k +1 = c. (5.21)If c ≪
1, then c/r is small and r is almost constant. This case is analyzed inthe previous section and now we focus on the case when c/r cannot be neglected.Then either c . r or r < c < r .When c . r Eq.(5.5) takes the form m √ r − a + k √ r − b + √ r − c = ( N − r. (5.22)The ratios a/r and b/r are small since m, k ≫
1. Hence we expand the radicalsin powers of these ratios up to quadratic terms and solve the resulting equation.The answer is r = 12 1 − c √ − c , r − c r = 1 − c − c , max( a , b ) < c ≤ . (5.23)It is reasonable that r → / c → c ≥ r Eq.(5.7) takes the form m √ r − a + k √ r − b − √ r − c = ( N − r. (5.24) HAPTER 5. UNIVERSALITY OF MANY-QUBIT ENTANGLEMENT r = 12 1 − c √ − c , r − c r = 3 c − − c , < c < . (5.25)As one would expect, r → ∞ at c → / r = 12 1 − c √ − c , max( a , b ) < c < . (5.26)The question at issue is when (5.26) gives a required accuracy in the asymmetricregion r < c < r . We compare it with the numerical solutions of (5.22) and(5.24) for the values ( m = 8 , k = 10 , a/b = 0 . , r ≈ .
34) in Fig.5.4, where thesolid line is the plot of (5.26) and the dashed line is the numerical solution. Re-markably, the approximate solution is in a perfect agreement with the numericalone in the asymmetric region.
However, there are W state that are outside the realm of the model sketched inthe previous subsection. These are states with few (at most three) coefficientsclose to the first critical value r ∼ / √
3. In this case these coefficients are notsmall and the resulting r should has a different behavior.Notice, two coefficients cannot exceed the first critical value simultaneously.But we can construct W states whose coefficients depend on a free parameter insuch a way that at one value of the free parameter the last coefficient exceedsthe first critical value and at another value of the free parameter the precedingcoefficient exceeds the first critical value. Below we construct an illustrativeexample of a such state and analyze its entanglement diameter.An example is the 19-qubit four-parameter W state with coefficients c = · · · = c ≡ a, c = · · · = c ≡ b, c ≡ c, c ≡ d. (5.27)For the normalized states we can use free parameters ϕ, k and c as follows a = cos ϕ k (1 − c ) , b = sin ϕ k (1 − c ) , d = k − k (1 − c ) . Now we analyze the function r ( c ) at k = 1 . , ϕ = π/ c coincides with its first critical value r ( c ) at c ≈ .
606 , that is, the solution of the system7 q r − a + 10 q r − b + q r − d = 17 r and r = c is r = c ≈ . r ( c ) should range from r ( c ) to infinity when c ranges from r ( c ) to 1/2 and should has a vertical asymptote at c → / HAPTER 5. UNIVERSALITY OF MANY-QUBIT ENTANGLEMENT Figure 5.4: (Color online) Graphic illustrations of the function r ( c ) for the three-and four-parameter W states. The solid curve is the approximate solution (5.26).The dashed curve is the joined numerical solution of Eqs. (5.22) and (5.24). Allremaining coefficients are well away from the first critical value ( ≈ .
58) when c varies within the range of definition in this case. Accordingly, the state is inthe symmetric region when 0 < c < .
58 and in the asymmetric region when0 . < c < . c is ≈ .
606 and for thelast coefficient d is ≈ .
59 which is attained at c = 0 . . < c < .
606 and in the asymmetric regionotherwise. Remarkably, the three curves coincide when c > . d coincides with its first critical value r ( d ) at d ≈ . q r − a + 10 q r − b + q r − c = 17 r and r = d is r = d ≈ . c ≈ . r shouldincrease when d ranges from r ( d ) to d max . But the maximum value of d is less than the second critical value since d = d ( c = 0) = ( k − /k =4 / < /
2. Therefore r should be bounded above in the interval [ r ( d ) , d max ]and attain a maximum at d max . As d is a decreasing function on c , r shouldattain a maximum at c = 0 and then decrease when c ranges from 0 to0.45657. HAPTER 5. UNIVERSALITY OF MANY-QUBIT ENTANGLEMENT d < r ( d ) and c < r ( c ). Hence r ( c ) should be minimal and nearly constant when 0 . < c < . c dependence of the function r ( c ).It agrees completely with the above analyze.The main point is that all the three curves coincide when c > r ( c ). In thenext section we will show that this is not accidental and the curves must coincide.In this context the equation (5.26) is a surprising result. The quantity r , as wellas the maximal product overlap Λ max , depends from c only. The rest of the stateparameters appear in (5.26) in the combination | ψ | − c and drop out by thenormalization condition! c g Figure 5.5: (Color online) The plots of the function Λ max ( c ). The solid line isthe approximate solution (5.28), the dashed and dotted lines are the numericalsolutions for the states (5.21) and (5.27), respectively. The curves may havedifferent behaviors when c N < r , but coincide when c N ≥ r .Furthermore, we can derive an analytic expression for the maximal overlap.Using approximations (5.16) one obtainsΛ ( c ) = (1 − c ) e − (1 − c ) / (1 − c ) . (5.28)The behavior of the function Λ max ( c ) is shown in Fig.5.5. The solid line isthe curve (5.28), the dashed curve is the numerical solution for the state (5.21)and the dotted line is the numerical computation for the state (5.27). They allcoincide when c > r ( c ).For highly entangled states the maximal product overlap ranges from its lowerto the upper bound when c ranges from r to r . On the other hand the Blochvector b of N th qubit is collinear with axis z and b z = 1 − c . Thus Λ max is a one-variable function on b z and one can vary the entanglement of the multiqubit W HAPTER 5. UNIVERSALITY OF MANY-QUBIT ENTANGLEMENT − < b z < − r , N ≫
1. These qubitsare just spectators, they should appear in the W state, but have no influence onthe entanglement of the state.
The results of the previous sections are based on the fact that the entanglementdiameter r is bounded below. In the symmetric region it is rigidly bound by thefollowing theorem. Theorem 1. If r is a solution of Eq.(5.5), then14 ≤ r ≤ . (5.29) Proof.
Note that c i r = r − c i r ! − r − c i r ! ≤ − r − c i r ! . By summing over i the above inequality and using (5.5) and the normalizationcondition one obtains 1 r ≤ n − n + 2) = 4 . Hence r ≥ /
4. Next, from x ≤ √ x for 0 ≤ x ≤ n X i =1 (cid:18) − c i r (cid:19) ≤ n X i =1 r − c i r , or n − r ≤ n − , that is, r < / max of arbi-trary N-qubit W states in the symmetric region. Indeed, in this region c i ∼ /N and therefore c i /r ≪
1. Then one can expand the radicals in (5.5) and obtain N − r ≈ N − , which generalizes (5.5) and (5.6) to arbitrary W states with c N ≪ c i are small( ∼ / √ N ). Then the ratios c i /r are small since r isbounded below ( ∼ /
2) and we can keep first nonvanishing orders of these ratios.
