6 Bipartite entanglement Entanglement is a fundamental concept in quantum information theory, considered by many to be a quintessential characteristic that distinguishes quantum systems from their classical counterparts. Informally speaking, a state of a collection of registers X1,..., Xn is said to be entangled when it is not possible to specify the correlations that exist among the registers in classical terms. When it is possible to describe these correlations in classical terms, the registers are said to be in a separable state. Entanglement among two or more registers is therefore synonymous with a lack of separability. This chapter introduces notions associated with bipartite entanglement, in which correlations between precisely two registers (or two collections of registers) are considered. Topics to be discussed include the property of separability, which is applicable not only to states but also to channels and measurements; aspects of entanglement manipulation and quantification; and a discussion of operational phenomena associated with entanglement, including teleportation, dense coding, and non-classical correlations among measurements on separated systems. 6.1 Separability This section introduces the notion of separability, which is applicable to states, channels, and measurements on bipartite systems. It is possible to define a multipartite variant of this concept, but only bipartite separability is considered in this book. 6.1.1 Separable operators and states The property of separability for operators acting on bipartite tensor product spaces is defined as follows. 6.1 Separability 311 Definition 6.1 For any choice of complex Euclidean spaces and , X Y the set Sep( : ) is defined as the set containing all positive semidefinite X Y operators R Pos( ) for which there exists an alphabet Σ and two ∈ X ⊗ Y collections of positive semidefinite operators, P : a Σ Pos( ) and Q : a Σ Pos( ), (6.1) { a ∈ } ⊂ X { a ∈ } ⊂ Y such that R = P Q . (6.2) a ⊗ a a Σ X∈ Elements of the set Sep( : ) are called separable operators. X Y Remark It must be stressed that separability is defined with respect to a particular tensor product structure of the underlying complex Euclidean space of a given operator, as the previous definition reflects. When the term separable operator is used, one must therefore make this tensor product structure known (if it is not implicit). An operator R Pos( ) ∈ X ⊗ Y ⊗ Z may, for instance, be an element of Sep( : ) but not Sep( : ). X Y ⊗ Z X ⊗ Y Z By restricting the definition above to density operators, one obtains a definition of separable states. Definition 6.2 Let and be complex Euclidean spaces. One defines X Y SepD( : ) = Sep( : ) D( ). (6.3) X Y X Y ∩ X ⊗ Y Elements of the set SepD( : ) are called separable states (or separable X Y density operators). Convex properties of separable operators and states The sets Sep( : ) and SepD( : ) possess various properties relating to X Y X Y convexity, a few of which will now be observed. Proposition 6.3 For every choice of complex Euclidean spaces and , X Y the set SepD( : ) is convex, and the set Sep( : ) is a convex cone. X Y X Y Proof It will first be proved that Sep( : ) is a convex cone. It suffices to X Y prove that Sep( : ) is closed under addition as well as multiplication by X Y any nonnegative real number. To this end, assume that R ,R Sep( : ) 0 1 ∈ X Y are separable operators and λ 0 is a nonnegative real number. One may ≥ write R = P Q and R = P Q (6.4) 0 a ⊗ a 1 a ⊗ a a Σ a Σ X∈ 0 X∈ 1 312 Bipartite entanglement for disjoint alphabets, Σ0 and Σ1, and two collections of positive semidefinite operators, Pa : a Σ0 Σ1 Pos( ), { ∈ ∪ } ⊂ X (6.5) Q : a Σ Σ Pos( ). { a ∈ 0 ∪ 1} ⊂ Y It holds that R + R = P Q , (6.6) 0 1 a ⊗ a a Σ Σ ∈X0∪ 1 and therefore R + R Sep( : ). Moreover, it holds that 0 1 ∈ X Y λR = (λP ) Q . (6.7) 0 a ⊗ a a Σ X∈ 0 As λP Pos( ) for every positive semidefinite operator P Pos( ), it ∈ X ∈ X follows that λR Sep( : ). 