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arXiv:2112.15183 · Physical Review A

Class of Bell-diagonal entanglement witnesses in C4⊗C4 : Optimization and the spanning property

Anindita Bera, Filip Wudarski, Gniewomir Sarbicki, Dariusz Chruściński

Published May 2, 2022• 11 citations
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Abstract

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.

DOI
10.1103/physreva.105.052401

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Two classes of Bell-diagonal indecomposable entanglement witnesses in C4\ensuremath⨂C4{\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}C4\ensuremath⨂C4 are considered. Within the first class, we find a generalization of the well-known Choi witness from C3\ensuremath⨂C3{\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}C3\ensuremath⨂C3, while the second one contains the reduction map. Interestingly, contrary to the C3\ensuremath⨂C3{\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}C3\ensuremath⨂C3 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.
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 Hm,n{\mathcal{H}}_{m,n}Hm,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 Hm,n{\mathcal{H}}_{m,n}Hm,n​ with $m,n\ensuremath{\ge}3$.
It has been shown that finding generic Bell-state diagonal-entanglement witnesses for ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{d}_{n}$ systems reduces to linear programming if the feasible region is a polygon by itself, and it can be solved approximately via linear programming if the feasible region is encircled by a polygon. Since solving linear programming for the generic case is difficult, multiqubit, $2\ensuremath{\bigotimes}N$ and $3\ensuremath{\bigotimes}3$ systems for the special case of generic Bell-state diagonal-entanglement witnesses for some particular choice of parameters have been considered. We obtain the optimal nondecomposable entanglement witness for a $3\ensuremath{\bigotimes}3$ system for some particular choice of parameters. By proving the optimality of the well-known reduction map and combining it with the optimal and nondecomposable $3\ensuremath{\bigotimes}3$ Bell-state diagonal-entanglement witnesses (named critical entanglement witnesses) the family of optimal and nondecomposable $3\ensuremath{\bigotimes}3$ Bell-state diagonal-entanglement witnesses has also been obtained. Using the approximately critical entanglement witnesses, some $3\ensuremath{\bigotimes}3$ bound entangled states are so detected. So the well-known Choi map as a particular case of the positive map in connection with this witness via Jamiolkowski isomorphism has been considered.
If one wants to establish optimality of a given bipartite entanglement witness, the current standard approach is to check whether it has the spanning property. Although this is not necessary for optimality, it is most often satisfied in practice, and for small enough dimensions or sufficiently structured witnesses this criterion can be checked by hand. In this work we introduce a novel characterization of the spanning property via entanglement-breaking channels, which in turn leads to a new sufficient criterion for optimality. This criterion amounts to just checking the kernel of some bipartite state. It is slightly weaker than the spanning property, but it is a lot easier to test for -- by hand as well as numerically -- and it applies to almost all witnesses which are known to have the spanning property. A second criterion is derived from this, where one can simply compute the expectation value of the given witness on a maximally entangled state. Finally, this approach implies new spectral constraints on witnesses as well as on positive maps.
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).