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Journal of Physics A Mathematical and Theoretical

Optimal approximate doubles

Siendong Huang

Published Oct 8, 2009• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1751-8113/42/44/445202

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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.
While all bipartite pure entangled states are known to generate correlations violating a Bell inequality, and are therefore nonlocal, the quantitative relation between pure state entanglement and nonlocality is poorly understood. In fact, some Bell inequalities are maximally violated by non-maximally entangled states and this phenomenon is also observed for other operational measures of nonlocality. In this work, we study a recently proposed measure of nonlocality defined as the probability that a pure state displays nonlocal correlations when subjected to random measurements. We first prove that this measure satisfies some natural properties for an operational measure of nonlocality. Then, we show that for pure states of two qubits the measure is monotonic with entanglement for all correlation two-outcome Bell inequalities: for all these inequalities, the more the state is entangled, the larger the probability to violate them when random measurements are performed. Finally, we extend our results to the multipartite setting.
Nonlocality and quantum entanglement constitute two special aspects of the quantum correlations existing in quantum systems, which are of paramount importance in quantum-information theory. Traditionally, they have been regarded as identical (equivalent, in fact, for pure two qubit states, that is, Gisin's Theorem ), yet they constitute different resources. Describing nonlocality by means of the violation of several Bell inequalities, we obtain by direct optimization those states of two qubits that maximally violate a Bell inequality, in terms of their degree of mixture as measured by either their participation ratio R = 1/Tr(ρ 2 ) or their maximum eigenvalue λ max . This optimum value is obtained as well, which coincides with previous results. Comparison with entanglement is performed too. An example of an application is given in the XY model. In this novel approximation, we also concentrate on the nonlocality for linear combinations of pure states of two qubits, providing a closed form for their maximal nonlocality measure. The case of Bell diagonal mixed states of two qubits is also extensively studied. Special attention concerning the connection between nonlocality and entanglement for mixed states of two qubits is paid to the so-called maximally entangled mixed states. Additional aspects for the case of two qubits are also described in detail. Since we deal with qubit systems, we will perform an analogous study for three qubits, employing similar tools. Relation between distillability and nonlocality is explored quantitatively for the whole space of states of three qubits. We finally extend our analysis to four-qubit systems, where nonlocality for generalized Greenberger–Horne–Zeilinger states of arbitrary number of parties is computed.
We propose a measure to quantify correlations in a bipartite quantum system of two quibits by assessing the minimum difference between outcome states of a subsystem by performing a local measurement on the other subsystem. This maximum similarity measure is a monotone function of the concurrence for pure states of two qubits; for mixed states it accounts for entanglement, dissonance, and classical correlations. Besides, we found a closed formula for evaluating the similarity degree of an arbitrary mix state of two two-dimensional systems.
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.
It is a well-known fact that both quantum entanglement and nonlocality (implied by the violation of Bell inequalities) constitute quantum correlations that cannot be arbitrarily shared among subsystems. They are both monogamous, albeit in a different fashion. In the present contribution we focus on nonlocality monogamy relations such as the Toner-Verstraete, the Seevinck, and a derived monogamy inequality for three parties and compare them with multipartite nonlocality measures for the whole set of pure states distributed according to the Haar measure. In this numerical endeavor, we also see that, although monogamy relations for nonlocality cannot exist for more than three parties, in practice the exploration of the whole set of states for different numbers of qubits will return effective bounds on the maximum value of all bipartite Bell violations among subsystems. Hence, we shed light on the effective nonlocality monogamy bounds in the multiqubit case.