HAPTER 5. UNIVERSALITY OF MANY-QUBIT ENTANGLEMENT r should has a lowerbound but has not an upper bound since r → ∞ at c → r . One may expectthat the lower bound of r in the asymmetric region coincides with the upperbound of r in the symmetric region. But the following theorem shows that thisis not the case. Theorem 2. If r is a solution of Eq.(5.7), then r ≥ . (5.30) Proof.
We use the same technique, namely1 r = N − X i c i r + c N r ≤ N − X i − r − c i r ! + c N r , or 1 r ≤ − r − c N r + c N r ≤ c N ≤ r. This bound, as well as bounds (5.29), is tight, for example, r → / c → / N ∼
10) than the symmetric one (5.15)( N ∼ r is greater in this case. Second, since c N is greater( c N >r ) the remaining coefficients should be smaller due to the normalization condi-tion. These two factors together make the ratio c i /r smaller. Hence the approxi-mate solution should has a better agreement with the exact one. Aside from that, r is a fast increasing function and goes to the infinity unlike to the symmetriccase. Hence the values of the coefficients c i become irrelevant when r ≫ Theorem 3. If c N = r , then r = 13 + O ( 1 N ) (5.31) Proof.
Note that on the boundary of the symmetric and asymmetric regions r = r = c N and therefore r ≥ /
3. Expanding the radicals in (5.3) in powersof c i /r one obtains N − − − c N c N + O ( 1 N ) = N − , HAPTER 5. UNIVERSALITY OF MANY-QUBIT ENTANGLEMENT N ≫
1) are involved the first critical value dependsneither the number of qubits nor the state parameters and is a constant, r ≈ / √ < c N < r all functions r ( c )must converge to the point r (1 / √ ≈ / √
3. This is the effect of the firstcritical value.3. All functions r ( c ) have the the same vertical asymptote, namely, r ( c ) → ∞ at c → / √
2. This is the effect of the second critical value.These statements together give no chance to differ markedly exact and ap-proximate solutions in the asymmetric region. In conclusion, when N ≫ b z among the z components of the Bloch vectors. Using approximations12 r − c i r ! ≈ e − c i / r , i = 1 , , · · · , N − ( N ≫
1) = ( b z e − bz bz , if 0 < b z < − b z , if b z < max is shown in Fig. 5.6, where the b z dependence of Λ max is plotted for N =10. The solid and dashed lines represent the interpolating function (5.32) andnumerical computation, respectively.We did not plotted numerical results for different states because differentcurves overlap and become indistinguishable. We failed to find the states forwhich the numerical results markedly differ from the plotted one provided N ≫ The main result of this work is the formula (5.32). First, it shows that sometimesthe characterization and manipulation of the entanglement of many qubit statesis a simple task, while the case of few or several qubits is a complicated problem.Second, it states that when N ≫ HAPTER 5. UNIVERSALITY OF MANY-QUBIT ENTANGLEMENT - - - - - b z g Figure 5.6: (Color online) The maximal product overlap Λ max as a function of z component of the Bloch vector b z . The solid line is the interpolating formula(5.32). The dashed line is the numerical computation for a 10-qubit W state.states. Then a question arises: Why do the maximal product overlaps of thedifferent W states far apart from the exceptional points have the same behavior?Perhaps the reason is that these states are all W-class states. Classification of en-tangled states explains that pure states can be probabilistically converted to oneanother within the same class by stochastic local operations and classical com-munication [49, 83, 102]. And one can assume that large-scale systems within thesame class have the feature, aside from the interconvertibility, that their entangle-ment is universal. An argument in favor of this assumption is that the geometricmeasure of entanglement [25], the relative entropy of entanglement [23] and thelogarithmic global robustness [89] are related by bounding inequalities and, more-over, the relative entropy of entanglement is an upper bound to entanglement ofdistillation. Hence it is unlikely that these measures may exhibit contradictingresults and each of them predicts its own and very different entanglement be-havior of large-scale W-states. If this argument is true, then entanglement oflarge-scale states within the same class is universal. However, states from thedifferent classes may exhibit different behaviors. By no means it is obvious, andprobably not true, that the maximal product overlap of GHZ-class states shouldhave a behavior similar to that of W states.Another possible explanation is that the universality of the maximal overlap oflarge scale W states is the inherent feature of the geometric entanglement measurerather than the inherent feature of quantum states. If it is indeed the case, thena reasonable question is the following: do the exceptional points really existor they are just the fabrication of the geometric entanglement measure? In thiscontext the second exceptional point is a fundamental quantity. Indeed, there arestates applicable for the perfect teleportation and dense coding and these statesall should possess the same amount of entanglement. Hence there is an specific HAPTER 5. UNIVERSALITY OF MANY-QUBIT ENTANGLEMENT hapter 6Non-strict inequality for Schmidtcoefficients of three-qubit states
In this chapter we analyze generalized Schmidt decomposition for three-qubitstates and establish a relation between those Schmidt coefficients [53].Tripartite entanglement is a difficult subject for physicists. Essential resultswere obtained in this field [38, 47, 49, 57], but fundamental problems remain un-solved. Two of them are the main obstacles to understand tripartite entanglementso well as bipartite entanglement.The first problem is the entanglement transformation problem. Its essence isthe set of necessary and sufficient conditions for transforming a given pure tripar-tite state to another pure tripartite state by local operations and classical com-munication. This problem is solved for bipartite systems [55] and therefore theconditions for bipartite entanglement transformation based on majorization givea concise answer to the questions: among given states which ones are more/lessentangled and which ones are incomparable? Unfortunately these problem is apuzzle in the case of tripartite systems.The second problem, closely related to the first one, is the notion of maximallyentangled states. This problem also is solved for bipartite systems and maximallyentangled two-qubit states are the Einstein-Podolsky-Rosen state [14] and its localunitary(LU) equivalents known as Bell states [15]. However, there is no clearand unique definition of a maximally entangled state in multipartite settings.Consequently it is impossible to introduce operational entanglement measuresbased on optimal rates of conversion between arbitrary states and maximallyentangled states [19, 20, 21].For bipartite systems these problems have been solved with the help of theSchmidt decomposition [45, 46]. Therefore its generalization to multipartitestates can solve difficult problems related to multipartite entanglement. Thisgeneralization for three qubits is done by Ac´ın et al [47], where it is shown thatan arbitrary pure state can be written as a linear combination of five productstates. Independently, Carteret et al developed a method for such a generaliza-85
