Exact algebraic separability criterion for two-qubit systems

TL;DR

This paper presents a simpler proof of the Hefei inequality, providing a necessary and sufficient algebraic separability criterion for all mixed two-qubit states, improving upon Bell-CHSH limitations.

Contribution

The paper offers a new, simpler derivation of the Hefei inequality that does not rely on the uncertainty relation, highlighting the role of the Peres-Hodrodecki condition.

Findings

The inequality is necessary and sufficient for two-qubit separability.

It accurately tests mixed states like Werner states.

The derivation introduces an analogy to the Dirac equation.

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.