aa r X i v : . [ qu a n t - ph ] F e b Book TitleBook EditorsIOS Press, 2003 Dynamic Symmetry Approachto Entanglement
Alexander Klyachko , Bilkent University, Turkey
Abstract.
In this lectures I explain a connection between geometric invariant theoryand entanglement, and give a number of examples how this approach works.
Keywords.
Entanglement, Dynamic symmetry, Geometric invariant theory
1. Physical background
Let me start with classical nonlinear equation d θdt = − ω sin θ, ω = gℓ (1)describing graceful swing of a clock pendulum in a corner of Victorian drawing room. Ithas double periodic solution θ ( t ) = θ ( t + T ) = θ ( t + iτ ) , with real period T , and imaginary one iτ . Out of this equation, carefully studied byLegendre, Abel, and Jacobi, stems the whole theory of elliptic functions.Physicists are less interested in mathematical subtleties, and usually shrink equation(1) to linear one d θdt = − ω θ, | θ | ≪ with simple harmonic solution θ = e ± iωt . This example outlines a general feature ofclassical mechanics, where linearity appears mainly as a useful approximation. Correspondence to: A. Klyachko, Bilkent University, 06800, Bilkent, Ankara, Turkey. Tel.: +90 312 2902115; Fax: +90 312 266 4579; E-mail: [email protected]
A. Klyachko / Entanglement
In striking contrast to this, quantum mechanics is intrinsically linear, and therefore moresimple then classical one, in the same way as analytic geometry of Descartes is simplerthen synthetic geometry of Euclid. As a price for its simplicity quantum mechanics runsinto enormous difficulties to manifest itself in a harsh macroscopic reality. This is whatmakes quantum phenomenology so tricky.Mathematicians encounter a similar problem when try to extract geometrical gistfrom a mess of coordinate calculations. In both cases the challenge is to cover formalbonds of mathematical skeleton with flesh of meaning.As we know from Klein’s
Erlangen program , the geometrical meaning rests uponinvariant quantities and properties (w.r. to a relevant structure group G ). This thesis ef-fectively reduces “elementary” geometry to invariant theory.As far as physics is concerned, we witnessed its progressive geometrization in thelast decades [65,25] . To name few examples: general relativity, gauge theories, fromelectro-weak interactions to chromodynamics, are all geometrical in their ideal essence.In this lectures, mostly based on preprint [32], I explain a connection between geometricinvariant theory and entanglement, and give a number of examples how this approachworks. One can find further applications in [33,34]. A background of a quantum system A is Hilbert space H A , called state space . Here, bydefault, the systems are expected to be finite : dim H A < ∞ . A pure state of the systemis given by unit vector ψ ∈ H A , or by projector operator | ψ ih ψ | , if the phase factoris irrelevant. Classical mixture ρ = P i p i | ψ i ih ψ i | of pure states called mixed state or density matrix . This is a nonnegative Hermitian operator ρ : H A → H A with unit trace Tr ρ = 1 . An observable of the system A is Hermitian operator X A : H A → H A . Actual mea-surement of X A upon the system in state ρ produces a random quantity x A ∈ Spec X A implicitly determined by expectations h f ( x A ) i ρ = Tr ( ρf ( X A )) = h ψ | f ( X A ) | ψ i for arbitrary function f ( x ) on Spec X A (the second equation holds for pure state ψ ).The measurement process puts the system into an eigenstate ψ λ with the observed eigen-value λ ∈ Spec X A . Occasionally we use ambiguous notation | λ i for the eigenstate witheigenvalue λ . The linearity of quantum mechanics is embedded from the outset in
Schrödinger equa-tion describing time evolution of the system i ~ dψdt = H A ψ (2)where H A : H A → H A is the Hamiltonian of the system A . Being linear Schrödingerequation admits simple solution . Klyachko / Entanglement ψ ( t ) = U ( t ) ψ (0) , (3)where U ( t ) = exp (cid:16) − i ~ R t H A ( t ) dt (cid:17) is unitary evolution operator .Solutions of Schrödinger equation (2) form a linear space. This observation is asource of general superposition principle, which claims that a normalized linear combi-nation aψ + bϕ of realizable physical states ψ, ϕ is again a realizable physical state (with no recipe howto cook it). This may be the most important revelation about physical reality after atomichypothesis. It is extremely counterintuitive and implies, for example, that one can set thecelebrated Shcrödinger cat into the state ψ = | dead i + | alive i intermediate between death and life. As BBC put it: “In quantum mechanics it is not soeasy to be or not to be.” From the superposition principle it follows that state space of composite system AB splits into tensor product H AB = H A ⊗ H B of state spaces of the components, as opposed to direct product P AB = P A × P B ofconfiguration spaces in classical mechanics. The linearity imposes severe restrictions on possible manipulations with quantum states.Here is a couple of examples.
Let’s start with notorious claim
Theorem ([67], [12]) . An unknown quantum state can’t be duplicated.
Indeed the cloning process would be given by operator ψ ⊗ ( state of the Cloning Machine ) ψ ⊗ ψ ⊗ ( another state of the Machine ) which is quadratic in state vector ψ of the quantum system. As another application of linearity consider the following
Theorem.
No information on quantum system can be gained without destruction of itsstate.
A. Klyachko / Entanglement
Indeed the measurement process is described by linear operator U : ψ ini ⊗ Ψ ini ψ fin ⊗ Ψ fin , where ψ and Ψ are states of the system and the measurement device respectively. Theinitial state Ψ ini of the apparatus supposed to be fixed once and for all, so that the finalstate ψ fin ⊗ Ψ fin is a linear function of ψ ini . This is possible only if • ψ fin is linear in ψ ini and Ψ fin is independent of ψ ini , • or vice versa Ψ fin is linear in ψ ini and ψ fin is independent of ψ ini .In the former case the final state of the measurement device contains no information onthe system, while in the latter the unknown initial state ψ ini is completely erased in themeasurement process.Emmanuel Kant, who persistently defended absolute reality of unobservable “thing-in-itself”, or noumenon , as opposed to phenomenon , should be very pleased with thistheorem identifying noumenon with quantum state.The theorem suggests that complete separation of a system from a measuring ap-paratus is unlikely. As a rule the system remains entangled, with the measuring device,with two exceptions described above. Density matrix of composite system AB can be written as a linear combination of sepa-rable states ρ AB = X α a α ρ αA ⊗ ρ αB , (4)where ρ αA , ρ αB are mixed states of the components A, B respectively, and the coefficients a α are not necessarily positive. Its reduced matrices or marginal states may be definedby equations ρ A = P α a α Tr ( ρ αB ) ρ αA := Tr B ( ρ AB ) ,ρ B = P α a α Tr ( ρ αA ) ρ αB := Tr A ( ρ AB ) . The reduced states ρ A , ρ B are independent of the decomposition (4) and can be charac-terized intrinsically by the following property h X A i ρ AB = Tr ( ρ AB X A ) = Tr ( ρ A X A ) = h X A i ρ A , ∀ X A : H A → H A , (5)which tells that ρ A is a “visible” state of subsystem A . This justifies the terminology. Example 1.6.1.
