In a recent paper [quant-ph/9910066], Arens and Varadarajan gave a characterization of what they call EPR-states on a bipartite composite quantum system. By definition, such states imply perfect correlation between suitable pairs of observables in the two subsystems, and the task is to determine all such correlated pairs for a given state. In this note the argument is shortened and simplified, and at the same time extended to observables in general von Neumann algebras, which naturally arise in quantum field theory.
In a recent paper [quant-ph/9910066], Arens and Varadarajan gave a characterization of what they call EPR-states on a bipartite composite quantum system. By definition, such states imply perfect correlation between suitable pairs of observables in the two subsystems, and the task is to determine all such correlated pairs for a given state. In this note the argument is shortened and simplified, and at the same time extended to observables in general von Neumann algebras, which naturally arise in quantum field theory.
The identification and physical interpretation of arbitrary quantum correlations are not always effortless. Two features that can significantly influence the dispersion of the joint observable outcomes in a quantum bipartite system composed of systems I and II are: (a) All possible pairs of observables describing the composite are equally probable upon measurement, and (b) The absence of concurrence (positive reinforcement) between any of the observables within a particular system; implying that their associated operators do not commute. The so-called EPR states are known to observe (a). Here, we demonstrate in very general (but straightforward) terms that they also satisfy condition (b), a relevant technical fact often overlooked. As an illustration, we work out in detail the three-level systems, i.e., qutrits. Furthermore, given the special characteristics of EPR states (such as maximal entanglement, among others), one might intuitively expect the CHSH correlation, computed exclusively for the observables of qubit EPR states, to yield values greater than two, thereby violating Bell's inequality. We show such a prediction does not hold true. In fact, the combined properties of (a) and (b) lead to a more limited range of values for the CHSH measure, not surpassing the nonlocality threshold of two. The present constitutes an instructive example of the subtleties of quantum correlations.
In this paper, two conjectures which were proposed in [Phys. Rev. A \textbf{82}, 052122(2010)] on the correlations in a bipartite state ρAB are studied. If the mutual information $I\Pa{\rho^{AB}}$ between two quantum systems A and B before any measurement is considered as the total amount of correlations in the state ρAB, then it can be separated into two parts: classical correlations and quantum correlations. The so-called classical correlations $C\Pa{\rho^{AB}}$ in the state ρAB, defined by the maximizing mutual information between two quantum systems A and B after von Neumann measurements on system B, we show that it is upper bounded by the von Neumann entropies of both subsystems A and B, this answered the conjecture on the classical correlation. If the quantum correlations $Q\Pa{\rho^{AB}}$ in the state ρAB is defined by $Q\Pa{\rho^{AB}}= I\Pa{\rho^{AB}} - C\Pa{\rho^{AB}}$, we show also that it is upper bounded by the von Neumann entropy of subsystem B. It is also obtained that $Q\Pa{\rho^{AB}}$ is upper bounded by the von Neumann entropy of subsystem A for a class of states.
In this paper, two conjectures which were proposed in Luo et al (2010 Phys. Rev. A 82 052122 ) on the correlations in a bipartite state ρ AB are studied. If the mutual information I (ρ AB ) between two quantum systems A and B before any measurement is considered as the total amount of correlations in the state ρ AB , then it can be separated into two parts: classical correlations and quantum correlations. The so-called classical correlations C (ρ AB ) in the state ρ AB are defined by the maximizing mutual information between two quantum systems A and B after von Neumann measurements on system B . We show that it is upper bounded by the von Neumann entropies of both subsystems A and B , which answered the conjecture on the classical correlation. If the quantum correlations Q (ρ AB ) in the state ρ AB are defined by Q (ρ AB ) = I (ρ AB ) − C (ρ AB ), we also show that it is upper bounded by the von Neumann entropy of subsystem B . It is also found that Q (ρ AB ) is upper bounded by the von Neumann entropy of subsystem A for a class of states.
In this paper the notion of an EPR state for the composite S of two quantum systems S1,S2, relative to S2 and a set 𝒪 of bounded observables of S2, is introduced in the spirit of the classical examples of Einstein–Podolsky–Rosen and Bohm. We restrict ourselves mostly to EPR states of finite norm. The main results are contained in Theorems 3–6 and imply that if EPR states of finite norm relative to (S2,O) exist, then the elements of 𝒪 have discrete probability distributions and the Von Neumann algebra generated by 𝒪 is essentially imbeddable inside S1 by an antiunitary map. The EPR states then correspond to the different imbeddings and certain additional parameters, and are explicitly given by formulas which generalize the famous example of Bohm. If 𝒪 generates all bounded observables, S2 must be of finite dimension and can be imbedded inside S1 by an antiunitary map, and the EPR states relative to S2 are then in canonical bijection with the different imbeddings of S2 inside S1; moreover they are then given by formulas which are exactly those of the generalized Bohm states. The notion of EPR states of infinite norm is also explored and it is shown that the original state of Einstein–Podolsky–Rosen can be realized as a renormalized limit of EPR states of finite quantum systems considered by Weyl, Schwinger, and many others. Finally, a family of states of infinite norm generalizing the Einstein–Podolsky–Rosen example is explicitly given.
The nonlocality of quantum states on a bipartite system is tested by comparing probabilistic outcomes of two local observables of different subsystems. For a fixed observable A of the subsystem its optimal approximate double A ' of the other system is defined such that the probabilistic outcomes of A ' are almost similar to those of the fixed observable A . The case of σ-finite standard von Neumann algebras is considered and the optimal approximate double A ' of an observable A is explicitly determined. The connection between optimal approximate doubles and quantum correlations is explained. Inspired by quantum states with perfect correlation, like Einstein–Podolsky–Rosen states and Bohm states, the nonlocality power of an observable A for general quantum states is defined as the similarity that the outcomes of A look like the properties of the subsystem corresponding to A '. As an application of optimal approximate doubles, maximal Bell correlation of a pure entangled state on is found explicitly.
Typicality arguments for emergent spacetime rely on the Lubkin-Page bounds, which show that generic quantum states have vanishing correlations between subsystems. These bounds assume a tensor-product Hilbert space (a Type~I von~Neumann algebra), but the observable algebras in quantum field theory and quantum gravity are generically Type~II or Type~III, raising the question of whether the bounds survive. We prove that they do for all Type~II von~Neumann factors. For the hyperfinite Type~II1 factor with a tripartite decomposition R≅A⊗B⊗E, the mutual information between subsystems A and B vanishes as O((dAdB/dE)2) in finite-dimensional approximations, provided dAdB≤dE (Theorem~1). For Type~II∞ factors, including the gravitational algebras constructed via the crossed-product method by Witten and by Chandrasekaran, Longo, Penington, and Witten, the bound acquires an additional exponential suppression controlled by the Bekenstein-Hawking entropy (Theorem~2). We identify the obstructions to extending the result to Type~III factors and discuss the open question of whether the commutant of the observable algebra can serve as a natural thermal bath that tightens the bound further.