Entanglement of symmetric Werner states Hans Maassen, Radboud University (Nijmegen), QuSoft (Amsterdam) Workshop Mathematics of Quantum Information Theory, Lorentz Center Leiden, May 9, 2019. Collaboration with Burkhard K¨ ummerer, (Darmstadt) Discussions with Michael Walter, Freek Witteveen, Maris Ozols, Christian Majenz (QuSoft)
Statement of the problem We consider a quantum system consisting of n identical, but distinguishable subsystems (”particles”) described by Hilbert spaces of dimension d . A state on such a system is called a Werner state if it is invariant under the global unitary rotation of all the individual Hilbert spaces together. It is called symmetric if is invariant for permutation of the particles. A state is called entangled if it can not be written as a convex combination of product states. For a given symmetric Werner state, we want to find out if it is entangled or not. And also, whether there is a relation between entangement and extendability.
Motivation Entanglement is a central issue in quantum information theory. The study of n party-entanglement is considered difficult. It is complicated by the fact that the state space of n systems of size d has a large dimension: d 2 n − 1. The number of parameters is greatly reduced by restricting attention to the symmetric Werner states. The dimension d drops out entirely, and the number of parameters becomes (one less than) the number of possible partitions of the n particles. For example, for 2 quantum identical systems of arbitrary size d there is only one parameter. An advantage of this restraint is that we can lean on a vast body of results from classical mathematics: the representation theory of S n and SU ( d ), as pioneered by Frobenius, Schur, Weyl, Littlewood, . . . . But also some relatively recent work in multilinear algebra turns out to be relevant to our question. Extendability relates to the ‘monogamy’ property of entanglement.
Overview of the talk ◮ Entanglement of Werner states for n = 2; ◮ Symmetric Werner states on n particles; ◮ Separable symmetric Werner states and immanants; ◮ The case n = 3; ◮ The ‘shadow’ of the product states and its behaviour for general n ; ◮ Schur’s inequality and Lieb’s conjecture; ◮ The case n = 4; ◮ The case n = 5: Hope crashed. ◮ Extendability of symmetric Werner states ◮ A quantum de Finetti theorem of Christandl et al.
Symmetries in the two-particle case H = C d ⊗ C d . Symmetry group: u ∈ SU ( d ) acting as u ⊗ u . This action commutes with the ”Flip” operator: ϕ ⊗ ψ �→ ψ ⊗ ϕ . F : Eigenspaces H + and H − of F are invariant for SU ( d ): H + := span { ψ ⊗ ψ | ψ ∈ C d } � 1 � � � basis: { e i ⊗ e i | 0 ≤ i ≤ d } ∪ √ ( e i ⊗ e j + e j ⊗ e i ) � 0 ≤ i < j ≤ d . � 2 � � dim H + = d ( d + 1) d + 1 = ; 2 2 � 1 � � � basis of H − : √ ( e i ⊗ e j − e j ⊗ e i ) � 0 ≤ i < j ≤ d . � 2 � � dim H − = d ( d − 1) d = . 2 2
Werner states The group SU ( d ) acts irreducibly on H + and on H − . Hence the action of SU ( d ) has commutant { u ⊗ u | u ∈ SU ( d ) } ′ = { λ p + + µ p − | λ, µ ∈ C } = { F } ′′ , and the minimal nonzero symmetric projections are p ± := 1 l ± F = projection onto H ± . 2 The SU ( d )-symmetric states (i.e. Werner states) are convex combinations of � 1 l ± F � x �→ tr p ± x tr p ± = tr 2 x ω ± := (anti-)symmetric state: . dim H ± The Werner states are given by ω = λω + + (1 − λ ) ω − , 0 ≤ λ ≤ 1 . We note that a Werner state is fixed by specifying its value on the flip operator: ω ( F ) = λ − (1 − λ ) = 2 λ − 1 .
Entanglement of Werner states for n = 2 For x ∈ M d ⊗ M d , let Tx denote its average over the group SU ( d ): � ( u ⊗ u ) ∗ x ( u ⊗ u ) du , Tx := SU ( d ) where du denotes the Haar measure on SU ( d ). For states: T ∗ ϑ : x �→ ϑ ( Tx ): projection of ϑ onto the Werner states, T ∗ ϑ coincides with ϑ on F . Theorem A Werner state ω on M d ⊗ M d is separable iff ω ( F ) ≥ 0 . This the optimal Bell inequality for Werner states on two particles. entangled separable ω − ω + ω +
Proof. For any pure product state ψ ⊗ ϕ ∈ C d ⊗ C d the expectation of F is positive: � ψ ⊗ ϕ, F ( ψ ⊗ ϕ ) � = � ψ ⊗ ϕ, ϕ ⊗ ψ � = � ψ, ϕ �� ϕ, ψ � = |� ψ, ϕ �| 2 ≥ 0 . This inequality extends to all separable states by convexity. Conversely, suppose 0 ≤ ω ( F ) ≤ 1 for some Werner state ω , and choose unit vectors ψ, ϕ with |� ψ, ϕ �| 2 = ω ( F ) . Then the separable state � � � � � σ : x �→ ψ ⊗ ϕ, T ( x ) ψ ⊗ ϕ = ( u ⊗ u ) ψ ⊗ ϕ, x ( u ⊗ u ) ψ ⊗ ϕ du SU ( d ) is a Werner state, and coincides with ω on F . Hence ω = σ , which is separable.