Let’s identify pure state of two component system ψ = X ij ψ ij α i ⊗ β j ∈ H A ⊗ H B with its matrix [ ψ ij ] in orthonormal bases α i , β j of H A , H B . Then the reduced states of ψ in respective bases are given by matrices . Klyachko / Entanglement ρ A = ψ † ψ, ρ B = ψψ † , (6)which have the same non negative spectra Spec ρ A = Spec ρ B = λ (7)except extra zeros if dim H A = dim H B . The isospectrality implies so called Schmidtdecomposition ψ = X i p λ i ψ Ai ⊗ ψ Bi , (8)where ψ Ai , ψ Bi are eigenvectors of ρ A , ρ B with the same eigenvalue λ i .In striking contrast to the classical case marginals of a pure state ψ = ψ A ⊗ ψ B aremixed ones, i.e. as Srödinger put it “maximal knowledge of the whole does not necessar-ily includes the maximal knowledge of its parts” [58]. He coined the term entanglement just to describe this phenomenon. Von Neumann entropy of the marginal states providesa natural measure of entanglement E ( ψ ) = − Tr ( ρ A log ρ A ) = − Tr ( ρ B log ρ B ) = − X i λ i log λ i . (9)In equidimensional system dim H A = dim H B = n maximum of entanglement,equal to log n entangled bits (ebits), is attained for a state with scalar reduced matrices ρ A , ρ B . In the above discussion we tacitly suppose, following von Neumann, that all observable X A : H A → H A or what is the same all unitary transformations e itX A : H A → H A areequally accessible for manipulation with quantum states. However physical nature of thesystem may impose unavoidable constraints. Example 1.7.1.
The components of composite system H AB = H A ⊗ H B may be spa-tially separated by tens of kilometers, as in EPR pairs used in quantum cryptography. Insuch circumstances only local observations X A and X B are available. This may be evenmore compelling if the components are spacelike separated at the moment of measure-ment. Example 1.7.2.
Consider a system of N identical particles, each with space of internaldegrees of freedom H . By Pauli principle state space of such system shrinks to symmet-ric tensors S N H ⊂ H ⊗ N for bosons, and to skew symmetric tensors ∧ N H ⊂ H ⊗ N forfermions. This superselection rule imposes severe restricion on manipulation with quan-tum states, effectively reducing the accessible measurements to that of a single particle. Example 1.7.3.
State space H s of spin s system has dimension s + 1 . Measurementsupon such system are usually confined to spin projection onto a chosen direction. Theygenerate Lie algebra su (2) rather then full algebra of traceless operators su (2 s + 1) .This consideration led many researchers to the conclusion, that available observablesshould be included in description of any quantum system from the outset [24,16]. RobertHermann stated this thesis as follows: A. Klyachko / Entanglement “The basic principles of quantum mechanics seem to require the postulation of aLie algebra of observables and a representation of this algebra by skew-Hermitianoperators.”
We’ll refer to the Lie algebra L as algebra of obsevables and to the correspondinggroup G = exp( i L ) as dynamical symmetry group of the quantum system in question.Its state space H together with unitary representation of the dynamical group G : H issaid to be quantum dynamical system . In contrast to R. Hermann we treat L as algebraof Hermitian, rather then by skew-Hermitian operators, and include imaginary unit i inthe definition of Lie bracket [ X, Y ] = i ( XY − Y X ) .The choice of the algebra L depends on the measurements we are able to performover the system, or what is the same the Hamiltonians which are accessible for manipu-lation with quantum states.For example, if we are restricted to local measurements of a system consisting oftwo remote components
A, B with full access to the local degrees of freedom then thedynamical group is
SU( H A ) × SU( H B ) acting in H AB = H A ⊗ H B .In settings of Example 1.7.2 suppose that a single particle is described by dynamicalsystem G : H . Then ensemble of N identical particles corresponds to dynamical system G : S N H for bosons, and to G : ∧ N H for fermions.The dynamic group of spin system from Example 1.7.3 is SU(2) in its spin s repre-sentation H s .
2. Coherent states
Coherent states, first introduced by Schrödinger [57] in 1926, lapsed into obscurity fordecades until Glauber [22] recovered them in 1963 in connection with laser emission.He have to wait more then 40 years to win Nobel Prize in 2005 for three paper publishedin 1963-64.Later in 70th Perelomov [47,48] puts coherent states into general framework of dy-namic symmetry groups. We’ll use a similar approach for entanglement, and to warm uprecall here some basic facts about coherent states.
Let’s start with quantum oscillator , described by canonical pair of operators p , q , [ p, q ] = i ~ , generating Weyl-Heisenberg algebra W . This algebra has unique unitary irreduciblerepresentation, which can be realized in Fock space F spanned by orthonormal set of n -excitations states | n i on which dimensionless annihilation and creation operators a = q + ip √ ~ , a † = q − ip √ ~ , [ a, a † ] = 1 act by formulae a | n i = √ n | n − i , a † | n i = √ n + 1 | n + 1 i . . Klyachko / Entanglement A typical element from
Weyl-Heisenberg group W = exp W , up to a phase factor,is of the form D ( α ) = exp( αa † − α ∗ a ) for some α ∈ C . Action of this operator onvacuum | i produces state | α i := D ( α ) | i = exp (cid:18) − | α | (cid:19) X n ≥ α n √ n ! | n i , (10)known as Glauber coherent state . The number of excitations in this state has Poissondistribution with parameter | α | . In many respects its behavior is close to classical, e.g.Heisenberg’s uncertainty ∆ p ∆ q = ~ / for this state is minimal possible. In coordinaterepresentation q = x, p = i ~ ddx its time evolution is given by harmonic oscillation of Gaussian distribution of width √ ~ with amplitude | α |√ ~ . Therefor for big number of photons | α | ≫ coherent statesbehave classically. Recall also Glauber’s theorem [23] which claims that classical fieldor force excites quantum oscillator into a coherent state.We’ll return to these aspects of coherent states later, and focus now on their mathe-matical description Glauber coherent states = W -orbit of vacuum which sounds more suggestive then explicit equation (10). Let’s now turn to arbitrary quantum system A with dynamical symmetry group G =exp i L . By definition its Lie algebra L = Lie G is generated by all essential observables of the system (like p, q in the above example). To simplify the underling mathematicssuppose in addition that state space of the system H A is finite, and representation of G in H A is irreducible.To extend (10) to this general setting we have to understand the special role of thevacuum, which primary considered as a ground state of a system. For group-theoreticalapproach, however, another its property is more relevant: Vacuum is a state with maximal symmetry.
This may be also spelled out that vacuum is a most degenerate state of a system.
Symmetries of state ψ are given by its stabilizers G ψ = { g ∈ G | gψ = µ ( g ) ψ } , L ψ = { X ∈ L | Xψ = λ ( X ) ψ } (11)in the dynamical group G or in its Lie algebra L = Lie G . Here µ ( g ) and λ ( X ) arescalars. Looking back to the quantum oscillator, we see that some symmetries are ac- A. Klyachko / Entanglement tually hidden, and manifest themselves only in complexified algebra L c = L ⊗ C andgroup G c = exp L c . For example, stabilizer of vacuum | i in Weyl-Heisenberg algebra W is trivial W | i = scalars, while in complexified algebra W c it contains annihilationoperator, W c | i = C + C a . In the last case the stabilizer is big enough to recover thewhole dynamical algebra W c = W c | i + W c | i† . This decomposition, called complex polarization , gives a precise meaning for the maxi-mal degeneracy of a vacuum or a coherent state.
State ψ ∈ H is said to be coherent if L c = L cψ + L cψ † In finite dimensional case all such decompositions come from
Borel subalgebra , i.e. amaximal solvable subalgebra B ⊂ L c . The corresponding Borel subgroup B = exp B is a minimal subgroup of G c with compact factor G c /B . Typical example is subgroupof upper triangular matrices in SL ( n, C ) = complexification of SU ( n ) . It is a basicstructural fact that B + B † = L c , and therefore ψ is coherent ⇔ ψ is an eigenvector of B In representation theory eigenstate ψ of B is called highest vector , and the correspondingeigenvalue λ = λ ( X ) , Xψ = λ ( X ) ψ, X ∈ B is said to be highest weight .Here are the basic properties of coherent states. • For irreducible system G : H the highest vector ψ (=vacuum) is unique. • There is only one irreducible representation H = H λ with highest weight λ . • All coherent states are of the form ψ = gψ , g ∈ G . • Coherent state ψ in composite system H AB = H A ⊗ H B with dynamical group G AB = G A × G B splits into product ψ = ψ ⊗ ψ of coherent states of thecomponents. Remark.
