Paradoxes of measures of quantum entanglement and Bell's inequality violation in two-qubit systems

TL;DR

This paper reviews counterintuitive properties of quantum entanglement measures and Bell's inequality violation in two-qubit systems, highlighting ambiguities and the importance of operational measures through extensive simulations.

Contribution

It reveals ambiguities in entanglement measures and emphasizes the need for operationally-defined measures over standard formal ones in quantum systems.

Findings

Entanglement measures can order states ambiguously.

Robustness of entanglement varies in lossy systems.

Some mixed states are more entangled than pure states with same negativity.

Abstract

We review some counterintuitive properties of standard measures describing quantum entanglement and violation of Bell's inequality (often referred to as "nonlocality") in two-qubit systems. By comparing the nonlocality, negativity, concurrence, and relative entropy of entanglement, we show: (i) ambiguity in ordering states with the entanglement measures, (ii) ambiguity of robustness of entanglement in lossy systems and (iii) existence of two-qubit mixed states more entangled than pure states having the same negativity or nonlocality. To support our conclusions, we performed a Monte Carlo simulation of $10^6$ two-qubit states and calculated all the entanglement measures for them. Our demonstration of the relativity of entanglement measures implies also how desirable is to properly use an operationally-defined entanglement measure rather than to apply formally-defined standard measures. In fact, the problem of estimating the degree of entanglement of a bipartite system cannot be analyzed separately from the measurement process that changes the system and from the intended application of the generated entanglement.