HAPTER 6. GENERALIZED SCHMIDT DECOMPOSITION
HAPTER 6. GENERALIZED SCHMIDT DECOMPOSITION
In this section we derive GSD for three-qubit pure states in detail since the deriva-tion method is used in Sec.IV to compute the second variation of the maximalproduct overlap.For a three-qubit pure state | ψ i the maximal product overlap λ ( ψ ) is definedas Λ max ≡ λ ( ψ ) = max |h u u u | ψ i| , (6.1)where the maximum is over all tuples of vectors | u k i with k u k k = 1 , ( k = 1 , , λ ( ψ ) with constraints k u k k = 1 we form the auxiliaryfunction Λ given byΛ = |h u u u | ψ i| + α ( h u | u i −
1) + α ( h u | u i −
1) + α ( h u | u i − , (6.2)where the Lagrange multipliers α k enforce unit nature of the local vectors | u k i . HAPTER 6. GENERALIZED SCHMIDT DECOMPOSITION | u k i and α k , that is | u k i → | u k i + | δu k i ; α k → α k + δα k , and compute the resulting variation of Λ. Hereafter δ Λand δ n Λ mean the full and the n th variation of Λ, respectively.First we consider the first variation and require that δ Λ = 0. Then thevanishing of the partial derivatives of Λ with respect to the Lagrange multipliers α k gives h u | u i − h u | u i − h u | u i − , (6.3)which are constraints on the local states | u i i .The vanishing of the partial derivatives of Λ with respect to these local statesgives h ψ | u u u i h u u | ψ i + α | u i = 0 h ψ | u u u i h u u | ψ i + α | u i = 0 (6.4) h ψ | u u u i h u u | ψ i + α | u i = 0and their Hermitian conjugates. From (6.4) it follows that α = α = α = − λ and therefore we can adjust phases of | u k i so that stationarity equations (6.4)become h u u | ψ i = λ | u i , h u u | ψ i = λ | u i , h u u | ψ i = λ | u i . (6.5)In the case of three-qubit states these equations are sufficient to constructGSD as follows. For each single-qubit state | u k i there is, up to an arbitrary phase,a unique single-qubit state | v k i orthogonal to it. Then from (6.5) it follows thatthe product states | u u v i , | u v u i , | v u u i are orthogonal to | ψ i and (6.5) can be written as h u | ψ i = λ | u u i + λ | v v i , h u | ψ i = λ | u u i + λ | v v i , (6.6) h u | ψ i = λ | u u i + λ | v v i . We choose the phases of | v k i such that λ , λ , λ ≥
0. Note that after this choicethe collective sign-flip of | v k i ’s does not change anything and we will use thisfreedom in a little while.The state ψ can be written as a linear combination of five product states asfollows | ψ i = λ | u u u i + λ | u v v i + λ | v u v i + λ | v v u i + λ | v v v i , (6.7)where λ is a complex number. It has two constraints, first λ ≥ | λ | and second − π/ ≤ Arg( λ ) ≤ π/
2, which can be achieved by the simultaneous change ofthe signs of the local states | v k i .Sometimes one relabels | u i → | i , | v i → | i for the simplicity. We leave (6.7)as is and refer to as GSD for three-qubits. HAPTER 6. GENERALIZED SCHMIDT DECOMPOSITION In this section we show that for a given state | ψ i the canonical form (6.7) is notunique except rare cases and additional relations are needed to single out theSchmidt decomposition from the useless canonical forms.Consider a three-parameter family of W type states [49] given by | w ( a, b, c ) i = a | i + b | i + c | i , (6.8)where parameters a, b, c are all positive since their phases can be eliminated byappropriate LU transformations. Stationarity equations (6.5) of this state havethree simple solutions and one special solution which exists if and only if param-eters a, b, c can form a triangle [37].The three simple solutions are | u (1) i = | i , | u (1) i = | i , | u (1) i = | i , λ (1) = a ; (6.9) | u (2) i = | i , | u (2) i = | i , | u (2) i = | i , λ (2) = b ; (6.10) | u (3) i = | i , | u (3) i = | i , | u (3) i = | i , λ (3) = c ; (6.11)where numbers within brackets mark solutions.The fourth nontrivial solution is | u (4) i = a √ r a | i + √ r b r c | i S , | u (4) i b √ r b | i + √ r a r c | i S | u (4) i = c √ r c | i + √ r a r b | i S , λ (4) = abc S (6.12)where r a = b + c − a ,r b = a + c − b , (6.13) r c = a + b − c and S is the area of the triangle ( a, b, c ).At r a r b r c = 0 the special solution reduces to a trivial solution. Note thatabsolute values of these quantities | r a | , | r b | , | r c | are magnitudes of Bloch vectorsof the first, second and third qubits, respectively and r a r b r c = 0 means that someof one-particle reduced densities is a multiple of the unit matrix. In other words,the states with a completely mixed subsystems appear at the edge of the specialsolution and viceversa.These four solutions of (6.5) give the following four canonical forms for thestate (6.8) HAPTER 6. GENERALIZED SCHMIDT DECOMPOSITION | w ( a, b, c ) i = (6.14) λ (1) | u (1) u (1) u (1) i + b | v (1) u (1) v (1) i + c | v (1) v (1) u (1) i ; | w ( a, b, c ) i = (6.15) λ (2) | u (2) u (2) u (2) i + c | u (2) u (2) u (2) i + a | v (2) v (2) u (2) i ; | w ( a, b, c ) i = (6.16) λ (3) | u (3) u (3) u (3) i + b | u (2) v (2) v (2) i + a | v (3) u (3) v (3) i ; | w ( a, b, c ) i = (6.17) λ (4) | u (4) u (4) u (4) i + i √ r a r b r c S | v (4) v (4) v (4) i + ar a S | u (4) v (4) v (4) i + br b S | v (4) u (4) v (4) i + cr c S | u (4) u (4) v (4) i . Now which of these canonical forms is a right decomposition?It is easy to clarify this question in this particular case since we have allsolutions of the stationarity equations (6.5) and can single out the one whoselargest coefficient is the dominant eigenvalue of (6.5).The answer is [37]:1. if r a < λ (1) is the maximal eigenvalue of (6.5), but λ (2), λ (3), λ (4) are not.2. if r b < λ (2) is the maximal eigenvalue of (6.5), but λ (1), λ (3), λ (4) are not.3. if r c < λ (3) is the maximal eigenvalue of (6.5), but λ (1), λ (2), λ (4) are not.4. otherwise only λ (4) is the maximal eigenvalue of (6.5), but λ (1), λ (2), λ (3) are not.However, we are unable to solve (6.5) for generic states and single out the maximaleigenvalue in this way. Also we are not forced to compare all eigenvalues of (6.5)to see whether the largest