Entanglement optimizing mixtures of two-qubit states
aa r X i v : . [ qu a n t - ph ] A ug Entanglement optimizing mixtures of two-qubit states
K. V. Shuddhodan, ∗ M. S. Ramkarthik, † and Arul Lakshminarayan ‡ Department of Physics, Indian Institute of Technology Madras, Chennai, 600036, India
Entanglement in incoherent mixtures of pure states of two qubits is considered viathe concurrence measure. A set of pure states is optimal if the concurrence for any mixture of them is the weighted sum of the concurrences of the generating states.When two or three pure real states are mixed it is shown that 28 .
5% and 5 .
12% ofthe cases respectively, are optimal. Conditions that are obeyed by the pure statesgenerating such optimally entangled mixtures are derived. For four or more purestates it is shown that there are no such sets of real states. The implications of theseon superposition of two or more dimerized states is discussed. A corollary of theseresults also show in how many cases rebit concurrence can be the same as that ofqubit concurrence. PACS numbers: 03.67.-a, 03.65.Bg, 03.67.Mn
I. INTRODUCTION
Entanglement properties of pure and mixed quantum states have been the subject of in-tense and extensive study in the recent past [1]. Of these, entanglement in qubits or spin-1/2systems have dominated due to their use as fundamental objects in quantum computations.For an arbitrary state of two qubits the concurrence measure (or its square, called tangle)introduced by Hill and Wootters [2] is simply calculable from the density matrix and is ameasure of entanglement. To be precise the entanglement of formation [3] is a monotonicfunction of the concurrence. The concurrence measure has been extensively applied in manyphysical contexts, for instance in the study of quantum phase transitions [4]. In a collectionof qubits, concurrence measures the entanglement present within any chosen pair. Thus dueto the monogamy property of entanglement [5] it is reasonable to expect that states withlarge multipartite entanglement have low or vanishing concurrence. In fact for a randomstate with more than six qubits the probability that a chosen pair has nonzero concurrenceis vanishingly small, most of the entanglement is of the multipartite kind [6, 7].To elaborate on this property, consider a mixed state of two qubits ρ = k X i =1 p i | ψ i ih ψ i | , k > , (1)where p i ≥ P i p i = 1 and the projectors are arbitrary, in particular, | ψ i ih ψ i | need not be ∗ rs˙[email protected] † [email protected] ‡ [email protected] orthogonal. The convexity of concurrence [8] implies that C ( ρ ) ≤ k X i =1 p i C ( ψ i ) (2)where C is the concurrence function, an entanglement monotone [2]. Thus the maximumthat C ( ρ ) can attain is the weighted sum of the concurrence of the extremal (pure) states. Ifthere exists a set {| ψ i , · · · , ψ k i} such that equality is obtained in Eq. (2) for any arbitraryset of weights { p i } , it is referred to herein as optimal. However note that such a propertywill be specific to the concurrence measure of entanglement.For states that are real in the standard basis, it is shown that a very large fractionof states made by incoherently superposing 2 two-qubit states optimize their entanglement.This property is analyzed in detail in this paper and conditions to be satisfied by the extremalstates such that the resultant density matrix is optimal are derived. Any real density matrixcan be tested for optimality of its diagonal decomposition using the inequalities derived.These are also generalized beyond two states, and it is shown that for more than three realstates, not one optimal decomposition exists.The relevance to superposed dimers will be studied in section III. The relation betweenentanglement and superpostion of quantum states is an interesting one [9]. Indeed super-position of states with a tensor product structure is necessary for entanglement, however ofcourse this is not sufficient. There is a significant amount of literature establishing boundson various entanglement measures for the superposition in terms of the entanglement in thestates that are being so superposed [9–15]. II. RANK-WISE STUDY OF OPTIMIZING MIXTURES
If two pure and real states | ψ i and | ψ i are chosen at random, it is shown in thispaper that in 28 .
5% of cases the resultant entanglement in p | ψ ih ψ | + (1 − p ) | ψ ih ψ | isthe maximum possible, namely equality holds in Eq. (2). This implies that on superposing2 two-qubit states, 28 .
5% of the states will remain entangled optimally, as defined in theintroduction. The fraction 0 .
285 of optimal pairs is interesting and strong evidence that itis actually ( π − / .
