The entanglement of states on n-independent C∗ subalgebras is considered, and equivalent conditions are given for C∗ subalgebras to be independent.

1. Introduction

Quantum correlations have been one of the most hottest subjects in the last two decades, many scholars devoted to the study [1–11]. In this paper, the most special correlation between quantum systems is investigated, that is “independence,” which is closely related to the entanglement of the states.

In quantum mechanics, the state entanglement is the property of two particles with a common origin whereby a measurement on one of the particles determines not only its quantum state but also the quantum state of the other particle as well, which is characterized as follows.

Definition 1.1.

Let ℋ1 and ℋ2 be Hilbert spaces and ℋ=ℋ1⊗ℋ2, and a state ϕ of B(ℋ1⊗ℋ2)=B(ℋ1)⊗B(ℋ2) is called to be separable if it is a convex combination of product states ϕi1⊗ϕi2, that is
(1.1)ϕ(A⊗B)=∑i=1npiϕi1(A)ϕi2(B),where∑i=1npi=1,pi≥0.
Otherwise, ϕ is called entanglement.

In the algebraic quantum theory, the observable is represented by an adjoint operator in a C* algebra. Naturally an interesting problem is raised.

Problem 1.

In a C* algebra 𝒜, what is the condition of the commuting C* subalgebras 𝒜1 and 𝒜2, under which the entanglement of states can be considered, where 𝒜i is the C* algebra generated by an observable Ai.

Notice that the problem is to seek the condition of commuting C* algebras 𝒜1 and 𝒜2, such that C*(𝒜1,𝒜2)=𝒜1⊗𝒜2 holds true.

In fact the condition is that 𝒜1 and 𝒜2 are C* independent.

Haag and Kastler [12] introduced a notion called statistical independence. If 𝒜 and ℬ represent the algebras generated by the observables associated with two quantum subsystems, the statistical independence of 𝒜 and ℬ can be construed as follows: any two partial states on the two subsystems can be realized by the same preparation procedure. The statistical independence of 𝒜 and ℬ in the category of C* algebras is called C* independence, which is defined as follows: if for any state ϕ1 on 𝒜 and ϕ2 on ℬ, there is a state ϕ on C*(𝒜,ℬ), such that ϕ|𝒜=ϕ1 and ϕ|ℬ=ϕ2, where C*(𝒜,ℬ) denotes the unital C* algebra generated by 𝒜,ℬ.

Roos [13] gave a characterization of C* independence. He showed if 𝒜 and ℬ are commuting subalgebras of a C* algebra, then 𝒜 and ℬ are C* independent, if and only if 0≠A∈𝒜 and 0≠B∈ℬ imply that AB≠0. Later, Florig and Summers [14] studied the relation between C* independence and W* independence and showed if 𝒜 and ℬ are commuting subalgebras of a σ-finite W* algebra 𝒲, then they are C* independent if and only if they are W* independent, where 𝒜 and ℬ are W* independent if for every normal state ϕ1 on 𝒜 and every normal state ϕ2 on ℬ, there exists a normal state ϕ on 𝒲, such that ϕ|𝒜=ϕ1 and ϕ|ℬ=ϕ2. Bunce and Hamhalter [15] gave some equivalent conditions by the faithfulness of the state on the C* algebras. Jin et al. in [16–18] proved if 𝒜 and ℬ are C* subalgebras of 𝒜, they are C* independent if and only if for any observables A∈𝒜 and B∈ℬ, W(A,B)=W(A)×W(B), where W(A,B) is the joint numerical range of operator tuple (A,B), if and only if for any observables A∈𝒜 and B∈ℬ, σ(A,B)=σ(A)×σ(B), where σ(A,B) is the joint spectrum. By the theorem, we know the states on C*(𝒜,ℬ) are “almost” separable (i.e., not entangled), which will be characterized and more detailed in the following.

The entanglement of n(n≥3) particles is very important, and many problems are different from the case of two particles, such as “the maximal entangled pure states” [1]. To consider the entanglement of n particles in a C* algebra, we introduce the independence of nC* subalgebras as follows.

Definition 1.2.

Let (𝒜i)i=1n be commuting unital C* subalgebras of a C* algebra 𝒜, where 𝒜i is generated by the observable Ai, and they are C* independent if for any state ϕi on 𝒜i, there is a state ϕ on 𝒜, such that ϕ|𝒜i=ϕi.

To study the entanglement of independent quantum systems, equivalent conditions are given for n quantum systems to be independent.

Theorem 1.3.

Let (𝒜i)i=1n be commuting unital C* subalgebras of a C* algebra 𝒜, where 𝒜i is generated by the observable Ai, then the following statements are equivalent.

(𝒜i)i=1n are C* independent.

For any Ai∈𝒜i,
(1.2)W(A1,…,An)=W(A1)×⋯×W(An),

where W(A1,…,An) denotes the joint numerical range.

For any Ai∈𝒜i, Ai≠0(i=1,…,n) imply that ∏i=1nAi≠0.

