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arXiv:1006.3059 · Physical Review A

Constructing optimal entanglement witnesses. II. Witnessing entanglement in 4N×4N systems

Dariusz Chruściński, Justyna Pytel

Published Nov 15, 2010• 32 citations
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Abstract

We provide a class of optimal nondecomposable entanglement witnesses for $4N\ifmmode\times\else\texttimes\fi{}4N$ composite quantum systems or, equivalently, another construction of nondecomposable positive maps in the algebra of $4N\ifmmode\times\else\texttimes\fi{}4N$ complex matrices. This construction provides natural generalization of the Robertson map. It is shown that their structural physical approximations give rise to entanglement breaking channels.

DOI
10.1103/physreva.82.052310

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We provide a class of optimal nondecomposable entanglement witnesses for 4N\ifmmode×\else\texttimes\fi4N4N\ifmmode\times\else\texttimes\fi{}4N4N\ifmmode×\else\texttimes\fi4N composite quantum systems or, equivalently, another construction of nondecomposable positive maps in the algebra of 4N\ifmmode×\else\texttimes\fi4N4N\ifmmode\times\else\texttimes\fi{}4N4N\ifmmode×\else\texttimes\fi4N complex matrices. This construction provides natural generalization of the Robertson map. It is shown that their structural physical approximations give rise to entanglement breaking channels.
We provide a class of indecomposable entanglement witnesses. In $4\ifmmode\times\else\texttimes\fi{}4$ case, it reproduces the well-known Breuer-Hall witness. We prove that these witnesses are optimal and atomic, i.e., they are able to detect the ``weakest'' quantum entanglement encoded into states with positive partial transposition. Equivalently, we provide a construction of indecomposable atomic maps in the algebra of $2k\ifmmode\times\else\texttimes\fi{}2k$ complex matrices. It is shown that their structural physical approximations give rise to entanglement breaking channels. This result supports recent conjecture by Korbicz et al. [Phys. Rev. A 78, 062105 (2008)].
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We provide a recurrent construction of entanglement witnesses for a bipartite systems living in a Hilbert space corresponding to 2N2N2N qubits (NNN qubits in each subsystem). Our construction provides a method of generalization of the Robertson map that naturally meshes with 2N2N2N-qubit systems, i.e., its structure respects the 22N{2}^{2N}22N growth of the state space. We prove that for N>1N>1N>1 these witnesses are indecomposable and optimal. As a byproduct we provide a family of PPT (Positive Partial Transpose) entangled states.
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).