1% of cases this gives rise tooptimally entangled states. It is also then shown that for four or more states there is not even one set of pure real states, such that all their mixtures are optimal. These generalizationsare of relevance when more than 2 two-qubit states are superposed.The reader is first reminded of the procedure to find the concurrence in ρ , a given stateof two qubits [2]. The spin-flipped state ˜ ρ = σ y ⊗ σ y ρ ∗ σ y ⊗ σ y is found, where the complexconjugation is done in the standard basis. Then the matrix ρ ˜ ρ is diagonalized and haspositive eigenvalues µ ≥ µ ≥ µ ≥ µ . The concurrence C ( ρ ) is max(0 , √ µ − √ µ −√ µ − √ µ ).This somewhat involved definition of the concurrence renders it opaque for considerationsof optimality. However it is possible to express the concurrence of ρ = P ki =1 p i | ψ i ih ψ i | moreexplicitly in terms of the the states | ψ i i and the weights p i . Restrict to the case k ≤
4, thatis the size of the generating set of states is not larger than the maximum rank of ρ . Notethat if any ρ is expressed in its eigenbasis this is not a restriction at all. It is now shownthat the eigenvalues of ρ ˜ ρ are the same as that of r ′ r ′∗ where r ′ ij = √ p i p j r ij , and r ij = h ψ i | σ y ⊗ σ y | ψ ∗ j i . (3)Thus rather than using the density matrix directly, the pure states comprising a particularensemble are used. Note that | r | and | r | are the concurrences, C ( ψ ) and C ( ψ ), of thepure states | ψ i and | ψ i respectively.For convenience the eigenvalue equation for ˜ ρρ is considered, whose right eigenvectorscan be written in the nonorthogonal, sub-normalized basis of the extremal states as | ν i = P i α i | ψ ′ i i , where | ψ ′ i i = √ p i | ψ i i . Also, writing ρ = P ki =1 | ψ ′ i ih ψ ′ i | and using the fact that h ψ ′ m | ˜ ρρ | ν i = µ h ψ ′ m | ν i results in k X i =1 ( k X j =1 ˜ ρ ′ mj t ′ ji − µt ′ mi ) α i = 0 , ≤ m ≤ k, (4)where t ′ ji = √ p j p i t ji = √ p j p i h ψ j | ψ i i (5)is a matrix of inner products (the Gram matrix) and˜ ρ ′ mj = √ p m p j ˜ ρ mj = √ p m p j h ψ m | ˜ ρ | ψ j i . (6)Note also that since, ρ ∗ = P p i | ψ ∗ i ih ψ ∗ i | ,˜ ρ ′ mj = h ψ ′ m | ˜ ρ | ψ ′ j i = h ψ ′ m | σ y ⊗ σ y ( X p i | ψ ∗ i ih ψ ∗ i | ) σ y ⊗ σ y | ψ ′ j i (7)the following matrix identity is readily derived: ˜ ρ ′ = r ′ r ′∗ = r ′ r ′† , where r and r ′ are definedabove in Eq. (3).For the k equations in Eq.(4) to have non-trivial solutions det(Λ) = 0, whereΛ mi = ( ˜ ρ ′ t ′ ) mi − µ ( t ′ ) mi , (8)which further implies that det( t ′ ) det( r ′ r ′∗ − µI ) = 0 . (9)As the number of vectors | ψ i i are no larger in number than the dimensionality of the Hilbertspace, we assume them to be independent and therefore det( t ′ ) = p · · · p k det( t ) = 0. Thusdet( r ′ r ′∗ − µI ) = 0 . (10)and hence the characteristic polynomials of ρ ˜ ρ and r ′ r ′∗ are identical.If the state ρ is real in the computational basis then the eigenvalue problem of r ′ r ′∗ isthat of r ′ , whose eigenvalues are the square of the eigenvalues of r ′ , which are indicatedas λ . The expression for concurrence is derived and the conditions for optimality are nowconsidered case-by-case starting from k = 2. A. Rank-2 density matrices: k = 2
For the case of mixtures of two real pure states, k = 2, the above considerations lead tothe following characteristic equation of r ′ (as defined in Eq. (3)): λ − ξλ + p p χ = 0 (11)where ξ = p r + p r , and χ = ( r r − r ). The eigenvalues are therefore λ ± = 12 ( ξ ± p ξ − p p χ ) (12)If χ > ξ > λ + > λ − >
0, and C ( ρ ) = λ + − λ − . Alternatively if χ > ξ < λ − < λ + <
0, and C ( ρ ) = | λ − | − | λ + | . Thus if χ >
