Entangled State Vectors

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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 ωA on R by

ωA(X) ≡ ω(A∗XA)/ω(A∗A) —– (2)

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

ω = ∑i=1nλiωi —– (3)

Then, letting λAi ≡ ω(A∗A)−1ωi(A∗A)λi, ωA is again a convex combination

ωA = ∑i=1n λAiωAi —– (4)

Moreover, it is not difficult to see that the map ω → ωA is weak-∗ continuous at any point ρ such that ρ(A∗A) ≠ 0. Indeed, let O1 = N(ρA : X1,…,Xn, δ) be a weak-∗ neighborhood of ρA. Then, taking O2 = N(ρ : AA,AX1A,…,AXnA, δ) and ω ∈ O2, we have

|ρ(AXiA) − ω(AXiA)| < δ —– (5)

for i = 1,…,n, and

|ρ(AA) − ω(AA)| < δ —– (6)

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

A(Xi) − ωA(Xi)| < O(δ) ≤ δ —– (7)

for an appropriate choice of δ. That is, ωA ∈ O1 forall ω ∈ O2 and ω → ωA 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, ωA 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, ωA is again separable.

Proposition: Let R1,R2 be nonabelian von Neumann algebras such that R1 ⊆ R′2. If x is cyclic for R1, then ωx 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 ωAx = (ωx)A is a nonseparable state for R12. This, by the preceding lemma, entails that ωx 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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7 thoughts on “Entangled State Vectors

  1. “More subtle forms of nonlocality”. This is an interesting statement. It tells me that ‘nonlocality’ does not indicate a specfic situation, buy rather it is a term for a category of types. Can you describe in relative layman terms what this can mean?

    • This non-classical aspect of quantum theory, its essential wholeness, is also connected to what could be called the “form” of the wave function. In classical systems it is generally possible to separate the system’s description into that of various spatially separate subsystems in interaction. In quantum theory, however, the wave function is not spatially separable in this way, into a product of subsystems, and so the special form of the wave function plays a significant role.

      While a quantum mechanical wave function is not truly separable into a simple product of different contributions it is, to a good approximation, possible to treat a system of, for example, two alpha particles and four electrons, as being more or less reducible to a description of two helium atoms with a weak interaction between them. However, in certain other circumstances a collection of helium atoms–alpha particles and electrons–will act in a markedly different way, as a single collective or coherent state–a superfluid. Likewise, under certain conditions, a gas of electrons will also act as a single coherent whole.

      The nature of these coherent and condensed states is reflected in the very special forms of their wave functions and the fact that they preserve their internal correlations over macroscopic distances. This feature is totally novel to the quantum theory and does not occur in any classical system. This form of the wave function exercises a sort of dynamical holism over the system, ensuring the active, dynamical correlation of every particle within the coherent system, no matter how far apart they may be from each other. Thats subtle form of nonlocality.

  2. That is absolutly excellent. I appreciate your willingness to break these down.

    My philosophical work involves the traversing what we could call “classical” philosophy, what I call conventional philosophy, to make allowances for what we could call “quantum”Arena of philosophy.

    I have a fairly good understanding of general concepts of quantum theory, and these seem to develop well for my work. But because I I am not a mathematician Nora physicist, I can’t translate the mathematical formulas that are involved in those discussions. So I’m eager to understand more thoroughly with more involved mathematical formulas such as you talk about in your blog, really mean for translating them to regular word descriptikns.

    🤘🏽

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