Entangled State Vectors

Let R₁, R₂ be von Neumann algebras on H such that R₁ ⊆ R′₂. Recall that a state ω of R₁₂ is called a normal product state just in case ω is normal, and there are states ω₁ of R₁ and ω₂ of R₂ such that

ω(AB) = ω1(A)ω2(B) —– (1)

∀ A ∈ R₁, B ∈ R₂. Werner, in dealing with the case of B(Cⁿ) ⊗ B(Cⁿ), defined a density operator D to be classically correlated — the term separable is now more commonly used — just in case D can be approximated in trace norm by convex combinations of density operators of form D₁ ⊗ D₂. Although Werner’s definition of nonseparable states directly generalizes the traditional notion of pure entangled states, he showed that a nonseparable mixed state need not violate a Bell inequality; thus, Bell correlation is in general a sufficient, though not necessary condition for a state’s being non-separable. On the other hand, it has since been shown that nonseparable states often possess more subtle forms of nonlocality, which may be indicated by measurements more general than the single ideal measurements which can indicate Bell correlation.

In terms of the linear functional representation of states, Werner’s separable states are those in the norm closed convex hull of the product states of B(Cⁿ) ⊗ B(Cⁿ). However, in case of the more general setup — i.e., R₁ ⊆ R′₂, where R₁, R₂ are arbitrary von Neumann algebras on H — the choice of topology on the normal state space of R₁₂ will yield in general different definitions of separability. Moreover, it has been argued that norm convergence of a sequence of states can never be verified in the laboratory, and as a result, the appropriate notion of physical approximation is given by the (weaker) weak-∗ topology. And the weak-∗ and norm topologies do not generally coincide even on the normal state space.

For the next proposition, then, we will suppose that the separable states of R₁₂ are those normal states in the weak-∗ closed convex hull of the normal product states. Note that β(ω) = 1 if ω is a product state, and since β is a convex function on the state space, β(ω) = 1 if ω is a convex combination of product states. Furthermore, since β is lower semicontinuous in the weak-∗ topology, β(ω) = 1 for any separable state. Conversely, any Bell correlated state must be nonseparable.

We now introduce some notation that will aid us in the proof of our result. For a state ω of the von Neumann algebra R and an operator A ∈ R, define the state ω^A on R by

ω^A(X) ≡ ω(A∗XA)/ω(A∗A) —– (2)

if ω(A∗A) ≠ 0, and let ω^A = ω otherwise. Suppose now that ω(A∗A) ≠ 0 and ω is a convex combination of states:

ω = ∑_i=1ⁿλ_iω_i —– (3)

Then, letting λ^A_i ≡ ω(A∗A)⁻¹ω_i(A∗A)λ_i, ω^A is again a convex combination

ω^A = ∑_i=1ⁿ λ^A_iω^A_i —– (4)

Moreover, it is not difficult to see that the map ω → ω^A is weak-∗ continuous at any point ρ such that ρ(A∗A) ≠ 0. Indeed, let O₁ = N(ρ^A : X₁,…,X_n, δ) be a weak-∗ neighborhood of ρ_A. Then, taking O₂ = N(ρ : A^∗A,A^∗X₁A,…,A^∗X_nA, δ) and ω ∈ O₂, we have

|ρ(A^∗X_iA) − ω(A^∗X_iA)| < δ —– (5)

for i = 1,…,n, and

|ρ(A^∗A) − ω(A^∗A)| < δ —– (6)

By choosing δ < ρ(A^∗A) ≠ 0, we also have ω(A^∗A) ≠ 0, and thus

|ρ^A(X_i) − ω^A(X_i)| < O(δ) ≤ δ —– (7)

for an appropriate choice of δ. That is, ω^A ∈ O₁ forall ω ∈ O₂ and ω → ω^A is weak-∗ continuous at ρ.

Specializing to the case where R₁ ⊆ R′₂, and R₁₂ = {R₁ ∪ R₂}”, it is clear from the above that for any normal product state ω of R₁₂ and for A ∈ R₁, ω^A is again a normal product state. The same is true if ω is a convex combination of normal product states, or the weak-∗ limit of such combinations. Summarizing the results of this discussion in the following lemma:

Lemma: For any separable state ω of R₁₂ and any A ∈ R₁, ω^A is again separable.

Proposition: Let R₁,R₂ be nonabelian von Neumann algebras such that R₁ ⊆ R′₂. If x is cyclic for R₁, then ω_x is nonseparable across R₁₂.

Proof: There is a normal state ρ of R₁₂ such that β(ρ) = √2. But since all normal states are in the (norm) closed convex hull of vector states, and since β is norm continuous and convex, there is a vector v ∈ S such that β(v) > 1. By the continuity of β (on S), there is an open neighborhood O of v in S such that β(y) > 1 ∀ y∈O. Since x is cyclic for R₁,there is an A ∈ R₁ such that Ax ∈ O. Thus, β(Ax) > 1 which entails that ω_Ax = (ω_x)^A is a nonseparable state for R₁₂. This, by the preceding lemma, entails that ω_x is nonseparable.

Note that if R₁ has at least one cyclic vector x ∈ S, then R₁ has a dense set of cyclic vectors in S. Since each of the corresponding vector states is nonseparable across R₁₂, Proposition shows that if R₁ has a cyclic vector, then the (open) set of vectors inducing nonseparable states across R₁₂ is dense in S. On the other hand, since the existence of a cyclic vector for R₁ is not invariant under isomorphisms of R₁₂, Proposition does not entail that if R₁ has a cyclic vector, then there is a norm dense set of nonseparable states in the entire normal state space of R₁₂. Indeed, if we let R₁ = B(C²) ⊗ I, R₂ = I ⊗ B(C²), then any entangled state vector is cyclic for R₁; but, the set of nonseparable states of B(C²) ⊗ B(C²) is not norm dense. However, if in addition to R₁ or R₂ having a cyclic vector, R₁₂ has a separating vector (as is often the case in quantum field theory), then all normal states of R₁₂ are vector states, and it follows that the nonseparable states will be norm dense in the entire normal state space of R₁₂.

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