0 ∈ X Y The fact that SepD( : ) is convex follows from the fact that it is equal X Y to the intersection of two convex sets, Sep( : ) and D( ). X Y X ⊗ Y The next proposition, when combined with the previous one, implies that Sep( : ) is equal to the cone generated by SepD( : ). X Y X Y Proposition 6.4 Let be a complex Euclidean space, let Pos( ) be Z A ⊆ Z a cone, and assume that = D( ) is nonempty. It holds that B A ∩ Z = cone( ). (6.8) A B Proof Suppose first that ρ and λ 0. It follows that λρ by virtue ∈ B ≥ ∈ A of the fact that and is a cone, and therefore B ⊆ A A cone( ) . (6.9) B ⊆ A Now suppose that P . If P = 0, then one has that P = λρ for λ = 0 ∈ A and ρ being chosen arbitrarily. If P = 0, then consider the density ∈ B 6 operator ρ = P/ Tr(P ). It holds that ρ because 1/ Tr(P ) > 0 and is ∈ A A a cone, and therefore ρ . As P = λρ for λ = Tr(P ) > 0, it follows that ∈ B P cone( ). Therefore, ∈ B cone( ), (6.10) A ⊆ B which completes the proof. Two equivalent ways of specifying separable states are provided by the next proposition, which is a straightforward consequence of the spectral theorem. 6.1 Separability 313 Proposition 6.5 Let ξ D( ) be a density operator, for complex ∈ X ⊗ Y Euclidean spaces and . The following statements are equivalent: X Y 1. ξ SepD( : ). ∈ X Y 2. There exists an alphabet Σ, collections of states ρ : a Σ D( ) { a ∈ } ⊆ X and σ : a Σ D( ), and a probability vector p (Σ), such that { a ∈ } ⊆ Y ∈ P ξ = p(a) ρ σ . (6.11) a ⊗ a a Σ X∈ 3. There exists an alphabet Σ, collections of unit vectors x : a Σ { a ∈ } ⊂ X and y : a Σ , and a probability vector p (Σ), such that { a ∈ } ⊂ Y ∈ P ξ = p(a) x x∗ y y∗. (6.12) a a ⊗ a a a Σ X∈ Proof The third statement trivially implies the second, and it is immediate that the second statement implies the first, as SepD( : ) is convex and X Y ρ σ SepD( : ) for each a Σ. It remains to prove that the first a ⊗ a ∈ X Y ∈ statement implies the third. Let ξ SepD( : ). As ξ Sep( : ), one may write ∈ X Y ∈ X Y ξ = P Q (6.13) b ⊗ b b Γ X∈ for some choice of an alphabet Γ and collections P : b Γ Pos( ) and { b ∈ } ⊂ X Q : b Γ Pos( ) of positive semidefinite operators. Let n = dim( ), { b ∈ } ⊂ Y X let m = dim( ), and consider spectral decompositions of these operators as Y follows: n m Pb = λj(Pb)ub,jub,j∗ and Qb = λk(Qb)vb,kvb,k∗ , (6.14) jX=1 kX=1 for each b Γ. Define Σ = Γ 1, . , n 1, . , m , and define ∈ × { } × { } p((b, j, k)) = λj(Pb)λk(Qb), x(b,j,k) = ub,j , (6.15) y(b,j,k) = vb,k , for every (b, j, k) Σ. A straightforward computation reveals that ∈ p(a) x x∗ y y∗ = P Q = ξ. (6.16) a a ⊗ a a b ⊗ b a Σ b Γ X∈ X∈ Moreover, each value p(a) is nonnegative, and because p(a) = Tr(ξ) = 1, (6.17) a Σ X∈ 314 Bipartite entanglement it follows that p is a probability vector. It has therefore been proved that statement 1 implies statement 3. By the equivalence of the first and second statements in the previous proposition, it holds that a given separable state ξ SepD( : ) represents ∈ X Y a classical probability distribution over independent quantum states of a pair of registers (X, Y); and in this sense the possible states of the registers X and Y, when considered in isolation, are classically correlated. For a separable state ξ SepD( : ), the expression (6.12) is generally ∈ X Y not unique—there may be many inequivalent ways that ξ can be expressed in this form. It is important to observe that an expression of this form cannot necessarily be obtained directly from a spectral decomposition of ξ. Indeed, for some choices of ξ SepD( : ) it may hold that every expression of ξ in ∈ X Y the form (6.12) requires that Σ has cardinality strictly larger than rank(ξ). An upper bound on the size of the alphabet Σ required for an expression of the form (6.12) to exist may, however, be obtained from Carath´eodory’s theorem (Theorem 1.9). Proposition 6.6 Let ξ SepD( : ) be a separable state, for and ∈ X Y X being complex Euclidean spaces. There exists an alphabet Σ such that Y Σ rank(ξ)2, two collections of unit vectors x : a Σ and | | ≤ { a ∈ } ⊂ X y : a Σ , and a probability vector p (Σ) such that { a ∈ } ⊂ Y ∈ P ξ = p(a)x x∗ y y∗. (6.18) a a ⊗ a a a Σ X∈ Proof By Proposition 6.5 it holds that SepD( : ) = conv xx∗ yy∗ : x ( ), y ( ) , (6.19) X Y ⊗ ∈ S X ∈ S Y from which it follows that ξ is contained in the set conv xx∗ yy∗ : x ( ), y ( ), im(xx∗ yy∗) im(ξ) . (6.20) ⊗ ∈ S X ∈ S Y ⊗ ⊆ Every density operator ρ D( ) satisfying im(ρ) im(ξ) is contained ∈ X ⊗ Y ⊆ in the real affine subspace H Herm( ) : im(H) im(ξ), Tr(H) = 1 (6.21) ∈ X ⊗ Y ⊆ of dimension rank(ξ)2 1, and therefore the proposition follows directly from − Carath´eodory’s theorem.