0, irrespective of thesign of ξ , we have that C ( ρ ) = p ξ − p p χ < | ξ | < p C ( ψ ) + p C ( ψ ) , (13)confirming the convexity of concurrence.The case χ < ξ > λ − < < λ + and λ + > | λ − | . It follows that C ( ρ ) = λ + − | λ − | = λ + + λ − = ξ . Combining a similaranalysis of the case ξ < χ <
0, irrespective of the sign of ξC ( ρ ) = | ξ | = | p r + p r | . (14)This brings us to the possibility that if r r > C ( ρ ) = p C ( ψ )+ p C ( ψ ) . Thus when | ψ i and | ψ i satisfy the conditions that r r > r r − r < any arbitrary mixtureof these pure states has the maximum entanglement which is their average entanglement.This is the first set of optimality conditions that we derive.The set of such optimal states is a subset from pairs of real states. Each real state of twoqubits is characterized by 4 real coefficients, say x i , i = 1 , . . . ,
4. The normalization conditionmeans that there is a isomorphism between these and the 3-sphere x + x + x + x = 1.Apart from the fact that states differing by a sign are really the same (thus the space is aprojective space) the states maybe thought of as points in S . Thus a pair of real states isa point on the manifold S × S , and the set of optimal states forms a subset therein whosefractional volume is of natural interest.Assume that the real states of two qubits are distributed uniformly on S , namely choosethe Haar measure. Equivalently, the probability density of random real pure states [16] oftwo qubits is given by P ( { x i } ) = 1 π δ X i =1 x i − ! , (15)where x i are the state components in a generic basis such as the computational one. Thefraction of optimal states f is then the following integral on S × S , written in terms ofthe ambient space components ( x i , y i ) of R × R : f = Z Θ( r r )Θ( r − r r ) P ( { x i } ) P ( { y i } ) Y i =1 dx i dy i . (16)Here r = h ψ | σ y ⊗ σ y | ψ i = 2( x x − x x ), r = h ψ | σ y ⊗ σ y | ψ i = 2( y y − y y ), r = h ψ | σ y ⊗ σ y | ψ i = x y + y x − x y − x y , and Θ is the Heaviside step function thatis 1 if its argument is positive and zero otherwise. An exact evaluation of this integral seemspossible and equal to ( π − / x i and y i distributed according to Eq. (15) andchecking to see if the optimality condition is satisfied. The initial choice of vectors is done bysimply taking 4 numbers from any zero centered normally distributed set and normalizingthem. This procedure gives the fraction f to be approximately 0 . π − / B. Rank-3 density matrices: k = 3
Now we consider the general setting of mixing three real pure states, k = 3, which leadsto the cubic equation for the eigenvalues of r ′ , (defined in Eq. (3)): f ( λ ) = λ + ξ λ + ξ λ + ξ = 0 , (17)where the coefficients ξ i are ξ = − X i p i r ii , ξ = X i = j p i p j ( r ii r jj − r ij ) , ξ = − p p p det( r ) . (18)Now we state two Lemmas which are key to understanding the nature of the roots of cubicequations and the possibility of optimal states in the case k = 3. Lemma 1. If p ( x ) is a cubic in x , p ( x ) = x + ax + bx + c with real coefficients, has realroots, and is such that a, b, c < then p ( x ) = 0 has two negative roots and one positive rootwith the positive root being greater than the other two in modulus.Proof. Since c < b < p ′ ( x ) has one positive and one negative root. Hence all the roots of p ( x ) cannot be positivesince the roots of p ′ ( x ) have to lie between the roots of p ( x ) by Cauchy’s mean value theoremfor differentiable functions. Hence the polynomial has two negative and one positive root.Observe that a < Lemma 2. If p ( x ) be a cubic in x , p ( x ) = x + ax + bx + c with real coefficients and hasreal roots, and is such that b < and a, c > then the cubic has two positive roots and onenegative root with the negative root being greater than the other two in modulus.Proof. The proof of this lemma is on similar lines to the previous one.
