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arXiv:1107.0505 · Physical Review A

Optimal decomposable witnesses without the spanning property

Remigiusz Augusiak, Gniewomir Sarbicki, Maciej Lewenstein

Published Nov 21, 2011• 24 citations

Abstract

One of the unsolved problems in the characterization of the optimal entanglement witnesses is the existence of optimal witnesses acting on bipartite Hilbert spaces ${\mathcal{H}}_{m,n}={\mathbb{C}}^{m}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$ such that the product vectors obeying $\ensuremath{\langle}e,f|W|e,f\ensuremath{\rangle}=0$ do not span ${\mathcal{H}}_{m,n}$. So far, the only known examples of such witnesses were found among indecomposable witnesses, one of them being the witness corresponding to the Choi map. However, it remains an open question whether decomposable witnesses exist without the property of spanning. Here we answer this question affirmatively, providing systematic examples of such witnesses. Then, we generalize some of the recently obtained results on the characterization of $2\ensuremath{\bigotimes}n$ optimal decomposable witnesses [R. Augusiak et al., J. Phys. A 44, 212001 (2011)] to finite-dimensional Hilbert spaces ${\mathcal{H}}_{m,n}$ with $m,n\ensuremath{\ge}3$.

DOI: 10.1103/physreva.84.052323

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One of the unsolved problems in the characterization of the optimal entanglement witnesses is the existence of optimal witnesses acting on bipartite Hilbert spaces

{\mathcal{H}}_{m,n}={\mathbb{C}}^{m}\ensuremath{\bigotimes}{\mathbb{C}}^{n}

such that the product vectors obeying

\ensuremath{\langle}e,f|W|e,f\ensuremath{\rangle}=0

do not span

{\mathcal{H}}_{m,n}

. So far, the only known examples of such witnesses were found among indecomposable witnesses, one of them being the witness corresponding to the Choi map. However, it remains an open question whether decomposable witnesses exist without the property of spanning. Here we answer this question affirmatively, providing systematic examples of such witnesses. Then, we generalize some of the recently obtained results on the characterization of

2\ensuremath{\bigotimes}n

optimal decomposable witnesses [R. Augusiak et al., J. Phys. A 44, 212001 (2011)] to finite-dimensional Hilbert spaces

{\mathcal{H}}_{m,n}

with

m,n\ensuremath{\ge}3

.

Two classes of Bell-diagonal indecomposable entanglement witnesses in ${\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}$ are considered. Within the first class, we find a generalization of the well-known Choi witness from ${\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$, while the second one contains the reduction map. Interestingly, contrary to the ${\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$ case, the generalized Choi witnesses are no longer optimal. We perform an optimization procedure of finding spanning vectors that eventually gives rise to optimal witnesses. Operators from the second class turn out to be optimal, however, without the spanning property. This analysis sheds light onto the intricate structure of optimal entanglement witnesses.

Optimal PPTES witnesses for states in

\mathbb C^n\otimes \mathbb C^n

In this paper, we present a characterization of optimal entanglement witnesses in terms of positive maps and then provide a general method of checking optimality of entanglement witnesses. Applying it, we obtain indecomposable optimal witnesses that have no spanning property. These also provide examples that support a recent conjecture saying that the so-called structural physical approximations to optimal positive maps (optimal entanglement witnesses) give entanglement breaking maps (separable states).

Entanglement witnesses (EWs) constitute one of the most important entanglement detectors in quantum systems. Nevertheless, their complete characterization, in particular with respect to the notion of optimality, is still missing, even in the decomposable case. Here we show that for any qubit–qunit decomposable EW (DEW) W , the three statements are equivalent: (i) the set of product vectors obeying ⟨ e , f | W | e , f ⟩ = 0 spans the corresponding Hilbert space, (ii) W is optimal, and (iii) W = Q Γ , with Q denoting a positive operator supported on a completely entangled subspace (CES) and Γ standing for the partial transposition. While implications (i)⇒(ii) and (ii)⇒(iii) are known, here we prove that (iii) implies (i). This is a consequence of a more general fact saying that product vectors orthogonal to any CES in span after partial conjugation the whole space. On the other hand, already in the case of the Hilbert space, there exist DEWs for which (iii) does not imply (i). Consequently, either (i) does not imply (ii) or (ii) does not imply (iii), and the above transparent characterization, obeyed by qubit–qunit DEWs, does not hold in general.