coefficient of a given decomposition is the maximalproduct overlap. We can just require that it is a truly maximum of the productoverlap instead and obtain a criteria which shows whether the largest coefficientof a given canonical form is the maximal product overlap of the state. This willbe done in next sections. HAPTER 6. GENERALIZED SCHMIDT DECOMPOSITION In this section we compute the second variation of the maximal product overlap.We compute it at stationary points to single out truly maximums and thereforewe use the results coming from the vanishing of the first variation. Straightfor-ward calculation gives δ Λ = λ |h δu | u i + h δu | u i + h δu | u i| (6.18) − λ (cid:0) k δu k + k δu k + k δu k (cid:1) + λ ( h δu | u i h δu | u i + h δu | u i h δu | u i + h δu | u i h δu | u i + cc)+ λ (cid:0) λ h δu | v i h δu | v i + λ h δu | v i h δu | v i + λ h δu | v i h δu | v i + cc (cid:1) + δα δ || u || + δα δ || u || + δα δ || u || , where cc means complex conjugate.Using the identity k δu k k ≡ |h δu k | u k i| + |h δu k | v k i| it can be rewritten as δ Λ = − λ (cid:0) |h δu | v i| + |h δu | v i| + |h δu | v i| (cid:1) (6.19)+ λ (cid:0) λ h δu | v i h δu | v i + λ h δu | v i h δu | v i + λ h δu | v i h δu | v i + cc (cid:1) + λ (cid:0) δ || u || δ || u || + δ || u || δ || u || ) + δ || u || δ || u || (cid:1) + δα δ || u || + δα δ || u || + δα δ || u || . From (6.3) it follows that terms containing δ || u k || vanish and the second variationtakes the form δ Λ = − λ (cid:0) |h δu | v i| + |h δu | v i| + |h δu | v i| (cid:1) (6.20)+ λ (cid:0) λ h δu | v i h δu | v i + λ h δu | v i h δu | v i + λ h δu | v i h δu | v i + cc (cid:1) . From h δu i | v i i h δu j | v j i ≤ | h δu i | v i i h δu j | v j i | it follows that δ Λ ≤ − λ X i,j =1 | h δu i | v i i || h δu j | v j i | A ij , (6.21)where the real and symmetric matrix A is given by A = λ − λ − λ − λ λ − λ − λ − λ λ . (6.22)Note that the inequality (6.21) can be saturated when vectors | δu k i are all mul-tiples of vectors v k and therefore (6.21) gives the least upper bound of δ Λ. HAPTER 6. GENERALIZED SCHMIDT DECOMPOSITION In this section we derive a non-strict inequality for the Schmidt coefficients.The condition δ Λ ≤ A is positivewhich means thattr( A ) ≥ , (tr( A )) − tr( A ) ≥ , det( A ) ≥ , (6.23)where tr and det mean the trace and the determinant of a matrix, respectively.The first condition tr( A ) = 3 λ > A )) − tr( A ) = 6 λ − λ + λ + λ ) > λ is the largest coefficient. But the third condition det( A ) ≥ λ ≥ λ + λ + λ + 2 λ λ λ λ . (6.24)This is a new and unexpected relation which says that nondiagonal coefficientsall together are bounded above by the quantity depending only on the largestcoefficient and therefore they should be small.Let us consider some particular cases. First consider the case when some ofnondiagonal coefficients, namely λ , vanishes. Then (6.24) reduces to the λ ≥ λ + λ , λ = 0 . (6.25)The solution (6.9) and the canonical form (6.15) present this case. This happenswhen a quantum state is a linear combination of three product states and itsamplitudes in a computational basis satisfy (6.25). Then the largest amplitude isthe largest Schmidt coefficient and GSD is achieved by a simple flipping of localstates. Similarly, the solution (6.10) with the form (6.16) and solution (6.11)with the form (6.17) are the cases λ = 0 and λ = 0, respectively.Conversely, when amplitudes of a three-term state in a computational basisdo not satisfy (6.25) it appears a special solution (6.12) which creates a newfactoizable basis. In this basis new amplitudes of the state given by (6.17) satisfy(6.24). Indeed, 4( abc ) ≥ ( ar a ) + ( br b ) + ( cr c ) + r a r b r c , (6.26)which can be checked using triangle inequalities. This means that if amplitudesof the state were not satisfying (6.24) in the initial basis from product states thenit appears a special solution giving rise to a new basis from product states andin this final basis amplitudes do satisfy (6.24).In conclusion, (6.24) clearly indicates whether a given canonical form is GSDor not and this is its main advantage. HAPTER 6. GENERALIZED SCHMIDT DECOMPOSITION λ ≥ λ, λ = λ = λ ≡ λ (6.27)and we are looking for the states with λ = 2 λ . The W state is a such state, thisis easy to see by setting a = b = c in (6.17). These substitutions yield λ ( W ) = 2 λ ( W ) = √ | λ ( W ) | , (6.28)which shows that (6.24) is indeed a non-strict inequality and gives the least upperbound for the nondiagonal coefficients. In this section we show that another inequality is needed to specify uniquelythe Schmidt coefficients of three-qubit states. To prove this statement let usassume the converse. Then (6.24) is a necessary and sufficient condition andGSD coefficients should satisfy only (6.24) and λ ≥ | λ | . Consider symmetricstates and put λ = λ = λ = λ which yields λ ≥ λ . Then it exists a statesuch that λ = | λ | = 2 λ and its GSD is given by | ψ contr i = 1 √
11 (2 | i + | i + | i + | i + 2 | i ) . (6.29)This is a wrong GSD. Indeed, λ ( wrong ) = 411 , but it is shown in Ref.[39] that absolute minimum of λ over three-qubit purestates is 4/9 and this minimum is reached at the W state. Hence no three qubitstate exists for which λ < /
9. For the sake of clarity we present the maximalproduct overlap and nearest product state for the state (6.29) λ ( right ) = 14 + 3 √ , | u u u i = (cos θ | i +sin θ | i ) ⊗ , tan θ = 1+ √ , (6.30)which can be derived by usual maximization tools.This example shows that conditions λ ≥ λ and λ ≥ | λ | are insufficient andanother relation should exist and this new relation should give bounds for the lastSchmidt coefficient. We know that when all nondiagonal coefficients vanish theupper bound is | λ (max) | = λ (known as GHZ state) and when all nondiagonal HAPTER 6. GENERALIZED SCHMIDT DECOMPOSITION | λ (max) | = λ / √ | λ | it exists an upper bound depending on the remainingcoefficients and this upper bound gives those particular bounds at GHZ and Wstates, respectively.We can derive this upper in some simple cases, for instance, when λ = λ = 0and the state is | ψ simple i = λ | i + λ | i + λ | i , (6.31)where λ is positive as its phase is meaningless in this case.The stationarity equations (6.5) for the state (6.31) have a relevant solutiongiven by | u i = λ | i + λ | i p λ + λ , | u i = | i , | u i = | i , λ ′ = q λ + λ . (6.32)From this solution it follows that (6.31) is a right decomposition if and only if λ ≥ λ ′ , that is λ ≥ λ + λ , λ = λ = 0 . (6.33)This inequality gives the least upper bound for the last Schmidt coefficient whentwo nondiagonal coefficients vanish. Unfortunately the tools used in this workwere unable to