Case 1.
Now suppose we have ξ , ξ < ξ < λ > ρ ˜ ρ = ( σ y ⊗ σ y ρ ) are λ , − λ , − λ .Hence the concurrence of ρ is C ( ρ ) = max(0 , λ + λ + λ ) = max(0 , − ξ ) = | ξ | . Thus C ( ρ ) = p r + p r + p r . (19)If we have r ii > i = 1 , , C ( ψ i ) = r ii . Also if ξ < p i ’s can bearbitrary positive reals bounded by 1 we have each term p i p j ( r ii r jj − r ij ) < r ii r jj − r ij < , i = j is a necessary condition. Thus we have that if r ii > , r ii r jj − r ij < , i = j (20)and det( r ) = X cyc r ( r r − r ) − r r r − r r r ) > {| ψ i , | ψ i , | ψ i} will be optimally entangled, that is C ( ρ ) = X i =1 p i C ( ψ i ) (22) Case 2.
Similarly suppose we have ξ , ξ > ξ < λ < ρ ˜ ρ = ( σ y ⊗ σ y ρ ) are − λ , λ , λ . Hence theconcurrence of ρ C ( ρ ) = max(0 , − ( λ + λ + λ )) = max(0 , ξ ) = ξ , since ξ >
0. Thus wehave if r ii < , r ii r jj − r ij < , i = j (23)and det( r ) = X cyc r ( r r − r ) − r r r − r r r ) < {| ψ i , | ψ i , | ψ i} is optimally entangled. These condi-tions that are derived for optimality of entanglement for all mixtures are both necessary andsufficient. They can be compactly stated as, r ii det( r ) > , [ r ] ii < , i = 1 , , , (25)where [ r ] ii are the principal minors of r .Once again we can therefore write the fraction of triples of pure real states of two qubitswhose arbitrary mixtures are optimally entangled in terms of an integral such as in Eq. (16),which when evaluated using a procedure extending the previously described one, gives thefraction f = 0 . . p i muststill be optimal. Thus when we go from mixtures of rank-2 to rank-3 we see a drastic dropin the percentage of states that lead to optimal entanglement. C. Rank-4 real density matrices are never optimal: k ≥ Incoherently superposing four of more pure and real states leads to a qualitatively differentbehavior, as shown below. When k = 4 the rank of the eigenvalue problem for ρ ˜ ρ is full,in the sense that it is the dimensionality of the Hilbert space. From our discussion aboveit is clear that we have a quartic polynomial whose constant term is det( r ). For optimalitywe have to have the concurrence evaluating to the trace of r ′ , we need to have either oneeigenvalue positive and three negative eigenvalues, all of them smaller than the positive onein modulus, or one negative eigenvalue higher in modulus than three positive eigenvalues.In either case this implies that det( r ) <
0. However it is easy to prove that det( r ) > r ij = h ψ i | σ y ⊗ σ y | ψ j i = X mn F im h m | σ y ⊗ σ y | n i F Tnj . (26)Here F im = h ψ i | m i , and | m i is any real orthogonal basis, for instance it could be thecomputational one. Thereforedet( r ) = det[ σ y ⊗ σ y ] det( F ) = det( F ) > , (27)the final inequality following from the reality of the transformation functions. Thereforeunlike the rank-deficient cases the sign of det( r ) is always positive and this rules out theexistence of even one real quadruplet such that any arbitrary mixture of these remainsoptimally entangled. This obviously implies the non-existence of even one set of real optimalstate for k >
4. It is necessary to have complex states in the ensemble for optimizing theentanglement in this case.Entanglement in real qubits have been studied earlier by also restricting the Hilbertspace to the space of reals, the so-called case of “rebits” [17]. In this case minimizationof the entanglement is also carried out only over the real ensembles that are realizationsof the density matrix, unlike in the current paper, where we have used the usual formulafor concurrence. The rebit formula for concurrence is | tr( σ y ⊗ σ y ρ ) | which in terms of thequantities introduced in this paper is | P i p i r ii | . In the case when k = 2 this is also theactual concurrence if we only require that χ = r r − r <
0, relaxing the conditionthat r r >
0. This is true in about 78 .
5% ( π/ k = 2. Thusstated in terms of rebits, the present work also implies that for 78 .