Coherent state theory, in the form given by Perelomov [48], is a physical equiv-alent of Kirillov–Kostant orbit method [31] in representation theory.The complexified group play crucial role in our study. Its operational interpretationmay vary. Here is a couple of examples. . Klyachko / Entanglement Example 2.4.1.
Spin systems.
For system of spin s (see example 1.7.3) coherent stateshave definite spin projection s onto some direction ψ is coherent ⇐⇒ ψ = | s i . Complexification of spin group
SU(2) is group of unimodular matrices
SL(2 , C ) . Thelatter is locally isomorphic to Lorentz group and controls relativistic transformation ofspin states in a moving frame.
Example 2.4.2.
For two component system H AB = H A ⊗ H B with full access to localdegrees of freedom the coherent states are decomposable ones ψ AB is coherent ⇐⇒ ψ AB = ψ A ⊗ ψ B . The dynamical group of this system is G = SU( H A ) × SU( H B ) , see example 1.7.1.Its complexification G c = SL( H A ) × SL( H B ) has an important quantum informationalinterpretation as group of invertible Stochastic Local Operations assisted with ClassicalCommunication (SLOCC transformations), see [61]. These are essentialy LOCC opera-tions with postselection. Let’s define total variance of state ψ by equation D ( ψ ) = X i h ψ | X i | ψ i − h ψ | X i | ψ i (12)where X i ∈ L form an orthonormal basis of the Lie algebra of essential observableswith respect to its invariant metric (for spin group SU(2) one can take for the basis spinprojector operators J x , J y , J z ). The total variance is independent of the basis X i , hence G -invariant. It measures the total level of quantum fluctuations of the system in state ψ .The first sum in (12) contains well known Casimir operator C = X i X i which commutes with G and hence acts as a scalar in every irreducible representation.Specifically Theorem 2.5.1.
The Casimir operator C acts in irreducible representation H λ of highestweight λ as multiplication by scalar C λ = h λ, λ + 2 δ i . One can use two dual bases X i and X j of L , with respect to invariant bilinear form B ( X i , X j ) = δ ij to construct the Casimir operator C = X i X i X i . For example, take basis of L consisting of orthonormal basis H i of Cartan subalgebra h ⊂ L and its rootvectors X α ∈ L normalized by condition B ( X α , X − α ) = 1 . Then the dual basis is obtained by substitution X α X − α and hence0 A. Klyachko / Entanglement C = X i H i + X α =root X α X − α = X i H i + X α> H α + 2 X α> X − α X α , where in the last equation we use commutation relation [ X α , X − α ] = H α . Applying this to the highest vector ψ ∈ H of weight λ , which by definition is annihilated by all operators X α , α > and Hψ = λ ( H ) ψ , H ∈ h , we get Cψ = X i λ ( H i ) ψ + X α> λ ( H α ) ψ = h λ, λ + 2 δ i ψ, (13)where δ = P α> α is the sum of positive roots and h∗ , ∗i is the invariant form B translated to the dualspace h ∗ . Hence Casimir operator C acts as scalar C λ = h λ, λ +2 δ i in irreducible representation with highestweight λ . For spin s representation H s of SU(2) the Casimir is equal to square of the total moment C = J = J x + J y + J z = s ( s + 1) . Hence D ( ψ ) = h λ, λ + 2 δ i − X i h ψ | X i | ψ i . (14) Theorem 2.6.1 (Delbourgo and Fox [11]) . State ψ is coherent iff its total variance isminimal possible, and in this case D ( ψ ) = h λ, δ i . Let ρ = | ψ ih ψ | be pure state and ρ L be its orthogonal projection into subalgebra L ⊂ Herm( H ) ofalgebra of all Hermitian operators in H with trace metric ( X, Y ) = Tr( X · Y ) . By definition we have h ψ | X | ψ i = Tr H ( ρX ) = Tr H ( ρ L X ) , ∀ X ∈ L . Choose a Cartan subalgebra h ⊂ L containing ρ L . Then h ψ | X i | ψ i = Tr H ( ρ L X i ) = 0 for X i ⊥ h and wecan restrict the sum in (14) to orthonormal basis H i of Cartan subalgebra h ⊂ L for which by the definition ofhighest weight h ψ | H | ψ i ≤ λ ( H ) with equality for the highest vector ψ only. Hence X i h ψ | X i | ψ i = X i h ψ | H i | ψ i ≤ X i λ ( H i ) = h λ, λ i , (15)and therefore D ( ψ ) ≥ h λ, λ + 2 δ i − h λ, λ i = h λ, δ i , with equality for coherent states only. The theorem supports the thesis that coherent states are closest to classical ones, cf. n ◦ minimal uncertainty ∆ p ∆ q = ~ / instead. Example 2.6.1.
For coherent state of spin s system Theorem 2.6.1 gives D ( ψ ) = s .Hence amplitude of quantum fluctuations √ s for such state is of smaller order then spin s , which by Example 2.4.1 has a definite direction. Therefor for s → ∞ such state lookslike a classical rigid body rotating around the spin axis. . Klyachko / Entanglement There is another useful description of coherent states by a system of quadratic equations.
Example 2.7.1.
Consider two component system H AB = H A ⊗ H B with full accessto local degrees of freedom G = SU ( H A ) ⊗ SU ( H B ) . Coherent states in this case arejust separable states ψ = ψ A ⊗ ψ B with density matrix ρ = | ψ ih ψ | of rank one. Suchmatrices can be characterized by vanishing of all minors of order two. Hence coherentstates of two component system can be described by a system of quadratic equations.It turns out that a similar description holds for arbitrary irreducible system G : H λ with highest weight λ , see [37]. Theorem 2.7.1.
State ψ ∈ H λ is coherent iff ψ ⊗ ψ is eigenvector of the Casimir operator C with eigenvalue h λ + 2 δ, λ i C ( ψ ⊗ ψ ) = h λ + 2 δ, λ i ( ψ ⊗ ψ ) . (16) Indeed, if ψ is highest vector of weight λ then ψ ⊗ ψ is a highest vector of weight λ and equation (16)follows from (13).Vice versa, in terms of orthonormal basis X i of Lie algebra L = Lie G the Casimir operator in thedoublet H λ ⊗ H λ looks as follows C = X i ( X i ⊗ ⊗ X i ) = X i X i ⊗ ⊗ X i + 2 X i X i ⊗ X i . Hence under conditions of the theorem h λ + 2 δ, λ i = h ψ ⊗ ψ | C | ψ ⊗ ψ i = 2 h λ + 2 δ, λ i + 2 X i h ψ | X i | ψ i . It follows that X i h ψ | X i | ψ i = h λ, λ i and hence by inequality (15) state ψ is coherent. The above calculation show that equation (16) is equivalent to X i X i ψ ⊗ X i ψ = h λ, λ i ψ ⊗ ψ, (17)which in turn amounts to a system of quadratic equations on the components of a coher-ent state ψ . Example 2.7.2.
For spin s system the theorem tells that state ψ is coherent iff ψ ⊗ ψ hasdefinite spin s . Equations (17) amounts to J x ψ ⊗ J x ψ + J y ψ ⊗ J y ψ + J z ψ ⊗ J z ψ = s ψ ⊗ ψ.
3. Entanglement
From a thought experiment for testing the very basic principles of quantum mechan-ics in its earlier years [15,58] entanglement nowadays is growing into an important tool A. Klyachko / Entanglement for quantum information processing. Surprisingly enough so far there is no agreementamong the experts on the very definition and the origin of entanglement, except unani-mous conviction in its fundamental nature and in necessity of its better understanding.Here we discuss a novel approach to entanglement [32], based on dynamical sym-metry group, which puts it into a broader context, eventually applicable to all quantumsystems. This sheds new light on known results providing for them a unified concep-tual framework, opens a new prospect for further development of the subject, reveals itsdeep and unexpected connections with other branches of physics and mathematics, andprovides an insight on conditions in which entangled state can be stable.