find the least upper bound of | λ | for generic states. The main result of this work is the inequality (6.24). Its role is to separate outthree-qubit Schmidt coefficients from the set of four positive and one complexnumbers. As is explained in above section, it is a necessary but not a sufficientcondition and another inequality should exist to complete the task.It is likely that the three nondiagonal elements together define bounds forthe last Schmidt coefficients in the missed inequality. Then the nondiagonalcoefficients are not just extra terms in GSD, but the ones which can show someimportant features of tripartite entanglement unknown so far.Another application of the derived non-strict inequality is that it can give us ahint how do we extend Nielsen’s protocol or operational entanglement measuresto three-qubit states. For instance, in bipartite case the protocol relies on in-equalities quadratic on Schmidt coefficients. In three-qubit case such a theoremshould include cubic relations as is evident from (6.24). hapter 7Summary
In this chapter we list the main results of the thesis. We have developed a method to derive algebraic equations for the geo-metric measure of entanglement of three-qubit pure states. Owing to it we havepresented the first calculation of the geometric measure of entanglement in awide range of three-qubit systems, including the general class of W states andstates which are symmetric under the permutation of two qubits. Additionally,we have shown that the nearest separable states are not necessarily unique, andhighly entangled states are surrounded by a one-parametric set of equally distantseparable states. We have derived an explicit expression for the geometric measure of en-tanglement for three-qubit states that are linear combinations of four orthogonalproduct states and thus have Schmidt rank 4. Any pure three-qubit state can bewritten in terms of five preassigned orthogonal product states via Schmidt de-composition. Thus the states discussed here are more general states compared tothe well-known Greenberger–Horne–Zeilinger and W states that have less rank.In fact, just a single step is needed to compute analytically the geometric measurefor five-parameter states and thereby to get the answer for arbitrary three-qubitstates. Using derived analytic expressions we have established that the geometricmeasure for three-qubit states has three different expressions depending on therange of definition in parameter space. Each expression of the measure has itsown geometrically meaningful interpretation. The states that lie on joint surfacesseparating different ranges of definition, designated as shared states, are quantumchannels for perfect teleportation and dense coding. Hence we have found acriterion which shows whether or not a given state can be be applied as a dualquantum channel. The Groverian measures are analytically computed in various types ofthree-qubit states and the final results are also expressed in terms of local-unitary95
HAPTER 7. SUMMARY We have developed a powerful method to compute analytically entangle-ment measures of multipartite systems. The method is based on duality whichconsists of two bijections. The first one creates a map between highly entangledn-qubit quantum states and n-dimensional unit vectors. The second one does thesame between n-dimensional unit vectors and n-part product states. In this waywe have obtained a double map or duality. The main advantage of the map isthat, if one knows any of the three vectors, then one instantly finds the othertwo. We have found the nearest product states for arbitrary generalized Wstates of n qubits, and shown that the nearest product state is essentially uniqueif the W state is highly entangled. It is specified by a unit vector in Eu-clidean n-dimensional space. We have used this duality between unit vectorsand highly entangled W states to find the geometric measure of entanglementof such states. The duality map reveals two critical values for quantum stateparameters. The first critical value separates symmetric and asymmetric entan-gled regions of highly entangled states, while the second one separates highly andslightly entangled states. We have shown that when N ≫ We have derived an interpolating formula for the geometric measure ofentanglement even if neither the number of particles nor the most of state pa-rameters are known. The importance of the interpolating formula in quantuminformation is threefold. First, it connects two quantities, namely the Bloch vec-tor and maximal product overlap, that can be easily estimated in experiments.Second, it is an example of how we compute entanglement of a quantum state withmany unknowns. Third, if the Bloch vector varies within the allowable domainthen maximal product overlap ranges from its lower to its upper bounds. Thenone can prepare the W state with the given maximal product overlap bringinginto the position the Bloch vector.
HAPTER 7. SUMMARY We have derived a non-strict inequality between three-qubit Schmidt coef-ficients, where the largest coefficient defines the least upper bound for the threenondiagonal coefficients or, equivalently, the three nondiagonal coefficients to-gether define the greatest lower bound for the largest coefficient. In addition, wehave shown the existence of another inequality which should establish an upperbound for the remaining Schmidt coefficient. The role of the inequalities is toseparate out three-qubit Schmidt coefficients from the set of four positive andone complex numbers. Another application of the derived non-strict inequality isthat it can give us a hint how do we extend entanglement transforming protocolsor operational entanglement measures to three-qubit states. cknowledgments