5% of pairs of real purestates that are mixed, the rebit entanglement coincides with the usual qubit entanglement.Similar generalizations to the case of triples of states gives about 48 . r ii have the same sign. However it follows from the aboveconsiderations that when we take a mixture of four (or more) real states, the resultant rebitentanglement is always suboptimal to the usual concurrence obtained on the full complexHilbert space. III. DIMERIZED STATES AND OPTIMALITY
As an application of the study of optimizing mixtures of two qubits, the problem ofentanglement sharing in superpositions of states with a dimerized structure is now takenup. If there are many pure states of 2 N qubits such that qubits 1 and 2 are entangled onlywith each other, 3 and 4 with each other and so on, and each of these pairs are in purestates, then an implication of the previous section is that superposing such “dimerized”states results in rather robust entanglement, especially if only two or three such states aresuperposed. On adding more such states, the entanglement in the pairs comes down due tothe lack of optimality, and will lead to more global, or multipartite entanglement. Considerthe, in general unnormalized, state: | ψ i = k X i =1 a i ⊗ Nj =1 | φ ij i ; where | φ ij i = α ij | i j + β ij | i j + γ ij | i j + δ ij | i j , (28)and P i | a i | = 1. Thus the state is a superposition of k states, labelled by i , each ofwhich has N pairs of entangled two qubit (normalized) pure states, labelled by j . Notwo pairs are entangled with each other. Such superpositions arise in many context, forexample in the Resonating Valence Bond states [18]. However note here that the “dimers”superposed are of the same kind, that is, the entangled pairs of particles are the same. MostHamiltonian systems have some form of time-reversal (anti-unitary) symmetry that renderstheir eigenstates real.Henceforth | φ i i , the state of the first entangled pair in the i th state is referred to as | ψ i i .For simplicity consider the case when k = 2. Let a = cos( θ ) and a = sin( θ ). As thesuperposed states are not orthogonal, there is the normalization factor N , for the state | ψ i in Eq. (28) which is N = 1 / p θµ , where µ = N Y j =1 ( α j α j + β j β j + γ j γ j + δ j δ j ) . (29)The reduced density matrix of any two qubits that are entangled in the original states | ψ ij i , which without loss of generality can be taken as the first two qubits, is ρ = N (cid:0) cos θ | ψ ih ψ | + sin θ | ψ ih ψ | + µ sin θ cos θ ( | ψ ih ψ | + | ψ ih ψ | ) (cid:1) . (30)Here µ is defined as µ = N Y j =2 ( α j α j + β j β j + γ j γ j + δ j δ j ) . (31)In all generality this is all that can be said about ρ , however for most states the interfer-ence or coherence term is negligible, due to the typical smallness of µ and µ . Thus theapproximation ρ ≈ ρ , where ρ = cos θ | ψ ih ψ | + sin θ | ψ ih ψ | , (32)is a good one.To estimate the typical value of µ and µ consider the Hilbert space of each entangledpair consisting of two qubits and take for the distribution of the coefficients the uniformor Haar measure of Eq. (15). The averages of µ and µ are both zero in this ensemble.However the second moments are nonzero and can be shown to be h µ i = 4 − N , h µ i = 4 − ( N − . This follows on observing that µ and µ in Eqs. (29, 31) are N and N − x i and y i are distributed according to the measure in Eq. (15): * X i =1 x i y i ! + = X i =1 (cid:10) x i y i (cid:11) = 1 / . (33)The first equality follows as the odd powers average to zero, and the second equality followsas the ensemble average of each of the x i is 1 /