Everybody knows, and nobody understand what is entanglement. Here are some virtualanswers to the question borrowed from Dagmar Bruß collection [6]: • J. Bell: . . . a correlation that is stronger then any classical correlation. • D. Mermin: . . . a correlation that contradicts the theory of elements of reality. • A. Peres: . . . a trick that quantum magicians use to produce phenomena that can-not be imitated by classical magicians. • C. Bennet: . . . a resource that enables quantum teleportation. • P. Shor: . . . a global structure of wave function that allows the faster algorithms. • A. Ekert: . . . a tool for secure communication. • Horodecki family: . . . the need for first application of positive maps in physics.
This list should be enhanced with extensively cited Schrödinger’s definition given in n ◦ gedanken experiment [15]. While the latter authors were amazed by nonlocal nature of correla-tions between the involved particles, J. Bell was the first to note that the correlationsthemselves, puting aside the nonlocality, are inconsistent with classical realism. Sincethen Bell’s inequalities are produced in industrial quantities and remain the main tool fortesting “genuine” entanglement. Note however that in some cases LOCC operations cantransform a classical state into nonclassical one [54]. Besides in a sense every quantumsystem of dimension at least three is nonclassical, see n ◦ Decay of a spin zero state into two components of spin 1/2 subjects to a strong corre-lation between spin projections of the components, caused by conservation of moment.The correlation creates an apparent information channel between the components, actingbeyond their light cones.Let me emphasize that quantum mechanics refuted the possibility that the spin pro-jection have been fixed at the moment of decay, rather then at the moment of measure-ment. Otherwise two spatially separated observers can see the same event like burst of asupernova simultaneously even if they are spacelike separated, see [50]. There is no such“event” or “physical reality” in the Bohm version of EPR experiment. . Klyachko / Entanglement This paradox, recognized in early years of quantum mechanics [15,3], nowadays hasmany applications, but no intuitive explanation. It is so disturbing that sometimes physi-cists just ignore it. For example, one of the finest recent book justifies QFT commutationrelations as follows [69]:
A basic relativistic principle states that if two spacetime points are spacelike withrespect to each other then no signal can propagate between them, and hence themeasurement of an observable at one of the points cannot influence the measurementof another observable at the other point.
Experiments with EPR pairs tell just the opposite [1,19]. I am not in position to commentthis nonlocality phenomenon, and therefor turn to less involved
Bell’s approach , limitedto the quantum correlations per se . Let’s start with classical marginal problem which asks for existence of a “body” in R n with given projections onto some coordinate subspaces R I ⊂ R n , I ⊂ { , , . . . , n } , i.e.existence of probability density p ( x ) = p ( x , x , . . . , x n ) with given marginal distribu-tions p I ( x I ) = Z R J p ( x ) dx J , J = { , , . . . , n }\ I. In discrete version the classical MP amounts to calculation of an image of a multidi-mensional symplex, say ∆ = { p ijk ≥ | P p ijk = 1 } , under a linear map like π : R ℓmn → R ℓm ⊕ R mn ⊕ R nℓ ,p ijk ( p ij , p jk , p ki ) ,p ij = X k p ijk , p jk = X i p ijk , p ki = X j p ijk . The image π (∆) is convex hull of π ( Vertices ∆) . So the classical MP amounts to calcu-lation of facets of a convex hull. In high dimensions this may be a computational night-mare [17,52].
Example 3.3.1.
Classical realism.
Let X i : H A → H A be observables of quantum sys-tem A . Actual measurement of X i produces random quantity x i with values in Spec ( X i ) and density p i ( x i ) implicitly determined by expectations h f ( x i ) i = h ψ | f ( X i ) | ψ i for all functions f on spectrum Spec ( X i ) . For commuting observables X i , i ∈ I therandom variables x i , i ∈ I have joint distribution p I ( x I ) defined by similar equation h f ( x I ) i = h ψ | f ( X I ) | ψ i , ∀ f. (18) A. Klyachko / Entanglement
Classical realism postulates existence of a hidden joint distribution of all variables x i .This amounts to compatibility of the marginal distributions (18) for commuting sets ofobservables X I . Bell inequalities , designed to test classical realism, stem from the clas-sical marginal problem.
Example 3.3.2.
Observations of disjoint components of two qubit system H A ⊗ H B always commute. Let A i , B j be spin projection operators in sites A, B onto directions i, j . Their observed values a i , b j = ± satisfy inequality a b + a b + a b − a b + 2 ≥ . Indeed product of the monomials ± a i b j in LHS is equal to − . Hence one of the mono-mials is equal to +1 and sum of the rest is ≥ − .If all the observables have a hidden joint distribution then taking the expectations wearrive at Clauser-Horne-Shimony-Holt inequality for testing “classical realism” h ψ | A B | ψ i + h ψ | A B | ψ i + h ψ | A B | ψ i − h ψ | A B | ψ i + 2 ≥ . (19)All other marginal constraints can be obtained from it by spin flips A i
7→ ± A i . Example 3.3.3.
For three qubits with two measurements per site the marginal constraintsamounts to 53856 independent inequalities, see [53].Bell’s inequalities make it impossible to model quantum mechanics by classicalmeans. In particular, there is no way to reduce quantum computation to classical one.
Here I’ll give an account of nonclassical states in spin 1 system. Its optical version, called biphoton , is a hot topic both for theoretical and experimental studies [59,28,64]. The so-called neutrally polarised state of biphoton routinely treated as entangled, since a beamsplitter can transform it into a EPR pair of photons. This is the simplest one componentsystem which manifests entanglement.Spin 1 state space may be identified with complexification of Euclidean space E H = E ⊗ C , where spin group SU (2) , locally isomorphic to
SO (3) , acts via rotations of E . Hilbertspace H inherits from E bilinear scalar and cross products, to be denoted by ( x, y ) and [ x, y ] respectively. Its Hermitian metric is given by h x | y i = ( x ∗ , y ) where star meanscomplex conjugation. In this model spin projection operator onto direction ℓ ∈ E isgiven by equation J ℓ ψ = i [ ℓ, ψ ] . It has real eigenstate | i = ℓ and two complex conjugate ones | ± i = √ ( m ± in ) ,where ( ℓ, m, n ) is orthonormal basis of E . The latter states are coherent , see Example2.4.1. They may be identified with isotropic vectors . Klyachko / Entanglement ψ is coherent ⇐⇒ ( ψ, ψ ) = 0 . Their properties are drastically different from real vectors ℓ ∈ E called completelyentangled spin states. They may be characterized mathematically as follows ψ is completely entangled ⇐⇒ [ ψ, ψ ∗ ] = 0 . Recall from Example 2.4.1 that Lorentz group, being complexification of
SO (3) , pre-serves the bilinear form ( x, y ) . Therefore it transforms a coherent state into another co-herent state. This however fails for completely entangled states.Every noncoherent state can be transformed into completely entangled one by aLorentz boost. In this respect Lorentz group plays rôle similar to SLOCC transform fortwo qubits which allows to filter out a nonseparable state into a completely entangledBell state, cf. Example 2.4.2.By a rotation every spin 1 state can be put into the canonical form ψ = m cos ϕ + in sin ϕ, ≤ ϕ ≤ π . (20)The angle ϕ , or generalized concurrence µ ( ψ ) = cos 2 ϕ , is unique intrinsic parameterof spin 1 state. The extreme values ϕ = 0 , π/ correspond to completely entangled andcoherent states respectively.Observe that J ℓ ψ = − [ ℓ, [ ℓ, ψ ]] = ψ − ( ℓ, ψ ) ℓ so that S ℓ = 2 J ℓ − ψ ψ − ℓ, ψ ) ℓ is reflection in plane orthogonal to ℓ . Hence S ℓ = 1 and operators S ℓ and S m commuteiff ℓ ⊥ m .Consider now a cyclic quintuplet of unit vectors ℓ i ∈ E , i mod 5 , such that ℓ i ⊥ ℓ i +1 , and call it pentagram . Put S i := S ℓ i . Then [ S i , S i +1 ] = 0 and for all possiblevalues s i = ± of observable S i the following inequality holds s s + s s + s s + s s + s s + 3 ≥ . (21)Indeed product of the monomials s i s i +1 is equal to +1 , hence at least one of them is +1 ,and the sum of the rest is ≥ − .Being commutative, observables S i , S i +1 have a joint distribution. If all S i wouldhave a hidden joint distribution then taking average of (21) one get Bell’s type inequality h ψ | S S | ψ i + h ψ | S S | ψ i + h ψ | S S | ψ i + h ψ | S S | ψ i + h ψ | S S | ψ i +3 ≥ (22)for testing classical realism. Note that all marginal constraints can be obtained from thisinequality by flips S i
7→ ± S i . Using equation S i = 1 − | ℓ i ih ℓ i | one can recast it intogeometrical form X i mod 5 |h ℓ i , ψ i| ≤ ⇐⇒ X i mod 5 cos α i ≤ , α i = c ℓ i ψ. (23) A. Klyachko / Entanglement
Completely entangled spin states easily violate this inequality. Say for regular pentagramand ψ ∈ E directed along its axis of symmetry one gets X i mod 5 cos α i = 5 cos π/
51 + cos π/ ≈ . > . We’ll see below that in a smaller extend every non-coherent spin state violates inequality(23) for an appropriate pentagram. The coherent states, on the contrary, pass this test forany pentagram.To prove these claims write inequality (23) in the form h ψ | A | ψ i ≤ , A = X i mod 5 | ℓ i ih ℓ i | , and observe the following properties of spectrum λ ≥ λ ≥ λ ≥ of operator A .1. Tr A = λ + λ + λ = 5 .
