Entanglement estimation from Bell inequality violation

TL;DR

This paper investigates the relationship between entanglement measures and Bell inequality violations in two-qubit states, identifying extremal states and proposing an efficient optical measurement method.

Contribution

It introduces a method to find extremal entangled states for given Bell violations and proposes a simplified optical measurement technique for experimental detection.

Findings

Identified extremal states with maximum and minimum entanglement for given Bell violations.

Established the range of entanglement accessible for two-qubit states violating CHSH.

Proposed a six-setting optical measurement scheme for efficient Bell violation detection.

Abstract

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.