For any Ai∈𝒜i,
(1.3)σ(A1,…,An)=σ(A1)×⋯×σ(An),

where σ(A1,…,An) denotes the joint spectrum.

Remark 1.4.

By the theorem, it is seen that the separable states on C*(A1,…,An) are dense in the state spaces, since the set of pure states corresponding to points of σ(A1)×⋯×σ(An) is a dense subset of the state spaces. In particular, if C*(A1,…,An) is finite dimensions as a Banach space, every state of
(1.4)C(σ(A1,…,An))=C(σ(A1))⊗⋯⊗C(σ(An))
is a convex combination of pure states, pure states correspond to points of σ(A1)×⋯×σ(An), and point masses are pure product states; thus there is not any entangled states on C*(A1,…,An), which gave us some hints that the entangled states are caused by those nonindependent and noncommutative observables.

2. Some Lemmas and Proof of the Theorem

It is a well-known result by Gelfand and Naimark that if 𝒜 is a unital commutative C* algebra and M𝒜 is its maximal ideal space, then 𝒜 is isometrical *-isomorphism with C(M𝒜). In case of (Ai)i=1n being commutative normal operators, denote by MC*(A) the maximal ideal space of C*(A), and the joint spectrum of A=(A1,…,An) is the set
(2.1)σC*(A)(A1,…,An)={(f(A1),…,f(An))∈ℂn;f∈MC*(A)}.
The joint numerical range of A=(A1,…,An) is the set
(2.2)W(A1,…,An)={(ϕ(A1),…,ϕ(An))∈ℂn;ϕisastateon𝒜}.
Notice that the joint numerical range of A=(A1,…,An) is the joint measurement of commuting observables Ai. It was shown in [19] that σ(A1,…,An) is a nonempty compact set in ℂn.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B19">20</xref>]).

Let A=(A1,…,An) be a commuting n-tuple of adjoint operators in a unital C* algebra 𝒜, then
(2.3)Ex(W(A1,…,An))⊆σ(A1,…,An),
where Ex(X) are the extreme points of the set X.

Proof of Theorem <xref ref-type="statement" rid="thm4">1.3</xref>.

(1) ⇒(2) Let Ai∈𝒜i, and it suffices to prove
(2.4)W(A1,…,An)⊇W(A1)×⋯×W(An).
By the definition,
(2.5)W(Ai)={ϕi(Ai);ϕiisastateon𝒜i}.
It follows by the condition of 𝒜i being C* independent that there is a state on 𝒜, such that ϕ|𝒜i=ϕi, that is ϕi(Ai)=ϕ(Ai), so
(2.6)(ϕ1(A1),…,ϕn(An))=(ϕ(A1),…,ϕ(An))∈W(A1,…,An),
thus
(2.7)W(A1,…,An)⊇W(A1)×⋯×W(An).

(2) ⇒(3) Let Ai∈𝒜i be any nonzero observables and λi=||Ai||, then λi∈Ex(W(Ai)), so
(2.8)(λ1,…,λn)∈Ex(W(A1,…,An)).
By Lemma 2.1,
(2.9)(λ1,…,λn)∈σ(A1,…,An),
so there is a multiplicative linear functional ϕ on C*(A1,…,An), such that
(2.10)ϕ(Ai)=λi,
therefore
(2.11)ϕ(A1⋯An)=ϕ(A1)⋯ϕ(An),
thus
(2.12)∥∏i=1nAi∥≥ϕ(∏i=1nAi)=∏i=1nϕ(Ai)≠0,
that is ∏i=1nAi≠0.

(3) ⇒(4) This easily follows by Urysohn’s lemma.

(4) ⇒(1) By the Gelfand-Naimark theorem, the C* algebra generated by the commuting observables Ai is isometrical *-isomorphism with C(σ(A1,…,An)),
(2.13)γ:C*(A1,…,An)→C(σ(A1,…,An)),
by condition (4), it has
(2.14)C(σ(A1,…,An))=C(σ(A1))⊗⋯⊗C(σ(An)),
where the tensor product is the minimal C* cross-norm.

Let ϕi be any state on 𝒜i, then ϕi·γ-1=ψi is a state on C(σ(Ai)), and it is seen that
(2.15)ψ=ψ1⊗⋯⊗ψn
is a state on C(σ(A1))⊗⋯⊗C(σ(An))=C(σ(A1,…,An)); thus the state ϕ=(ψ1⊗⋯⊗ψn)·γ is a state on C*(𝒜1,…,𝒜n), which satisfies ϕ|𝒜i=ϕi.

Acknowledgments

The work is supported by the China Natural Science Foundation (Grant no. 61201084, 61102149 and 71203042) and the Fundamental Research Funds for the Central Universities (Grant no. HIT NSRIF 2010057, HEUCF100604). The authors thank Professor W. Arveson, Professor E. Karimi, and Professor S. Gharibianthe for helpful discussions.

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