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arXiv:quant-ph/0310166 · International Journal of Quantum Information

BELL'S INEQUALITIES DETECT EFFICIENT ENTANGLEMENT

Antonio Acín, Nicolas Gisin, Lluís Masanes, Valerio Scarani

Published Mar 1, 2004• 38 citations
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Abstract

We review the status of Bell's inequalities in quantum information, stressing mainly the links with quantum key distribution and distillation of entanglement. We also prove that for all the eavesdropping attacks using one qubit, and for a family of attacks of two qubits, acting on half of a maximally entangled state of two qubits, the violation of a Bell inequality implies the possibility of an efficient secret-key extraction.

DOI
10.1142/s0219749904000043

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We review the status of Bell's inequalities in quantum information, stressing mainly the links with quantum key distribution and distillation of entanglement. We also prove that for all the eavesdropping attacks using one qubit, and for a family of attacks of two qubits, acting on half of a maximally entangled state of two qubits, the violation of a Bell inequality implies the possibility of an efficient secret-key extraction.
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Detection and quantification of entanglement in quantum resources are two key steps in the implementation of various quantum-information processing tasks. Here, we show that Bell-type inequalities are not only useful in verifying the presence of entanglement but can also be used to bound the entanglement of the underlying physical system. Our main tool consists of a family of Clauser-Horne-like Bell inequalities that cannot be violated maximally by any finite-dimensional maximally entangled state. Using these inequalities, we demonstrate the explicit construction of both lower and upper bounds on the concurrence for two-qubit states. The fact that these bounds arise from Bell-type inequalities also allows them to be obtained in a semi-device-independent manner, that is, with assumption of the dimension of the Hilbert space but without resorting to any knowledge of the actual measurements being performed on the individual subsystems.
The relation between Bell's inequalities with two two-outcome measurements per site and distillability is analyzed in systems of an arbitrary number of quantum bits. We observe that the violation of any of these inequalities by a quantum state implies that pure-state entanglement can be distilled from it. The corresponding distillation protocol may require that some of the parties join into several groups. We show that there exists a link between the amount of the Bell inequality violation and the size of the groups they have to form for distillation. Thus, a strong violation is always sufficient for full N-partite distillability. This result also allows for a security proof of multipartite quantum key distribution protocols.
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.