The Entanglement of Independent Quantum Systems
The entanglement of states on -independent subalgebras is considered, and equivalent conditions are given for subalgebras to be independent.
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 and be Hilbert spaces and , and a state of is called to be separable if it is a convex combination of product states , that is Otherwise, is called entanglement.
In the algebraic quantum theory, the observable is represented by an adjoint operator in a algebra. Naturally an interesting problem is raised.
Problem 1. In a algebra , what is the condition of the commuting subalgebras and , under which the entanglement of states can be considered, where is the algebra generated by an observable .
Notice that the problem is to seek the condition of commuting algebras and , such that holds true.
In fact the condition is that and are independent.
Haag and Kastler  introduced a notion called statistical independence. If and represent the algebras generated by the observables associated with two quantum subsystems, the statistical independence of and 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 and in the category of algebras is called independence, which is defined as follows: if for any state on and on , there is a state on , such that and , where denotes the unital algebra generated by .
Roos  gave a characterization of independence. He showed if and are commuting subalgebras of a algebra, then and are independent, if and only if and imply that . Later, Florig and Summers  studied the relation between independence and independence and showed if and are commuting subalgebras of a -finite algebra , then they are independent if and only if they are independent, where and are independent if for every normal state on and every normal state on , there exists a normal state on , such that and . Bunce and Hamhalter  gave some equivalent conditions by the faithfulness of the state on the algebras. Jin et al. in [16–18] proved if and are subalgebras of , they are independent if and only if for any observables and , , where is the joint numerical range of operator tuple , if and only if for any observables and , , where is the joint spectrum. By the theorem, we know the states on are “almost” separable (i.e., not entangled), which will be characterized and more detailed in the following.
The entanglement of particles is very important, and many problems are different from the case of two particles, such as “the maximal entangled pure states” . To consider the entanglement of particles in a algebra, we introduce the independence of subalgebras as follows.
Definition 1.2. Let be commuting unital subalgebras of a algebra , where is generated by the observable , and they are independent if for any state on , there is a state on , such that .
To study the entanglement of independent quantum systems, equivalent conditions are given for quantum systems to be independent.
Theorem 1.3. Let be commuting unital subalgebras of a algebra , where is generated by the observable , then the following statements are equivalent. (1) are independent.(2) For any , where denotes the joint numerical range.(3) For any , imply that .(4) For any , where denotes the joint spectrum.
Remark 1.4. By the theorem, it is seen that the separable states on are dense in the state spaces, since the set of pure states corresponding to points of is a dense subset of the state spaces. In particular, if is finite dimensions as a Banach space, every state of is a convex combination of pure states, pure states correspond to points of , and point masses are pure product states; thus there is not any entangled states on , 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 is a unital commutative algebra and is its maximal ideal space, then is isometrical -isomorphism with . In case of being commutative normal operators, denote by the maximal ideal space of , and the joint spectrum of is the set The joint numerical range of is the set Notice that the joint numerical range of is the joint measurement of commuting observables . It was shown in  that is a nonempty compact set in .
Lemma 2.1 (see ). Let be a commuting -tuple of adjoint operators in a unital algebra , then where are the extreme points of the set .
Proof of Theorem 1.3. (1) (2) Let , and it suffices to prove
By the definition,
It follows by the condition of being independent that there is a state on , such that , that is , so
(2) (3) Let be any nonzero observables and , then , so By Lemma 2.1, so there is a multiplicative linear functional on , such that therefore thus that is .
(3) (4) This easily follows by Urysohn’s lemma.
(4) (1) By the Gelfand-Naimark theorem, the algebra generated by the commuting observables is isometrical -isomorphism with , by condition (4), it has where the tensor product is the minimal cross-norm.
Let be any state on , then is a state on , and it is seen that is a state on ; thus the state is a state on , which satisfies .
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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