Towards measurable bounds on entanglement measures

TL;DR

This paper develops methods to establish measurable bounds on entanglement measures, especially concurrence, using positive maps and two-copy observables, improving detection capabilities for quantum entanglement.

Contribution

It introduces a general framework for constructing measurable lower bounds on concurrence from positive maps, extending previous bounds and including the transposition map case.

Findings

New bounds are positive where previous bounds are zero.

The method generalizes the Mintert--Buchleitner bound.

Upper bounds on all concurrences are provided.

Abstract

While the experimental detection of entanglement provides already quite a difficult task, experimental quantification of entanglement is even more challenging, and has not yet been studied thoroughly. In this paper we discuss several issues concerning bounds on concurrence measurable collectively on copies of a given quantum state. Firstly, we concentrate on the recent bound on concurrence by Mintert--Buchleitner [F. Mintert and A. Buchleitner, Phys. Rev. Lett. 98, 140505 (2007)]. Relating it to the reduction criterion for separability we provide yet another proof of the bound and point out some possibilities following from the proof which could lead to improvement of the bound. Then, relating concurrence to the generalized robustness of entanglement, we provide a method allowing for construction of lower bounds on concurrence from any positive map (not only the reduction one). All these quantities can be measured as mean values of some two--copy observables. In this sense the method generalizes the Mintert--Buchleitner bound and recovers it when the reduction map is used. As a particular case we investigate the bound obtained from the transposition map. Interestingly, comparison with MB bound performed on the class of 4\otimes 4 rotationally invariant states shows that the new bound is positive in regions in which the MB bound gives zero. Finally, we provide measurable upper bounds on the whole class of concurrences.