2. If the pentagram contains parallel vectors ℓ i k ℓ j then λ = λ = 2 , λ = 1 .3. For any pentagram with no parallel vectors(a) λ > ,(b) λ > ,(c) λ < . Proof. (1) Tr A = P i mod 5 Tr | ℓ i ih ℓ i | = 5 .(2) Let say ℓ = ± ℓ then ℓ , ℓ , ℓ form orthonormal basis of E . Hence A is sum ofidentical operator | ℓ ih ℓ | + | ℓ ih ℓ | + | ℓ ih ℓ | and projector | ℓ ih ℓ | + | ℓ ih ℓ | ontoplane < ℓ , ℓ > .(3a) Take unit vector x ∈ < ℓ , ℓ > ∩ < ℓ , ℓ > so that x = h ℓ , x i ℓ + h ℓ , x i ℓ = h ℓ , x i ℓ + h ℓ , x i ℓ . Then Ax = h ℓ , x i ℓ + h ℓ , x i ℓ + h ℓ , x i ℓ + h ℓ , x i ℓ + h ℓ , x i ℓ = 2 x + h ℓ , x i ℓ and λ ≥ h x | A | x i = 2 + |h x | ℓ i| > .(3b) This property is more subtle. It amounts to positivity of the form B ( x, y ) = h x | A − | y i = X i mod 5 h x | ℓ i ih ℓ i | y i − h x | y i . One can show that det B = 2 det A Y i Bell’s inequality h ψ | A | ψ i ≤ holds for coherent state ψ and any pen-tagram, while non-coherent state violates this inequality for some pentagram.Proof. Take ψ = m cos ϕ + in sin ϕ , ≤ ϕ ≤ π/ in canonical form (20). Then h ψ | A | ψ i = h m | A | m i cos ϕ + h n | A | n i sin ϕ. To violate Bell’s inequality we have to make the right hand side maximal. This happensfor m = | λ i , the eigenvector of A with maximal eigenvalue λ , and n = | λ i . Themaximal value thus obtained is h ψ | A | ψ i max = λ cos ϕ + λ sin ϕ = λ + λ λ − λ ϕ. (24)For coherent state ϕ = π/ we arrive at Bell’s inequality h ψ | A | ψ i max = λ + λ − λ ≤ which holds for all pentagrams by property (3b). The other part of the theorem followsfrom the following Claim. For every noncoherent state ≤ ϕ < π/ there exists pentagram s.t. h ψ | A | ψ i max = λ + λ λ − λ ϕ > . Indeed, for degenerate pentagram Π , containing parallel vectors, the correspondingoperator A has multiple eigenvalue λ = λ = 2 and simple one λ = 1 . In this caseequation (24) amounts to h ψ | A | ψ i max = 2 . Let e A be operator corresponding to a smallnondegenerate ε -perturbation e Π of pentagram Π , and e λ be its spectrum. Then for simpleeigenvalue λ we have by property (3b) ∆( λ ) = e λ − λ = O ( ε ) > , and hence ∆( λ + λ ) = ∆(5 − λ ) = O ( ε ) < . Hereafter O ( ε ) denote a quantity of exact order ε . The increment of multiple roots λ , λ is of smaller order ∆( λ ) = O ( √ ε ) > , ∆( λ ) = O ( √ ε ) < , ∆( λ − λ ) = O ( √ ε ) > , where the signs of the increments are derived from properties (3a) and (3c). As result ∆( h ψ | A | ψ i max ) = ∆ (cid:18) λ + λ λ − λ ϕ (cid:19) = O ( ε ) + O ( √ ε ) = O ( √ ε ) > , A. Klyachko / Entanglement provided cos 2 ϕ > and ε ≪ . Hence for noncoherent state Bell’s inequality fails: h ψ | e A | ψ i max > . Product of orthogonal reflections S i S i +1 in pentagram inequality (22) isa rotation by angle π in plane < ℓ i , ℓ i +1 > , i.e. S i S i +1 = 1 − J ℓ i ,ℓ i +1 ] , and the inequality can be written in the form h ψ | J ℓ ,ℓ ] | ψ i + h ψ | J ℓ ,ℓ ] | ψ i + h ψ | J ℓ ,ℓ ] | ψ i + h ψ | J ℓ ,ℓ ] | ψ i + h ψ | J ℓ ,ℓ ] | ψ i ≤ . Observe that ℓ i , ℓ i +1 , [ ℓ , ℓ i +1 ] are orthogonal and therefor J ℓ i + J ℓ i +1 + J ℓ i ,ℓ i +1 ] = 2 . This allows return to operators J i = J ℓ i h ψ | J | ψ i + h ψ | J | ψ i + h ψ | J | ψ i + h ψ | J | ψ i + h ψ | J | ψ i ≥ . The last inequality can be tested experimentally by measuring J and calculating the av-erage of J . Thus we managed to test classical realism in framework of spin 1 dynamicalsystem in which no two operators J ∈ su (2) commutes, cf. Example 3.3.1. The trick isthat squares of the operators may commute. The difference between coherent and entangled spin states disappearsfor the full group SU ( H ) . Hence with respect to this group all states are nonclassical,provided dim H ≥ , cf. [49]. Putting aside highly publicized philosophical aspects of entanglement, its physical man-ifestation usually associated with two phenomena: • violation of classical realism, • nonlocality. As we have seen above every state of a system of dimension ≥ with full dynamicalgroup SU ( H ) is nonclassical. Therefor violation of classical realism is a general featureof quantum mechanics in no way specific for entanglement.The nonlocality, understood as a correlation beyond light cones of the systems, isa more subtle and enigmatic effect. It tacitely presumes spatially separated componentsin the system. This premise eventually ended up with formal identification of entangledstates with nonseparable ones. The whole understanding of entanglement was formedunder heavy influence of two-qubits, or more generally two-components systems, forwhich Schmidt decomposition (8) gives a transparent description and quantification ofentanglement. However later on it became clear that entanglement does manifest itself insystems with no clearly separated components, e.g. • Entanglement in an ensemble of identical bosons or fermions [35,21,20,56,14,36,63,60,68,44]. . Klyachko / Entanglement • Single particle entanglement, or entanglement of internal degrees of freedom, see[7,30] and references therein.Nonlocality is meaningless for a condensate of identical bosons or fermions withstrongly overlapping wave functions. Nevertheless we still can distinguish coherent Bose-Einstein condensate of bosons Ψ = ψ N or Slater determinant for fermions Ψ = ψ ∧ ψ ∧ . . . ∧ ψ N from generic entangled states in these systems. Recall, thatentangled states of biphoton where extensively studied experimentally [59,28], and Bellinequalities can be violated in such simple system as spin 1 particle, see n ◦ not its indispensable constituent. See also [40,41].Lack of common ground already led to a controversy in understanding of entangle-ment in bosonic systems, see n ◦ SU ( H ) makes all states equivalent, see n ◦ superselection rules or symmetry breaking which reduce the dynamicalgroup to a subgroup G ⊂ SU ( H ) small enough to create intrinsical difference betweenstates. For example, entanglement in two component system H A ⊗ H B comes fromreduction of the dynamical group to SU( H A ) × SU( H B ) ⊂ SU( H A ⊗ H B ) . Thereforentanglement must be studied in the framework of quantum dynamical systems. Roughly speaking, we consider entanglement as a manifestation of quantum fluctuations in a state where they come to their extreme. Specifically, we look for states with maximaltotal variance D ( ψ ) = X i h ψ | X i | ψ i − h ψ | X i | ψ i = max . It follows from equation (14) that the maximum is attained for state ψ with zero expec-tation of all essential observables h ψ | X | ψ i = 0 , ∀ X ∈ L Entanglementequation (25)We use this condition as the definition of completely entangled state and refer to it as entanglement equation . Let’s outline its distinctive features. • Equation (25) tells that in completely entangled state the system is at the center of itsquantum fluctuations. • This ensure maximality of the total variance, i.e. overall level of quantum fluctuationsin the system. In this respect completely entangled states are opposite to coherent ones,and may be treated as extremely nonclassical . They should manifest purely quantum ef-fects, like violation of classical realism, to the utmost. • May be the main flaw of the conventional approach is lack of physical quantity associ-ated with entanglement. In contrast to this, we