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arXiv:1604.03282 · Modern Physics Letters A

Exact algebraic separability criterion for two-qubit systems

Kazuo Fujikawa, C. H. Oh

Published Mar 29, 2017• 1 citation

Abstract

A conceptually simpler proof of the separability criterion for two-qubit systems, which is referred to as “Hefei inequality” in literature, is presented. This inequality gives a necessary and sufficient separability criterion for any mixed two-qubit system unlike the Bell-CHSH inequality that cannot test the mixed-states such as the Werner state when regarded as a separability criterion. The original derivation of this inequality emphasized the uncertainty relation of complementary observables, but we show that the uncertainty relation does not play any role in the actual derivation and the Peres–Hodrodecki condition is solely responsible for the inequality. Our derivation, which contains technically novel aspects such as an analogy to the Dirac equation, sheds light on this inequality and on the fundamental issue to what extent the uncertainty relation can provide a test of entanglement. This separability criterion is illustrated for an exact treatment of the Werner state.

DOI: 10.1142/s0217732317500705

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We formulate an entanglement criterion using Peres-Horodecki positive partial transpose operations combined with the Schr\"odinger-Robertson uncertainty relation. We show that any pure entangled bipartite and tripartite state can be detected by experimentally measuring mean values and variances of specific observables. Those observables must satisfy a specific condition in order to be used, and we show their general form in the $2\ifmmode\times\else\texttimes\fi{}2$ (two qubits) dimension case. The criterion is applied on a variety of physical systems including bipartite and multipartite mixed states and reveals itself to be stronger than the Bell inequalities and other criteria. The criterion also work on continuous variable cat states and angular momentum states of the radiation field.

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