The chaotic state moves to the boundary as d → ∞ Curious fact: d →∞ τ d ⊗ τ d ( F ) = 0 . lim Indeed, d d 2 tr d ⊗ tr d ( F ) = 1 1 � τ d ⊗ τ d ( F ) = � e i ⊗ e j , F ( e i ⊗ e j ) � d 2 i , j =1 d d 1 � e i ⊗ e j , e j ⊗ e i � = 1 δ ij = 1 � � = d . d 2 d 2 i , j =1 i , j =1 entangled separable 1 1 1 · · · ω − ω + ω + 4 3 2
General n ∈ N : Schur-Weyl duality On the Hilbert space H := C d ⊗ C d ⊗ · · · ⊗ C d ( n times) there are representations of two groups: S n and SU ( d ): S n ∋ σ : π ( σ ) ψ 1 ⊗ ψ 2 ⊗ . . . ⊗ ψ n := ψ σ − 1 (1) ⊗ ψ σ − 1 (2) ⊗ · · · ⊗ ψ σ − 1 ( n ) π ′ ( u ) ψ 1 ⊗ ψ 2 ⊗ . . . ⊗ ψ n := u ψ 1 ⊗ u ψ 2 ⊗ · · · ⊗ u ψ n SU ( d ) ∋ u : The classical Schur-Weyl duality theorem states that these two group actions do not only commute, but the algebras they generate are actually each other’s commutant. In particular they have the same center: Z := Z ( n , d ) := π ( S n ) ′ ∩ π ′ ( SU ( d )) ′ . The minimal projections in this center cut both group representations into their irreducible components, and they are labeled by Young diagrams. Indeed we have π ( σ ) π ′ ( u ) ∼ � π λ ( σ ) ⊗ π ′ = λ ( u ) . λ ⊢ n In particular d n = � d ( λ ) d ′ ( λ ) . λ ⊢ n
The group algebra of S n Let A n denote the group algebra of S n : � f : S n → C to be viewed as f ( σ ) σ . σ ∈ S n Multiplication in A n is convolution: � f ( τ ) g ( τ − 1 σ ) . ( f ∗ g )( σ ) = τ ∈ S n The unit is δ e , where e is the identity element of S n . Adjoint operation: f ∗ ( σ ) = f ( σ − 1 ) . Every unitary representation of S n automatically extends to a representation of A n . In our case � π ( f ) : ψ 1 ⊗ . . . ⊗ ψ n �→ f ( σ ) ψ σ − 1 (1) ⊗ . . . ⊗ ψ σ − 1 n . σ ∈ S n
The (left) regular representation of S n We let f ∈ A n act on the Hilbert space l 2 ( S n ) by convolution on the left: h �→ f ∗ h . The trace is in this representation of a particularly simple form: � � trreg ( f ) := � δ σ , f ∗ δ σ � = ( f ∗ δ σ )( σ ) = n ! · f ( e ) , σ ∈ S n σ ∈ S n and will be called the regular trace. 1 The normalized version τ reg := n ! trreg is the regular trace state: τ reg : f �→ f ( e ) .
The center of the group algebra of S n Z n := A n ∩ A ′ n . We have f ∈ Z n if and only if for all σ, τ ∈ S n : f ( στ ) = f ( τσ ): the center consists of the class functions. Hence dim Z n = #(conjugacy classes of S n ) = #(partitions of n ) =: P ( n ) . On the other hand, since Z n is an abelian matrix algebra, it must be of the form � Z n = C p λ λ ⊢ n for some orthogonal set of minimal projections p λ in the center. These can be labeled by Young diagrams. The states on the center form a simplex with extreme points ω λ given by ω λ ( p µ ) = δ λµ .
minimal projections and irreducible representations The center of the algebra A n is spanned by the minimal projections: p λ ( σ − 1 ) = p λ ( σ ) , � p λ ∗ p µ = δ λµ p λ and p λ = δ e . λ ⊢ n They cut the algebra A = A n into factors p λ A : � � A = p λ A ≃ M d ( λ ) ⊗ 1 l d ( λ ) . λ ⊢ n λ ⊢ n Hence d ( λ ) 2 = tr ( p λ ) = n ! · p λ ( e ) .
The characters χ λ and χ ′ λ The character χ λ ( σ ) is the trace of σ in its irreducible representation π λ . � � χ λ ( σ ) := tr π λ ( σ ) χ ′ π ′ � � λ ( u ) := tr λ ( u ) . χ ′ ( λ ) is given by the Schur polynomials χ ′ λ (diag( x 1 , . . . , x d )) = s λ ( x 1 , . . . , x d ) . χ λ ( σ ) is directly related to the projection operator p λ : n ! d ( λ ) · p λ ( σ ) . χ λ ( σ ) =
Young frames The irreducible representations of S n (and hence also the minimal central projections and the characters) are labelled by Young frames with n boxes: λ = . (Hook length rule) n ! d ( λ ) = � hook lengths . For example: � � 5! 4 3 1 d = 4 × 3 × 2 = 5 hook lengths: . 2 1
Young vectors The irreducible representations of S n (and hence also the minimal central projections and the characters) can be constructed from Young frames with n boxes, for example: λ = . To λ we associate a unit vector ψ λ ∈ ( C d ) ⊗ n : ψ λ := ψ ⊗ ψ ⊗ ψ ⊗ ψ . Here, ψ (with height k ) is the antisymmetric product ε 1 ∧ ε 2 ∧ · · · ∧ ε k . in terms of the canonical basis ε 1 , ε 2 , . . . , ε d of C d .
Recommend
More recommend