consider entanglement as a manifestationof quantum fluctuations in a state where they come to their extreme. This, for example, A. Klyachko / Entanglement may help to understand stabilizing effect of environment on an entangled state, see [9]. • Entanglement equation (25) and the maximality of the total fluctuations plays an im-portant heuristic rôle, similar to variational principles in mechanics. It has also a trans-parent geometrical meaning discussed below in n ◦ • The total level of quantum fluctuations in irreducible system G : H λ varies in the range h λ, δ i ≤ D ( ψ ) ≤ h λ, λ + 2 δ i (26)with minimum attained at coherent states, and the maximum for completely entangled ones, see n ◦ s system this amounts to s ≤ D ( ψ ) ≤ s ( s + 1) . • Extremely high level of quantum fluctuations makes every completely entangled statemanifestly nonclassical, see Example 3.6.2 below. • The above definition make sense for any quantum system G : H and it is in conformitywith conventional one when the latter is applicable, e.g. for multi-component systems,see Example 3.6.3. For spin 1 system completely entangled spin states coincide with socalled neutrally polarized states of biphoton, see n ◦ • As expected, the definition is G -invariant, i.e. the dynamical group transforms com-pletely entangled state ψ into completely entangled one gψ , g ∈ G . There are few systems where completely entangled states fail to exist, e.g.in quantum system H with full dynamical group G = SU( H ) all states are coherent. Inthis case the total variance (12) still attains some maximum, but it doesn’t satisfy entan-glement equation (25). We use different terms maximally and completely entangled statesto distinguish these two possibilities and to stress conceptual, rather then quantitative,origin of genuine entanglement governed by equation (25). In most cases these notionsare equivalent, and all exceptions are actually known, see n ◦ G : H stable if it contains a completely entangled state, and unstable otherwise. Example 3.6.1. The conventional definition of entanglement explicitly refers to a com-posite system, which from our point of view is no more reasonable for entangled states,then for coherent ones. As an example let’s consider completely entangled state ψ ∈ H s of spin s system. According to the definition this means that average spin projection onto every direction ℓ should be zero: h ψ | J ℓ | ψ i = 0 . This certainly can’t happens for s = 1 / ,since in this case all states are coherent and have definite spin projection / onto somedirection. But for s ≥ such states do exist and will be described later in n ◦ ψ = | i for integral spin s , and ψ = 1 √ | + s i − | − s i ) for any s ≥ . They have extremely big fluctuations D ( ψ ) = s ( s + 1) , and therefor are manifestly nonclassical : average spin projection onto every direction is zero, while thestandard deviation p s ( s + 1) exceeds maximum of the spin projection s . . Klyachko / Entanglement Example 3.6.2. This consideration can be literally transferred to an arbitrary irreduciblesystem G : H λ , using inequality h λ, λ i < h λ, λ + 2 δ i instead of s < s ( s + 1) , to theeffect that a completely entangled state of any system is nonclassical. Example 3.6.3. Entanglement equation (25) implies that state of a multicomponent sys-tem, say ψ ∈ H ABC = H A ⊗ H B ⊗ H C , is completely entangled iff its marginals ρ A , ρ B , ρ C are scalar operators. This observation is in conformity with conventional ap-proach to entanglemnt [13], cf. also Example 1.6.1. From operational point of view state ψ ∈ H is entangled iff one can filter out from ψ a completely entangled state ψ using SLOCC operations. As we know from Example2.4.2 in standard quantum information settings SLOCC group coincide with complexifi-cation G c of the dydnamic group G . This leads us to the following Definition 3.7.1. State ψ ∈ H of dynamical system G : H is said to be entangled iff itcan be transformed into a completely entangled state ψ = gψ by complexified group G c (possibly asymptotically ψ = lim i g i ψ for some sequence g i ∈ G c ).In Geometric Invariant Theory such states ψ are called stable (or semistable if ψ can be reached only asymptotically). Their intrinsic characterization is one of the centralproblems both in Invariant Theory and in Quantum Information. Relation between thesetwo theories can be summarized in the following table, with some entries to be explainedbelow. D ICTIONARY Quantum Information Invariant Theory Entangled state Semistable vectorDisentangled state Unstable vectorSLOCC transform Action of complexified group G c Completely entangled state ψ prepared from ψ by SLOCC Minimal vector ψ in complex orbitof ψ State obtained from completelyentangled one by SLOCC Stable vectorCompletely entangled states can be characterized by the following theorem, known as Kempf–Ness unitary trick. A. Klyachko / Entanglement Theorem 3.7.2 (Kempf-Ness [29]) . State ψ ∈ H is completely entangled iff it has mini-mal length in its complex orbit | ψ | ≤ | g · ψ | , ∀ g ∈ G c . (27) Complex orbit G c ψ contains a completely entangled state iff it is closed. In this case thecompletely entangled state is unique up to action of G .3.7.3 Remark. Recall that entangled state ψ can be asympotically transformed bySLOCC into a completely entangled one. By Kempf-Ness theorem the question whenthis can be done effectively depends on whether the complex orbit of ψ is closed or not.The following result gives a necessary condition for this. Theorem (Matsushima [42]) . Complex stabilizer ( G c ) ψ of stable state ψ coincides withcomplexification of its compact stabilizer ( G ψ ) c . Square of length of the minimal vector in complex orbit µ ( ψ ) = inf g ∈ G c | gψ | , (28)provides a natural quantification of entanglement. It amounts to cos 2 ϕ for spin 1 state(20), to concurrence C ( ψ ) [26] in two qubits, and to square root of τ ( ψ ) forthree qubits (see below). We call it generalized concurrence . Evidently ≤ µ ( ψ ) ≤ .Equation µ ( ψ ) = 1 tells that ψ is already a minimal vector, hence completely entan-gled state.Nonvanishing of the generalized concurrence µ ( ψ ) > means that closure of com-plex orbit G c ψ doesn’t contains zero. Then the orbit of minimal dimension O ⊂ G c ψ is closed and nonzero. Hence by Kempf-Ness unitary trick it contains a completely en-tangled state ψ ∈ O which asymptotically can be obtained from ψ by action of thecomplexified dynamical group. Therefor by definition 3.7.1 µ ( ψ ) > ⇐⇒ ψ is entangled. The minimal value µ ( ψ ) = 0 corresponds to unstatable vectors that can asymptoticallyfall into zero under action of the complexified dynamical group. They form so-called null cone . It contains all coherent states, along with some others degenerate states, like W -state in three qubits, see Example 3.10.1.Noncoherent unstable states cause many controversies. There is unanimous agre-ment that coherent states are disentangled. In approach pursued in [63] all noncoherentstates are treated as entangled. Other researchers [21,20] argue that some noncoherentunstable bosonic states are actually disentangled. From our operational point of view allunstable states should be treated as disentangled, since they can’t be filtered out