4, which follows most easily from normalization. C on c u rr en c e p N = 4coherentincoherent 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0 0.2 0.4 0.6 0.8 1 C on c u rr en c e p N = 4coherentincoherent 0.05 0.1 0.15 0.2 0.25 0.3 0 0.2 0.4 0.6 0.8 1 C on c u rr en c e pN = 8 (OPTIMAL CASE)coherentincoherent 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.2 0.4 0.6 0.8 1 C on c u rr en c e p N = 8coherentincoherent FIG. 1: Concurrence of the first two qubits vs p = cos ( θ ) for the superposition of twostates ( k = 2) with N = 4 and N = 8 (see Eq. (28)) for two different realizations (left andright columns). The incoherent case is obtained on neglecting the interference terms in thesuperposition as in Eq. (32). The bottom left panel shows a case of optimality.Thus µ and µ are typically of the order of 2 − N and 2 − ( N − respectively. In practicefor most states with N > ρ obtained on dropping the interference term as in Eq. (32) is used or the actualreduced density matrix ρ (Eq. (30)) is used. This is illustrated in Fig. (1), where randomrealizations of the two-qubit states | φ ij i are used in Eq. (28) with k = 2. Two such realizationsare selected for the cases of N = 4 and N = 8 and the concurrence in the reduced densitymatrix of the first two qubits are calculated based on the exact state ρ (Eq. (30), referredto in the figure as “coherent”) and the approximation ρ (Eq. (32), referred to in the figureas “incoherent”). The concurrences are plotted as a function of the mixing between thestates that are superposed in Eq. (28). It is seen from the figure that when N = 8 whateverdifference persist between the entanglement in ρ and ρ is not visible, while it is for N = 4.Also from the N = 8 case, the realization on the right, illustrates the convexity of concurrenceas at p = 0 , p , thedensity matrix is an incoherent superposition of these (see Eq. (32)). However for N = 8the realization on the left is peculiar in that the concurrence for the intermediate values of p is just a linear interpolation of the concurrences of the pure states. In this case one hashit upon what is studied in this paper as an optimal pair of pure states.The approximate form of the reduced density matrix in Eq.(32) obtained by using theincoherent superposition condition holds also for complex states, however optimality resultsarise in the case of real state as studied in detail in the previous section. For k > p i are equal to | a i | . In termsof dimerized states, we see that superposing two of them leads to pairs of qubits that wereoriginally entangled with each other retaining much of it. About 28 .
5% of pairs of qubits0would have simply the weighted entanglements of the states before superposition. On theother hand superposing three dimerized states leads to a significant decrease in the fractionof robust dimers, which is now 5 . IV. DISCUSSIONS AND SUMMARY
This work has used a particular measure of entanglement of two-qubit density matrices,namely concurrence, and studied questions of optimality, as defined herein, mainly restrictingto the space of real states. Concurrence is only one measure of entanglement, but theentanglement of formation being a monotonic function of it renders it rather unique and assuch this measure has been used in very many studies. What exactly the issue of optimalitysays about the geometry of the quantum space of states [19] is an interesting question thatis not pursued here. It may also be interesting to study other measures of non-classicalcorrelation, such as discord [20] from this perspective.One possible application of optimality was discussed in the superposition of dimerizedstates. Such states are found in many quantum spin systems, such as the Majumdar Ghoshhamiltonian [21], and superpositions are relevant in the neighborhood of avoided crossingswhere it is known that interesting transformations of entanglement occur [22]. Resultsnot shown here indicate circumstances under which such intra-dimer entanglement can bebroken if a dimerized state is superposed with a non-dimerized, completely random state. Inparticular this results in dimer density matrices being very close to Werner states [23] andtherefore results in the entanglement of dimers vanishing when the random state componentis more than 2 /