I want to thank my coauthors Prof. Anthony Sudbery, Prof. Dae-Kil Parkand Dr. Sayatnova Tamaryan for the collaboration.I am grateful to my supervisor Dr. Lekdar Gevorgian who helped and sup-ported my research.Also I would like to thank Mrs. Eleonora Tameyan who was always near tomy office and always ready to help. 98 ppendix AMatrix O One can easily show that the elements of O defined in Eq.(2.6) are given by O = 12 ( u u ∗ + u ∗ u + u u ∗ + u ∗ u ) (A.1) O = 12 ( u u ∗ + u ∗ u − u u ∗ − u ∗ u ) O = | u | − | u | O = i u u ∗ + u u ∗ − u ∗ u − u ∗ u ) O = i u u ∗ + u ∗ u − u ∗ u − u u ∗ ) O = u u ∗ + u ∗ u O = u u ∗ + u ∗ u O = − i ( u u ∗ + u ∗ u ) O = i ( u u ∗ + u ∗ u )where u ij is element of the unitary matrix defined in Eq.(2.6). It is easy to prove OO T = O T O = , which indicates that O αβ is an element of O(3).99 ppendix BBloch representation If the density matrix associated from the pure state | ψ i in Eq.(2.12) is representedby Bloch form like Eq.(2.11), the explicit expressions for ~v i are ~v = λ λ cos ϕ λ λ sin ϕλ − λ − λ − λ − λ ~v = λ λ cos ϕ + 2 λ λ − λ λ sin ϕλ + λ + λ − λ − λ (B.1) ~v = λ λ cos ϕ + 2 λ λ − λ λ sin ϕλ + λ − λ + λ − λ and the components of h ( i ) are h (1)11 = 2 λ λ + 2 λ λ cos ϕ, h (1)22 = 2 λ λ − λ λ cos ϕ (B.2) h (1)33 = λ + λ − λ − λ + λ , h (1)12 = h (1)21 = − λ λ sin ϕh (1)13 = − λ λ + 2 λ λ cos ϕ, h (1)31 = − λ λ + 2 λ λ cos ϕh (1)23 = − λ λ sin ϕ, h (1)32 = − λ λ sin ϕh (2)11 = − h (2)22 = 2 λ λ , h (2)33 = λ − λ + λ − λ + λ h (2)12 = h (2)21 = 0 , h (2)13 = 2 λ λ cos ϕh (2)31 = − λ λ − λ λ cos ϕ, h (2)23 = 2 λ λ sin ϕh (2)32 = 2 λ λ sin ϕ. PPENDIX B. BLOCH REPRESENTATION h (3) αβ is obtained from h (2) αβ by exchanging λ with λ . The non-vanishing components of g αβγ are g = − g = − g = − g = 2 λ λ (B.3) g = − g = 2 λ λ , g = − g = 2 λ λ g = 2 λ λ cos ϕ, g = 2 λ λ sin ϕg = g = 2 λ λ sin ϕ, g = − λ λ − λ λ cos ϕg = 2 λ λ − λ λ cos ϕ, g = − λ λ + 2 λ λ cos ϕg = 2 λ λ sin ϕ, g = 2 λ λ − λ λ cos ϕg = 2 λ λ sin ϕ, g = λ − λ + λ + λ − λ . ppendix CGeometrical interpretation of theduality The nearest product state | u i ⊗ | u i ⊗ · · · ⊗ | u N i of the W state (5.1) can beparameterized as follows | u k i = sin θ k | i + cos θ k | i , ≤ θ k ≤ π , k = 1 , , ..., N, (B.1)where cos θ + cos θ + · · · + cos θ N = 1 . (B.2)Thus the angles cos θ k define a unit N-dimensional vector in Euclidean space.They satisfy the equalities1 r ≡ sin 2 θ c = sin 2 θ c = · · · = sin 2 θ N c N . (B.3)These equalities can be interpreted as trigonometric relations for the right trian-gles with hypotenuses r , angles 2 θ k , opposite legs c k and adjacent legs p r − c k .If 2 θ k > π/
2, then one takes the angle π − θ k instead. All of these trian-gles has the same hypotenuse r and therefore can be circumscribed by a sin-gle sphere with the diameter r . The final picture represents two inscribed N-dimensional pyramids with a common base and lateral sides c , c , ..., c N and p r − c , p r − c , p r − c N , respectively. The case N = 3 is illustrated inFig.C.1. 102 PPENDIX C. GEOMETRICAL INTERPRETATION
X YZ Z ODOC X C Y C Figure C.1: (Color online) The geometrical interpretation of the duality for three-qubit W states. Mutually perpendicular bold lines OX , OY and OZ are coor-dinate axes and −−→ OO ′ is an arbitrary direction. OC X , OC Y and OC Z are mirrorimages of the line OO ′ in respect to the three axes. The points C X , C Y and C Z are intersections of these lines with the sphere uniquely defined by the twoconditions: its center lies on the line OO ′ and its diameter OD ≡ r is the sumof the lateral sides of the upper pyramid (with the apex O and base C X C Y C Z ).Now the direction cosines (and sines) of the vector −−→ OD are coefficients of the localstates | u i i in a computational basis. And the lateral sides of the lower pyramid(with the apex D and base C X C Y C Z ) are the coefficients of a 3-qubit W-state inthe same basis. Thus each direction singles out a product state and a W stateand thereby establishes a correspondence among them. ibliography [1] R. P. Feynman, Simulating Physics with Computers , Int. J. Theor. Physics , 467 (1982).[2] D. Deutsch, Quantum Theory, the Church-Turing Principle and the Uni-versal Quantum Computer , Proc. R. Society Lond. A: , 97 (1985).[3] P. Shor,
Algorithms for Quantum Computation: Discrete Logarithm andFactoring , Proc. 35th Annual Symposium on Foundations of ComputerScience, 124-134 (1994).[4] L. M.K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sher-wood, and I. L. Chuang,
Experimental realization of Shor’s quantum factor-ing algorithm using nuclear magnetic resonance , Nature , 883 (2001).[5] A. K. Ekert,
Quantum cryptography based on Bells theorem , Phys. Rev.Lett , 661 (1991).[6] C. H. Bennett, F. Bessette, G. Brassard, L. Salvail and J. Smolin, Experi-mental Quantum Cryptography , J. Cryptology , 3 (1992).[7] C. H. Bennett and S. J. Wiesner, Communication via one- and two-particleoperators on Einstein-Podolsky-Rosen states , Phys. Rev. Lett , 2881(1992).[8] K. Mattle, H. Weinfurter, P. G. Kwiat and A. Zeilinger, Dense Coding inExperimental Quantum Communication , Phys. Rev. Lett. , 4656 (1996).[9] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A. Peres, and W. K.Wootters, Teleporting an unknown quantum state via dual classical andEinstein-Podolsky-Rosen channels , Phys. Rev. Lett. , 1895 (1993).[10] D. Boschi, S. Branca, F. De Martini, L. Hardy and S. Popescu, Experimen-tal Realization of Teleporting an Unknown Pure Quantum State via DualClassical and Einstein-Podolsky-Rosen Channels , Phys. Rev. Lett. , 1121(1998). 104 IBLIOGRAPHY
Efficient Classical Simulation of Slightly Entangled QuantumComputations , Phys. Rev. Lett. On the role of entanglement in quantum compu-tational speed-up , Proc. R. Soc. Lond. A , 2011 (2003).[13] C. Negrevergne, T. S. Mahesh, C. A. Ryan, M. Ditty, F. Cyr-Racine, W.Power, N. Boulant, T. Havel, D. G. Cory, and R. Laflamme,
BenchmarkingQuantum Control Methods on a 12-Qubit System , Phys. Rev. Lett. ,170501 (2006).[14] A. Einstein, B. Podolsky, N. Rosen, Can quantum mechanical descriptionof physical reality be considered complete?,
Phys. Rev. , 777 (1935).[15] J. S. Bell, On the einstein-podolsky-rosen paradox , Physics (Long IslandCity, N.Y.) , 195 (1964).[16] R. F. Werner, Quantum states with Einstein-Podolsky-Rosen correlationsadmitting a hidden-variable model , Phys. Rev. A , 4277 (1989).[17] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantumentanglement , Rev. Mod. Phys. , 865 (2009).[18] M.B. Plenio and S. Virmani, An introduction to entanglement measures ,Quant. Inf. Comp. , 1 (2007).[19] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Concen-trating partial entanglement by local operations , Phys. Rev. A , 2046(1996).[20] W. K. Wootters, Entanglement of Formation of an Arbitrary State of TwoQubits , Phys. Rev. Lett. , 2245 (1998).[21] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wootters, Mixed-state entanglement and quantum error correction , Phys. Rev. A ,3824(1996).[22] A. Shimony, Degree of entanglement , in D. M. Greenberg and A. Zeilinger(eds.), Fundamental problems in quantum theory: A conference held inhonor of J. A. Wheeler, Ann. N. Y. Acad. Sci. (1995) 675.[23] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight,
QuantifyingEntanglement , Phys. Rev. Lett. , 2275 (1997).[24] H. Barnum and N. Linden, Monotones and Invariants for Multi-particleQuantum States , J. Phys. A: Math. Gen. , 6787 (2001). IBLIOGRAPHY
Geometric measure of entanglement andapplication to bipartite and multipartite quantum states , Phys. Rev.