into acompletely entangled state even asymptotically. Therefore we accept the equivalence DISENTANGLED ⇐⇒ UNSTABLE ⇐⇒ NOT SEMISTABLE . . Klyachko / Entanglement The above controversy vanishes iff the null cone contains only coherent states, or equiv-alently dynamical group G acts transitively on unstable states. Spin one and two qubitssystems are the most notorious examples. They are low dimensional orthogonal systems with dynamical group SO ( n ) acting in H n = E n ⊗ C by Euclidean rotations. Null conein this case consists of isotropic vectors ( x, x ) = 0 , which are at the same time coherentstates, cf. n ◦ Theorem 3.8.1. Stable irreducible system G : H in which all unstable states are coher-ent is one of the following • Orthogonal system SO ( H ) : H , • Spinor representation of group Spin (7) of dimension 8, • Exceptional group G in its fundamental representation of dimension 7. The theorem can be deduced from Theorem 2.7.1 characterizing coherent states by quadratic equations.Indeed, the null cone is given by vanishing of all invariants. Hence in conditions of the theorem the fundamen-tal invariants should have degree two. For irreducible representation there is at most one invariant of degreetwo, the invariant metric ( x, y ) . Thus the problem reduces to description of subgroups G ⊂ SO ( H ) actingtransitively on isotropic cone ( x, x ) = 0 . The metric ( x, x ) is unique basic invariant of such system. Lookinginto the table in Vinberg-Popov book [62] we find only one indecomposable system with unique basic invariantof degree two not listed in the theorem: spinor representation of Spin (9) of dimension 16 studied by Igusa[27]. However, as we’ll see below, the action of this group Spin (9) on the isotropic cone is not transitive.Coherent states of decomposable irreducible system G A × G B : H A ⊗ H B are products ψ A ⊗ ψ B ofcoherent states of the components. Hence codimension of the cone of coherent states is at least d A d B − d A − d B + 1 = ( d A − d B − . As we’ve seen above, in conditions of the theorem the codimension should beequal to one, which is possible only for system of two qubits d A = d B = 2 , which is equivalent to orthogonalsystem of dimension four. One can also argue that projective quadric Q : ( x, x ) = 0 of dimension greater thentwo is indecomposable Q = X × Y . Both exceptional systems carry an invariant symmetric form ( x, y ) . Scalar square ( x, x ) generates the algebra of invarinats, and therefore the null cone consists of isotropicvectors ( x, x ) = 0 , as in the orthogonal case. These mysterious systems emerge alsoas exceptional holonomy groups of Riemann manifolds [2]. Their physical meaning isunclear.Élie Cartan [8] carefully studied coherent states in irreducible (half)spinor represen-tations of Spin ( n ) of dimension ν , ν = ⌊ n − ⌋ . He call them pure spinors . In generalthe cone of pure spinors is intersection of ν − (2 ν + 1) − (cid:0) ν +1 ν (cid:1) linear independentquadrics.For n < there are no equations, i.e. all states are coherent. In such systems there isno entanglement whatsoever, and we exclude them from the theorem. These systems arevery special and have a transparent physical interpretation. • For n = 3 spinor representation of dimension two identifies Spin (3) with SU (2) .Vector representation of SO (3) is just spin 1 system, studied in n ◦ • Two dimnensional halfspinor representations of Spin (4) identify this group with SU (2) × SU (2) and the orthogonal system of dimension 4 with two qubits. • For n = 5 spinor representation H of dimension 4 carries invariant simplecticform ω and identify Spin (5) with simplectic group Sp ( H , ω ) . The standardvector representation of SO (5) in this settings can be identified with the space ofskew symmetric forms in H modulo the defining form ω . A. Klyachko / Entanglement • For n = 6 halfsinor representations of dimension 4 identify Spin (6) with SU ( H ) and the orthogonal system of dimension 6 with SU ( H ) : ∧ H . Thisis a system of two fermions of rank 4. The previous group Spin (5) ≃ Sp ( H ) isjust a stabilizer of a generic state ω ∈ ∧ H .In the next case n = 7 coherent states are defined by single equation ( x, x ) = 0 andcoincide with unstable ones. Thus we arrive at the first special system Spin (7) : H .Stabilizer of a non isotropic spinor ψ ∈ H , ( ψ, ψ ) = 0 in Spin (7) is exceptionalgroup G and its representation in orthogonal complement to ψ gives the second system G : H . Alternatively it can be described as representation of automorphism group ofCayley octonic algebra in the space of purely imaginary octaves.Halfspinor representations of Spin (8) : H also carry invariant symmetric form ( x, y ) . It follows that Spin (8) acts on halfspinors as full group of orthogonal transforma-tions. Hence these representations are geometrically equivalent to the orthogonal system SO ( H ) : H . The equivalence is known as Cartan’s triality [8].Finally spinor representation of Spin (9) of dimension 16 also carries invariant sym-metric form ( x, y ) which is unique basic invariant of this representation. However ac-cording to Cartan’s formula the cone of pure spinors is intersection of 10 independentquadrics, hence differs from the null cone ( x, x ) = 0 . Spinor representations of two fold covering Spin (2 n ) of orthogonal group SO (2 n ) havea natural physical realization . Recall that all quadratic expressions in creation and an-nihilation operators a † i , a j , i, j = 1 . . . n in a system of fermions with n intrinsic de-grees of freedom form orthogonal Lie algebras so (2 n ) augmented by scalar operator (toavoid scalars one have to use ( a † i a i − a i a † i ) instead of a † i a i , a i a † i ). It acts in fermionicFock space F ( n ) , known as spinor representation of so (2 n ) . In difference with bosoniccase it has finite dimension dim F ( n ) = 2 n and splits into two halfspinor irreduciblecomponents F ( n ) = F ev ( n ) ⊕ F odd ( n ) , containing even and odd number of fermionsrespectively.For fermions of dimension n = 4 the halfspinors can be transformed into vectors bythe Cartan’s triality. This provides a physical interpretation of the orthogonal system ofdimension 8.To sum up, orthogonal systems of dimension n = 3 , , , have the following phys-ical description • n = 3 . Spin 1 system. • n = 4 . Two qubit system. • n = 6 . System of two fermions SU ( H ) : ∧ H of dimension . • n = 8 . System of fermions of dimension 4 with variable number of particles(either even or odd).The last example is fermionic analogue of a system of quantum oscillators n ◦ n ◦ ?? . . Klyachko / Entanglement Halfspinor representations of the next group Spin (10) was discussed as an intriguingpossibility, that quarks and leptons may be composed of five different species of fun-damental fermionic objects [69,66]. This is a very special system where all states areunstable, hence disentangled. In other words the null cone amounts to the whole statespace and there is no genuine entanglement governed by equation (25). Such systemsare opposite to those considered in the preceding section, where the null cone is as smallas possible. We call them unstable . There are very few types of such indecomposableirreducible dynamical systems [62,43]: • Unitary system SU ( H ) : H ; • Symplectic system Sp ( H ) : H ; • System of two fermions SU ( H ) : ∧ H of odd dimension dim H = 2 k + 1 ; • A halfspinor representation of dimension 16 of Spin (10) .All (half)spinor irreducible representations for n < fall into this category. There aremany more such composite systems, and their classification is also known due to M. Satoand T. Kimura [55]. Kempf–Ness theorem 3.7.2 identifies closed orbits of complexified group G c with com-pletely entangled states modulo action of G . Closed orbits can be separated by G -invariant polynomials. This leads to the following classical criterion of entanglement. Theorem 3.10.1 (Classical Criterion) . State ψ ∈ H is entangled iff it can be separatedfrom zero by G -invariant polynomial f ( ψ ) = f (0) , f ( gx ) = f ( x ) , ∀ g ∈ G, x ∈ H . Example 3.10.1. For two component system ψ ∈ H A ⊗ H B all invariants are polyno-mials in det[ ψ ij ] (no invariants for dim H A = dim H B ). Hence state is entangled iff det[ ψ ij ] = 0 . The generalized concurrence (28) related to this basic invariant by equation µ ( ψ ) = n | det[ ψ ij ] | /n . Unique basic invariant for 3-qubit is Cayley hyperdeterminant [ ? ,18] Det [ ψ ] = ( ψ ψ + ψ ψ + ψ ψ + ψ ψ ) − ψ ψ ψ ψ + ψ ψ ψ ψ + ψ ψ ψ ψ + ψ ψ ψ ψ + ψ ψ ψ ψ + ψ ψ ψ ψ )+4( ψ ψ ψ ψ + ψ ψ ψ ψ ) . related to 3-tangle [10] and generalized concurrence (28) by equations τ ( ψ ) = 4 | Det[ ψ ] | , µ ( ψ ) = p τ ( ψ ) . A. Klyachko / Entanglement One can check that Cayley hyperdeterminant vanishes for so called W-state W = | i + | i + | i√ which therefor is neither entangled nor coherent. This examples elucidate the nature of entanglement introduced here. Ittakes into account only those entangled states that spread over the whole system, anddisregards any entanglement supported in a smaller subsystem, very much like 3-tangledid. For example, absence of entanglement in two component system H A ⊗ H B for dim H A = dim H B reflects the fact that in this case every state belongs to a smallersubspace V A ⊗ V B , V A ⊂ H A , V B ⊂ H B as it follows from Schmidt decomposition (8).Entanglement of such states should be treated in the corresponding subsystems. The above examples, based on Theorem 3.10.1, shows that invariants are essential forunderstanding and quantifying of entanglement. Unfortunately finding invariants is atough job, and more then 100 years of study give no hope for a simple solution.There are few cases where all invariants are known, some of them were mentionedabove. In addition invariants and covariants of four qubits and three qutrits were foundrecently [39,4,5]. For five qubit only partial results are available [38]. See more on in-variants of qubits in [45,46]. For system of format × × the invariants are given in[51].Spin systems have an equivalent description in terms of binary forms , see Example3.11.2. Their invariants are described by theory of Binary Quantics , diligently pursuedby mathematicians from the second half of 19-th century. This is an amazingly difficultjob, and complete success was achieved by classics for s ≤ , the cases s = 5 / and being one of the crowning glories of the theory [43]. Modern authors advanced it up to s = 4 .Other classical results of invariant theory are still waiting physical interpretation andapplications. In a broader context Bryce S. DeWitt described the situation as follows: “Why should we not go directly to invariants? The whole of physics is contained inthem. The answer is that it would be fine if we could do it. But it is not easy.” Now, due to Hilbert’s insight, we know that the difficulty is rooted in a perverse desireto put geometry into Procrustean bed of algebra. He created Geometric Invariant Theory just to overcome it. Theorem 3.11.1 (Hilbert-Mumford Criterion [43]) . State ψ ∈ H is entangled iff everyobservable X ∈ L = Lie( G ) of the system in state ψ assumes a nonnegative value withpositive probability. By changing X to − X one deduces that X should assume nonpositive values aswell. So in entangled state no observable can be biased neither to strictly positive norto strictly negative values. Evidently completely entangled states with zero expectations h ψ | X | ψ i = 0 of all observables pass this test. . Klyachko / Entanglement Example 3.11.1. Let X = X A ⊗ ⊗ X B be observable of two qubit system H A ⊗H B with Spec X A = ± α, Spec X B = ± β, α ≥ β ≥ . Suppose that ψ is unstable and observable X assumes only strictly positive values instate ψ . Since those values are α ± β then the state is decomposable ψ = a | α i ⊗ | β i + b | α i ⊗ | − β i = | α i ⊗ ( a | β i + b | − β i ) , i.e. Hilbert-Mumford criterion characterizes entangled qubits.The general form of H-M criterion may shed some light on the nature of entangle-ment. However, it was originally designed for application to geometrical objects , likelinear subspaces or algebraic varieties of higher degree, and its efficacy entirely dependson our ability to express it in geometrical terms. Let’s give an example. Example 3.11.2. Stability of spin states. Spin s representation H s can be realized inspace of binary forms f ( x, y ) of degree d = 2 s H s = { f ( x, y ) | deg f = 2 s } in which SU (2) acts by linear substitutions f ( x, y ) f ( ax + by, cx + dy ) . To makeswap from physics to mathematics easier we denote by f ψ ( x, y ) the form correspondingto state ψ ∈ H s . Spin state ψ ∈ H s can be treated algebraically, physically, or geometri-cally according to the following equations ψ = µ = s X µ = − s a µ (cid:18) ss + µ (cid:19) x s + µ y s − µ = µ = s X µ = − s a µ (cid:18) ss + µ (cid:19) / | µ i = Y i ( α i x − β i y ) . The first one is purely algebraic, the second gives physical decomposition over eigen-states | µ i = (cid:18) ss + µ (cid:19) / x s + µ y s − µ , J z | µ i = µ | µ i of spin projector operator J z = (cid:16) x ∂∂x − y ∂∂y (cid:17) , and the last one is geometrical. Itdescribes form f ψ ( x, y ) in terms of configuration of its roots z i = ( β i : α i ) in Riemannsphere C ∪∞ = S (known also as Bloch sphere for spin / states, and Poincaré sphere for polarization of light).According to H-M criterion state ψ is unstable iff spin projections onto somedirection ℓ are strictly positive. By rotation we reduce the problem to z -component J z = (cid:16) x ∂∂x − y ∂∂y (cid:17) in which case the corresponding form f ψ ( x, y ) = X µ> a µ (cid:18) ss + µ (cid:19) / | µ i = X µ> a µ (cid:18) ss + µ (cid:19) x s + µ y s − µ A. Klyachko / Entanglement has root x = 0 of multiplicity more then s = d/ . As result we arrive at the followingcriterion of entanglement (=semistability) for spin states ψ is entangled ⇐⇒ no more then half of the roots f ψ ( x, y ) coinside. (29)One can show that if less then half of the roots coincide then the state is stable i.e. can betransformed into a completely entangled one by Lorentz group SL (2 , C ) acting on roots z i ∈ C ∪ ∞ by Möbius transformations z az + bcz + d . In terms of these roots entanglementequation (25) amounts to the following condition ψ completely entangled ⇐⇒ X i ( z i ) = 0 , (30)where parentheses denote unit vector ( z i ) ∈ S ⊂ E mapping into z i ∈ C ∪ ∞ understereographic projection. 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