3. In general the effect of superposition on entanglement has been studiedvigorously [9], as well as entanglement in the RVB states has been explored [24]. The resultsdiscussed in this paper may add in some small measure to the understanding of entanglementin such contexts.Real states are often obtained as eigenstates of time-reversal symmetric systems and thediscussion here will be of relevance to such systems, for instance to spin chains that havetime-reversal. Clearly when the states are complex the above approach to finding optimalityconditions does not work. In the case when k = 2 the optimality condition is found to bethe same as that for real states, namely 0 < r r /r < r is not restrictedto the reals). Thus it is conceivable that as this is the unit interval in the complex plane,the measure of optimal states is zero. However of course, there could be an infinity of these,for instance all real states are possible candidates. Acknowledgments
We thank Karol Zyczkowski, Steven Tomsovic, Suresh Govindarajan and V. Balakrishnanfor discussions. This work was in part supported by the DST project SR/S2/HEP-012/2009.1
Appendix: Evaluation of an integral for the optimal fraction f To recall, the integral is f = Z Θ( r r )Θ( r − r r ) P ( { x i } ) P ( { y i } ) Y i =1 dx i dy i , (A.1)where r ij = h ψ i | σ y ⊗ σ y | ψ j i , | ψ i i , = 1 , P are the uniform measures in Eq. (15). First, the particular Pauli matrix σ y that appears can be replaced by the other Pauli matrices. In particular the σ z matrix beingdiagonal, offers a simpler look. This replacement is quite easily seen to be equivalent tosome 45 ◦ rotations of the original variables.Also dropping the constraint on the product r r will be useful. If the resultant integralis denoted as f , then it is shown below that f is simply f − /
2. Next, it is proven that asfar as f is concerned, the two 4-vectors can be taken to be orthogonal. Decompose say | ψ i along the vector | ψ i and one orthogonal to it: | ψ i = cos( θ ) | ψ i + sin( θ ) | η i , (A.2)where h η | ψ i = 0. No additional phases are involved as the states are all real. A straight-forward calculation shows that r − r r = sin ( θ ) (cid:0) h ψ | σ z ⊗ σ z | η i − h η | σ z ⊗ σ z | η ih ψ | σ z ⊗ σ z | ψ i (cid:1) . (A.3)The quantity within the brackets is precisely the same combination as in the L.H.S., exceptthat instead of | ψ i i , the vectors are orthogonal. As sin ( θ ) has a constant positive sign, thisproves that we can consider the pairs, to begin with, as being orthogonal. In other words thesign of the combination r − r r is invariant under the Gram-Schmidt orthogonalizationprocess.The additional constraint of the vectors being orthonormal introduces an additional Diracdelta function term in the measure. Writing the fraction f as a ratio f n /f d , the numerator f n and the denominator f d are given by f n = 8 π Z Θ (cid:0) r − r r (cid:1) δ X i x i y i ! δ X i x i − ! δ X i y i − ! dx dy, (A.4)and f d = 8 π Z δ X i x i y i ! δ X i x i − ! δ X i y i − ! dx dy. (A.5)All the sums are from 1 to 4 and dx dy is the Euclidean eight dimensional volume ele-ment. The factor 8 /π is introduced for later convenience alone. To be further explicit thecombination r − r r =( x y − x y − x y + x y ) − ( x − x − x + x )( y − y − y + y ) . (A.6)Introducing a series of transformation to various two-dimensional polar coordinates (in the( x , x ) pair, the ( x , x ) pair, etc., as well as in the resulting radii) and performing two2delta function integrals corresponding to the normalizations results is: f n = 2 Z Θ (cid:2) (cos( α ) cos( β ) cos( θ ) − sin( α ) sin( β ) cos( φ )) − cos(2 α ) cos(2 β ) (cid:3) δ [cos( α ) cos( β ) cos( θ ) + sin( α ) sin( β ) cos( φ )] sin(2 α ) sin(2 β ) dαdβdθdφ, (A.7)and a corresponding integral for f d , only without the Heaviside theta function constraint.Here α, β ∈ [0 , π/
2] while θ, φ ∈ [0 , π ]. As a check, the integral can be easily done withouteither the Heaviside theta or the Dirac delta functions to give 8 π , which is exactly thefactor that follows from the normalization of the two Dirac delta normalization constraints;see Eq. (15), if one takes into account the factor 8 /π that is introduced in Eq. (A.4).Introducing variables v and u as cos( α ) cos( β ) cos( θ ) ± sin( α ) sin( β ) cos( φ ) respectively,allows the delta function integration over the v variable to be performed and results in: f n = Z Θ ( u − cos(2 α ) cos(2 β )) du dα dβ s − u ( α ) cos ( β ) s − u ( α ) sin ( β ) . (A.8)Notice that the given range of α and β , [0 , π/ can be divided into four equal squares,such that the Θ function constraint is effective only in [0 , π/ and in [ π/ , π/ , ascos(2 α ) cos(2 β ) is negative elsewhere. The range of the u integration is restricted dependingon α, β . Taking the range [0 , π/ f lln , ll for lower-lower,we have: f lln = Z π/ dα Z π/ β dβ Z u u du s − u c α c β s − u s α s β , (A.9)where β = sin − p / − s α , u = √ c α c β , and u = 2 s α s β , and s α stands for sin( α ) etc.. The limits of the integration are such that the Θ functionconstraint is satisfied as well as the square-roots are real numbers. The denominator fractioncan be similarly split up, and in fact f lld differs from the above in that β = 0 as well as u = 0. It is also not hard to see that f uun ( uu for upper-upper) is same as f lln and similarly for f d . In the other two regions, as the constraint is not operational, and because of symmetry,it follows that: f uld = f lud = f uln = f lun ≡ f ul . It then follows that f = 2 f ul + 2 f lln f ul + 2 f lld . (A.10)An evaluation of the three corresponding integrals is carried out numerically and results in f lln = 0 . ... , f ul = 1 / f lld = 0 . ... . It is remarked that standard softwares couldnot evaluate the integrals symbolically, however the numerical results are sufficient to givethe fraction f = 0 . ... , which is π/ . It is also easy to similarly seethat f lld = ( π − / f lln = ( π − / f we see that there are two Θconstraints, while we have considered only one, namely the second one. If the first constraintΘ( r r ) alone is present, it is easy to see that the integral is 1 /
2. This also follows from3the fact that r and r are both independent and uniformly distributed. We also stateparenthetically without proof that r is distributed according to the semi-circle distribution.Now, if r r is negative, then surely r − r r is positive, which is 50% of the time. Thusthe fraction of cases when r r is positive and r − r r is positive is f − /
2, whichis precisely the required fraction f . Thus we have evaluated f and presented sufficientevidence that it is actually ( π − /
4. It is not clear if it is only a coincidence that this isalso precisely f lld . The evaluation presented here may, by far, not be the “optimal” one, butis the best the authors could come up with. [1] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).[2] S. Hill and W. K. Wootters, Phys. Rev. Lett. , 5022 (1997); W. K. Wootters, Phys. Rev.Lett. , 2245 (1998).[3] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cam-bridge Univ. Press, 2000 Cambridge).[4] A. Osterloh, Luigi Amico, G. Falci, and Rosario Fazio, Nature , 608 (2002).[5] D. Bruß, Phys. Rev. A , 4344 (1999).[6] V. M. Kendon, Kae Nemoto, and W. J. Munro, J. Mod. Opt. , 1709 (2002).[7] A. J. Scott and C. M. Caves, J. Phys. A: Math. Gen. , 9553 (2003).[8] A. Uhlmann, Phys. Rev. A , 032307 (2000).[9] Noah Linden, Sandu Popescu and John A. Smolin, Phys. Rev. Lett. , 100502 (2006).[10] J. Nisert, N. J. Cerf, Phys. Rev. A , 042328 (2007).[11] Chang-shui Yu, X. X. Yi, and He-shan Song, Phys. Rev. A , 022332 (2007).[12] Yong-Cheng Ou, Heng Fan, Phys. Rev. A , 022320 (2007).[13] Gilad Gour, Aidan Roy, Phys. Rev. A , 012336 (2008).[14] Wei Song, Nai-Le Liu and Zeng-Bing Chen, Phys. Rev. A , 054303 (2007).[15] Andreas Osterloh, Jens Siewert and Armin Uhlmann, Phys. Rev. A , 032310 (2008).[16] T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey and S. S. Wong, Rev. Mod. Phys. , 385 (1981).[17] C. M. Caves, C. A. Fuchs and P. Rungta, Found. Phys. Lett. , 199 (2001).[18] P. W. Anderson, G. Baskaran, Z. Zou, and T. Hsu, Phys. Rev. Lett. , 2790 (1987).[19] I. Bengtsson and K. Zyczkowski, Geometry of Quantum States: An Introduction to QuantumEntanglement (Cambridge University, Cambridge, England, 2007).[20] H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001).[21] C. K. Majumdar and D. P. Ghosh, J. Math. Phys. 10, 1388 (1969); J. Math. Phys. 10, 1399(1969).[22] J. Karthik, A. Sharma, and A. Lakshminarayan, Phys. Rev. A 75, 022304 (2007).[23] R. F. Werner, Phys. Rev. A , 4277 (1989).[24] Anushya Chandran, Dagomir Kaszlikowski, Aditi Sen(De), Ujjwal Sen, and Vlatko Vedral,Phys. Rev. Lett. , 170502 (2007); Ravishankar Ramanathan, Dagomir Kaszlikowski, MarcinWiesniak, and Vlatko Vedral, Phys. Rev. B78