A 68
Counterexample to an additivity conjecture foroutput purity of quantum channels , J. Math. Phys. , 4353 (2002).[27] L. K. Grover, Quantum Mechanics Helps in Searching for a Needle in aHaystack , Phys. Rev. Lett. , 325 (1997).[28] O. Biham, M. A. Nielsen and T. J. Osborne, Entanglement monotone de-rived from Grover’s algorithm , Phys. Rev.
A 65 , 062312 (2002).[29] J. J. Hilling and A. Sudbery,
The geometric measure of multipartite en-tanglement and the singular values of a hypermatrix , J. Math. Phys. ,072102 (2010).[30] S. Tamaryan, A. Sudbery and L. Tamaryan, Duality and the geometricmeasure of entanglement of general multiqubit W states , Phys. Rev. A ,052319 (2010).[31] T.-C. Wei, M. Ericsson, P.M. Goldbart, and W.J. Munro, Connectionsbetween relative entropy of entanglement and geometric measure of entan-glement , Quant. Inf. Comput. , 252 (2004).[32] D. Cavalcanti, Connecting the generalized robustness and the geometricmeasure of entanglement , Phys. Rev. A , 044302 (2006).[33] M. Hayashi, D. Markham, M. Murao, M. Owari, and S. Virmani, Bounds onMultipartite Entangled Orthogonal State Discrimination Using Local Oper-ations and Classical Communication , Phys. Rev. Lett. , 040501 (2006).[34] O. G¨uhne, M. Reimpell and R.F. Werner, Estimating entanglement mea-sures in experiments , Phys. Rev. Lett. , 110502 (2007).[35] R. Or´us, Universal Geometric Entanglement Close to Quantum PhaseTransitions , Phys. Rev. Lett. , 130502 (2008).[36] R. Or´us, S. Dusuel, and J. Vidal,
Equivalence of critical scaling laws formany-body entanglement in the Lipkin-Meshkov-Glick model , Phys. Rev.Lett. , 025701 (2008).[37] L. Tamaryan, D.K. Park and S. Tamaryan,
Analytic Expressions for Geo-metric Measure of Three Qubit States , Phys. Rev.
A 77 , 022325 (2008).[38] H. A. Carteret, A. Higuchi, and A. Sudbery,
Multipartite generalisation ofthe Schmidt decomposition , J. Math. Phys. , 7932 (2000). IBLIOGRAPHY
Maximally entangled three-qubitstates via geometric measure of entanglement , Phys. Rev A , 052315(2009).[40] E. Jung, M. R. Hwang, D. K. Park, L. Tamaryan and S. Tamaryan, Three-Qubit Groverian Measure , Quant. Inf. Comp. , 0925 (2008).[41] M. Hayashi, D. Markham, M. Murao, M. Owari, and S. Virmani, Thegeometric measure of entanglement for a symmetric pure state with positiveamplitudes , J. Math. Phys. , 122104 (2009).[42] L. Tamaryan, D.K. Park, J.-W. Son, S. Tamaryan, Geometric Measureof Entanglement and Shared Quantum States , Phys. Rev.
A 78 , 032304(2008).[43] T.-C. Wei and S. Severini,
Matrix permanent and quantum entanglementof permutation invariant states , J. Math. Phys. , 092203 (2010).[44] R. H¨ubener, M. Kleinmann, T.-C. Wei, and O. G¨uhne, The geometric mea-sure of entanglement for symmetric states , Phys. Rev. A , 032324 (2009).[45] E. Schmidt, Zur theorie der linearen und nichtlinearen integralgleighungen ,Math. Ann. , 433 (1907).[46] A. Ekert and P. L. Knight, Entangled quantum systems and the Schmidtdecomposition , Am. J. Phys. , 415 (1995).[47] A. Ac´ın, A. Andrianov, L. Costa, E. Jan´e, J. I. Latorre and R. Tarrach, Generalized Schmidt Decomposition and Classification of Three-Quantum-Bit States , Phys. Rev. Lett. (2000) 1560 [quant-ph/0003050].[48] D. Greenberger, M. Horne and A. Zeilinger, Going Beyond Bell’s Theorem ,Phys. Today, August 1993, 24. arXiv:0712.0921[quant-ph][49] W. D¨ur, G. Vidal and J. I. Cirac,
Three qubits can be entangled in twoinequivalent ways , Phys Rev. A , 062314 (2000).[50] E. Jung, M. R. Hwang, H. Kim, M. S. Kim, D. K. Park, J. W. Son andS. Tamaryan, Reduced state uniquely defines Groverian measure of originalpure state , Phys. Rev.
A 77 , 062317 (2008).[51] L. Tamaryan, Z. Ohanyan and S. Tamaryan,
Universal behavior of the ge-ometric entanglement measure of many-qubit W states , Phys. Rev. A ,022309 (2010).[52] P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena ,Rev. Mod. Phys. , 435-479 (1977). IBLIOGRAPHY
Nonstrict inequality for Schmidt coefficients of three-qubitstates , Phys. Rev. A , 042333 (2013).[54] A. Sudbery, On local invariance of pure three-qubit states , J. Phys.
A 34 (2001) 643 [quant-ph/0001116].[55] M. A. Nielsen,
Conditions for a Class of Entanglement Transformations ,Phys. Rev. Lett. , 436 (1999).[56] N. Linden, S. Popescu and A. Sudbery, Nonlocal Parameters for Multipar-ticle Density Matrices , Phys. Rev. Lett. , 243 (1999).[57] V. Coffman, J. Kundu and W. K. Wootters, Distributed entanglement ,Phys. Rev. A , 052306 (2000).[58] N. Linden, S. Popescu and W. K. Wootters, Almost Every Pure State ofThree Qubits Is Completely Determined by Its Two-Particle Reduced Den-sity Matrices , Phys. Rev.Lett. , 207901 (2002).[59] B. R¨othlisberger, J. Lehmann, D. S. Saraga, Ph. Traber and D. Loss, HighlyEntangled Ground States in Tripartite Qubit Systems , Phys. Rev. Lett. ,100502 (2008), arXiv:0705.1710v1 [quant-ph].[60] Y. Shimoni, D. Shapira, and O. Biham,
Entangled quantum states generatedby Shors factoring algorithm , Phys. Rev. A , 062308 (2005).[61] Y. Shimoni, D. Shapira, and O. Biham, Characterization of pure quantumstates of multiple qubits using the Groverian entanglement measure , Phys.Rev. A , 062303 (2004).[62] P. Agrawal and A. Pati, Perfect teleportation and superdense coding withW states , Phys. Rev. A , 062320 (2006).[63] G. Vidal, D. Jonathan, and M. A. Nielsen, Approximate transformationsand robust manipulation of bipartite pure-state entanglement , Phys. Rev. A , 012304 2000.[64] R. A. Bertlmann and Ph. Krammer, Geometric entanglement witnesses andbound entanglement , Phys. Rev. A , 024303 (2008).[65] G. Vidal, Entanglement monotones , J. Mod. Opt. (2000) 355[quant-ph/9807077].[66] M. B. Plenio and V. Vedral, Bounds on relative entropy of entanglementfor multi-party systems , J. Phys.
