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Scientific Reports

Indecomposability of entanglement witnesses constructed from symmetric measurements

Katarzyna Siudzińska

Published Jun 24, 2022• 11 citations
Publisher

Abstract

We propose a family of positive maps constructed from a recently introduced class of symmetric measurements. These maps are used to define entanglement witnesses, which include other popular approaches with mutually unbiased bases and mutually unbiased measurements. A particular interest is given to indecomposable witnesses that can be used to detect entanglement of quantum states with positive partial transposition. We present several examples for different number of measurements.

DOI
10.1038/s41598-022-14920-5

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We propose a family of positive maps constructed from a recently introduced class of symmetric measurements. These maps are used to define entanglement witnesses, which include other popular approaches with mutually unbiased bases and mutually unbiased measurements. A particular interest is given to indecomposable witnesses that can be used to detect entanglement of quantum states with positive partial transposition. We present several examples for different number of measurements.
A broad class of informationally complete symmetric measurements is introduced. It can be understood as a common generalization of symmetric, informationally complete positive operator-valued measures and mutually unbiased bases. Additionally, it provides a natural way to define two more families of mutually unbiased symmetric measurement operators in any finite dimension. We show a general method of their construction, together with an example of an optimal measurement. Finally, we analyze the properties of symmetric measurements and provide applications in entropic relations and entanglement detection.
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Abstract From a practical perspective it is advantageous to develop methods that verify entanglement in quantum states with as few measurements as possible. In this paper we investigate the minimal number of mutually unbiased bases (MUBs) needed to detect bound entanglement in bipartite ( d × d ) -dimensional states, i.e. entangled states that are positive under partial transposition. In particular, we show that a class of entanglement witnesses (EWs) composed of MUBs can detect bound entanglement if the number of measurements is greater than d / 2 + 1 . This is a substantial improvement over other detection methods, requiring significantly fewer resources than either full quantum state tomography or measuring a complete set of d + 1 MUBs. Our approach is based on a partial characterisation of the (non-)decomposability of EWs. We show that non-decomposability is a universal property of MUBs, which holds regardless of the choice of complementary observables, and we find that both the number of measurements and the structure of the witness play an important role in the detection of bound entanglement.
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)].