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Bound entanglement can maximally violate Bell inequalities

Remigiusz Augusiak, Paweł Horodecki

Published May 30, 2004• 0 citations
OpenAlex

Abstract

It is shown that some four-qubit bound entangled states can maximally violate two-setting Bell inequality similar to standard CHSH inequality. Surprisingly this entanglement does not allow for secure key distillation, so neither entanglement nor violation of Bell inequalities implies quantum security. It is also pointed out how that kind of bound entanglement can be useful in reducing communication complexity.

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It is shown that some four-qubit bound entangled states can maximally violate two-setting Bell inequality similar to standard CHSH inequality. Surprisingly this entanglement does not allow for secure key distillation, so neither entanglement nor violation of Bell inequalities implies quantum security. It is also pointed out how that kind of bound entanglement can be useful in reducing communication complexity.
It is shown that Smolin four-qubit bound entangled states [J. A. Smolin, Phys. Rev. A 63, 032306 (2001)] can maximally violate the simple two-setting Bell inequality similar to the standard Clauser-Horne-Shimony-Holt (CHSH) inequality. The simplicity of the setting and the robustness of the entanglement make it promising for current experimental technology. On the other hand, the entanglement does not allow for secure key distillation, so neither entanglement nor maximal violation of Bell inequalities implies directly the presence of a quantum secure key. As a result, one concludes that two tasks---reducing of communication complexity and cryptography---are not (even qualitatively) equivalent in a quantum multipartite scenario.
We study the relation between violation of Bell inequalities and distillability properties of quantum states. Recently, Dür [Phys. Rev. Lett. 87, 230402 (2001)] has shown that there are some multiparticle bound entangled states which are nonseparable and nondistillable, that violate a Bell inequality. We prove that for all the states violating this inequality there exists at least one splitting of the parties into two groups such that some pure-state entanglement can be distilled, obtaining a connection between Bell inequalities and bipartite distillable entanglement.
In the case of bipartite two-qubit systems, we derive an analytical expression of bound Bell operator for any given pure state. Our result not only manifests some properties of Bell inequality, for example, which may be violated by any pure entangled state and only be maximally violated for a maximally entangled state, but also gives the explicit values of maximal violation for any pure state. Finally we point out that any mixed states which can produce maximal violation of Bell inequality must have a maximal concurrence value.
It is well known that the violation of Bell's inequality in the form given by Clauser, Horne, Shimony, and Holt (CHSH) in two-qubit systems requires entanglement, but not vice versa, i.e., there are entangled states which do not violate the CHSH inequality. Here we compare some standard entanglement measures with violations of the CHSH inequality (as given by the Horodecki measure) for two-qubit states generated by Monte Carlo simulations. We describe states that have extremal entanglement according to the negativity, concurrence, and relative entropy of entanglement for a given value of the CHSH violation. We explicitly find these extremal states by applying the generalized method of Lagrange multipliers based on the Karush-Kuhn-Tucker conditions. The found minimal and maximal states define the range of entanglement accessible for any two-qubit states that violate the CHSH inequality by the same amount. We also find extremal states for the concurrence versus negativity by considering only such states which do not violate the CHSH inequality. Furthermore, we describe an experimentally efficient linear-optical method to determine the highest Horodecki degree of the CHSH violation for arbitrary mixed states of two polarization qubits. By assuming to have access simultaneously to two copies of the states, our method requires only six discrete measurement settings instead of nine settings, which are usually considered.