A 34 (2001) 6997 [quant-ph/0010080].[67] S. Hill and W. K. Wootters,
Entanglement of a Pair of Quantum Bits ,Phys. Rev. Lett. (1997) 5022 [quant-ph/9703041]. IBLIOGRAPHY
Formation of Multipartite Entangle-ment Using Random Quantum Gates , Phys. Rev.
A 76 (2007) 022328,arXiv:0708.3481[quant-ph].[69] A. O. Pittenger and M. H. Rubin,
The geometry of entanglement wit-nesses and local detection of entanglement , Phys. Rev. A , 012327 (2003)[quant-ph/0207024v1].[70] J. Eisert and H. J. Briegel, Schmidt measure as a tool for quantifying multi-particle entanglement , Phys. Rev.
A 64 (2001) 022306 [quant-ph/0007081].[71] A. B. Zamolodchikov, “Irreversility” of the flux of the renormalization groupin a 2D field theory , JETP Lett. (1986) 730.[72] S. Tamaryan, , Completely mixed state is a critical point for three-qubitentanglement , Physics Letters A , 22242229(2011).[73] Y. Makhlin, Nonlocal properties of two-qubit gates and mixed states and op-timization of quantum computations , Quant. Info. Proc. , 243-252 (2002),[quant-ph/0002045].[74] D. Shapira, Y. Shimoni and O. Biham, Groverian measure of entanglementfor mixed states , Phys. Rev.
A73 , 044301 (2006).[75] L. Tamaryan, H. Kim, E. Jung, M.-R. Hwang, D.K. Park, and S. Tamaryan,
Toward an understanding of entanglement for generalized n-qubit W-states ,J. Phys. A: Math. Theor. , 475303 (2009).[76] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, andW. K. Wootters, Purification of Noisy Entanglement and Faithful Telepor-tation via Noisy Channels , Phys. Rev. Lett. , 722 (1996).[77] V. Vedral and M. Plenio, Entanglement measures and purification proce-dures , Phys. Rev. A , 1619 (1998).[78] C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W.Shor, J. A. Smolin, and W. K. Wootters, Quantum nonlocality withoutentanglement , Phys. Rev. A , 1070 (1999).[79] K. Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, Volume ofthe set of separable states , Phys. Rev. A , 883 (1998).[80] A. Harrow and M. Nielsen, Robustness of quantum gates in the presence ofnoise , Phys. Rev. A , 012308 (2003).[81] B. M. Terhal and K. G. H. Vollbrecht, Entanglement of Formation forIsotropic States , Phys. Rev. Lett. , 2625 (2000). IBLIOGRAPHY
Nonlocal Parameters for Multipar-ticle Density Matrices , Phys. Rev. Lett. , 243 (1999).[83] F. Verstraete, J. Dehaene, B. De Moor, and H. Verschelde, Four qubits canbe entangled in nine different ways , Phys. Rev. A , 052112 (2002).[84] P. Parashar and S. Rana, N-qubit W states are determined by their bipartitemarginals , Phys. Rev. A , 012319 (2009).[85] R. G. Unanyan, M. Fleischhauer, N. V. Vitanov, and K. Bergmann, En-tanglement generation by adiabatic navigation in the space of symmetricmultiparticle states , Phys. Rev. A , 042101 (2002).[86] M. Hayashi, D. Markham, M. Murao, M. Owari, and S. Virmani, Entangle-ment of multiparty-stabilizer, symmetric, and antisymmetric states , Phys.Rev. A, , 012104 (2008).[87] J. Martin, O. Giraud, P. A. Braun, D. Braun, and T. Bastin, Multi-qubit symmetric states with high geometric entanglement , Phys. Rev. A , 062347 (2010).[88] H. Zhu, L. Chen, and M. Hayashi, Additivity and non-additivity of multi-partite entanglement measures , New J. Phys. bf 12, 083002 (2010).[89] G. Vidal and R. Tarrach,
Robustness of entanglement , Phys. Rev. A ,141(1999).[90] H. H. Adamyan and G. Yu. Kryuchkyan, Continuous-variable entanglementof phase-locked light beams , Phys. Rev. A , 053814 (2004).[91] R. G. Unanyan, C. Ionescu, M. Fleischhauer, Many-particle entanglement inthe gaped antiferromagnetic Lipkin model , Phys. Rev. A , 022326 (2005).[92] S. S. Sharma and N. K. Sharma, Quantum coherences, K -way negativitiesand multipartite entanglement , Phys. Rev. A , 042117 (2008).[93] R. H¨ubener, M. Kleinmann, T.-C. Wei, C. G.-Guillen, and O. G¨uhne, Ge-ometric measure of entanglement for symmetric states , Phys. Rev. A ,032324 (2009).[94] N. Linden, S. Popescu, and W. K. Wootters, Almost Every Pure Stateof Three Qubits Is Completely Determined by Its Two-Particle ReducedDensity Matrices , Phys. Rev. Lett. , 207901 (2002).[95] A. Sen(De), U. Sen, M. Wiesniak, D. Kaszlikowski, and M. Zukowski, Mul-tiqubit W states lead to stronger nonclassicality than Greenberger-Horne-Zeilinger states , Phys. Rev. A , 062306 (2003). IBLIOGRAPHY
Characterizing Multi-particle Entanglement in Symmetric N-Qubit States via Negativity of Co-variance Matrices , Phys. Rev. Lett. , 060501 (2007).[97] X. Wang, Entanglement in the quantum Heisenberg XY model , Phys. Rev.A , 012313 (2001).[98] H. Mikami, Y. Li, K. Fukuoka, and T. Kobayashi, New High-EfficiencySource of a Three-Photon W State and its Full Characterization UsingQuantum State Tomography , Phys. Rev. Lett. , 150404 (2005).[99] D. Gonta and S. Fritzsche, Multipartite W states for chains of atoms con-veyed through an optical cavity , Phys. Rev. A , 022326 (2010).[100] Y. Shimoni, D. Shapira, and O. Biham, Characterization of pure quantumstates of multiple qubits using the Groverian entanglement measure , Phys.Rev. A , 062303 (2004).[101] A. K. Pati, Minimum classical bit for remote preparation and measurementof a qubit , Phys. Rev. A , 014302 (2000).[102] T. Bastin, S. Krins, P. Mathonet, M. Godefroid, L. Lamata, and E.Solano, Operational Families of Entanglement Classes for Symmetric N-Qubit States , Phys. Rev. Lett. , 070503 (2009).[103] H. Kim, M.-R. Hwang, E. Jung, and D.K. Park,
Difficulties in analyticcomputation for relative entropy of entanglement , Phys. Rev. A81