Lower bounds on concurrence and negativity from a trace inequality

Abstract

For bipartite quantum states we obtain lower bounds on two important entanglement measures, concurrence and negativity, studying the inequalities for the expectation value of a projector on some subspace of the Hilbert space. Several applications, including analysis of stability of entanglement under various perturbations of a state, are discussed.

Construction of genuinely multipartite entangled subspaces and the associated bounds on entanglement measures for mixed states

“…The extension of this witness to the GME case was obtained in Ref [12]. In connection with the (bipartite) witness also lower bounds on some entanglement measures for bipartite mixed states can be derived [22]. We aim to extend these bounds to the GME case and use the examples of GESs we construct to illustrate their application.…”

“…is known [48], [22] that the maximal first Schmidt coefficient over all states in this subspace is given by…”

“…Obtained in Ref. [22] lower bounds on the concurrence and the convex-roof extended negativity of an arbitrary bipartite mixed state ρ are determined by its overlap with some completely entangled subspace W of a d× d Hilbert space. For the concurrence the bound reads…”

“…It is given by p < 9/16 = 0.5625. Following Ref [22], it is also interesting to analyze the case when the expression for N is unknown and only the spectrum of the noise is given (robustness from spectrum). The immediate generalization of eq.…”

Tensor products of entangled subspaces and their properties from diagrammatic constructions

“…Let us apply Lemma 5 and condition (31) to the state ρ W (s 1 , d) ⊗ ρ W (s 2 , d), with both W 1 and W 2 chosen to be the antisymmetric subspace A of C d ⊗ C d , which has dimension equal to d(d − 1)/2. It is known [32] that the geometric measure G(A) = 1/2 (see also [33]). From Eq.…”