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arXiv:2305.14226

Bipartite entanglement detection by local generalized measurements

Maximilian Schumacher, G. Alber

Published May 23, 2023• 0 citations
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Abstract

Entanglement detection by local measurements, which can possibly be performed by far distant observers, are of particular interest for applications in quantum key distribution and quantum communication. In this paper sufficient conditions for arbitrary dimensional bipartite entanglement detection based on correlation matrices and joint probability distributions of such local measurements are investigated. In particular, their dependence on the nature of the local measurements is explored for typical bipartite quantum states and for measurements involving local orthonormal hermitian operators bases (LOOs) or generalized measurements based on informationally complete positive operator valued measures of the recently introduced $(N,M)$-type ($(N,M)$-POVMs) \cite{NMPOVM}. It is shown that symmetry properties of $(N,M)$-POVMs imply that sufficient conditions for bipartite entanglement detection exhibit peculiar scaling properties relating different equally efficient local entanglement detection scenarios. For correlation-matrix based bipartite local entanglement detection, for example, this has the consequence that LOOs and all informationally complete $(N,M)$-POVMs are equally powerful. With the help of a hit-and-run Monte-Carlo algorithm the effectiveness of local entanglement detection of typical bipartite quantum states is explored numerically. For this purpose Euclidean volume ratios between locally detectable entangled states and all bipartite quantum states are determined.

DOI
10.48550/arxiv.2305.14226

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Entanglement detection by local measurements, which can possibly be performed by far distant observers, are of particular interest for applications in quantum key distribution and quantum communication. In this paper sufficient conditions for arbitrary dimensional bipartite entanglement detection based on correlation matrices and joint probability distributions of such local measurements are investigated. In particular, their dependence on the nature of the local measurements is explored for typical bipartite quantum states and for measurements involving local orthonormal hermitian operators bases (LOOs) or generalized measurements based on informationally complete positive operator valued measures of the recently introduced (N,M)(N,M)(N,M)-type ((N,M)(N,M)(N,M)-POVMs) \cite{NMPOVM}. It is shown that symmetry properties of (N,M)(N,M)(N,M)-POVMs imply that sufficient conditions for bipartite entanglement detection exhibit peculiar scaling properties relating different equally efficient local entanglement detection scenarios. For correlation-matrix based bipartite local entanglement detection, for example, this has the consequence that LOOs and all informationally complete (N,M)(N,M)(N,M)-POVMs are equally powerful. With the help of a hit-and-run Monte-Carlo algorithm the effectiveness of local entanglement detection of typical bipartite quantum states is explored numerically. For this purpose Euclidean volume ratios between locally detectable entangled states and all bipartite quantum states are determined.
Motivated by the need for efficient local entanglement detection for applications in quantum information processing, sufficient conditions for arbitrary-dimensional local bipartite entanglement detection based on correlation matrices and joint probability distributions are investigated. In particular, their dependence on the nature of different classes of local measurements is explored for generalized measurements based on informationally complete (N,M)(N,M)(N,M) positive-operator-valued measures (POVMs) [Siudzinska, Phys. Rev. A 105, 042209 (2022)]. It is shown that symmetry properties of (N,M)(N,M)(N,M) POVMs necessarily imply that these sufficient conditions for bipartite entanglement detection exhibit characteristic scaling properties relating equivalent sufficient conditions. Based on these general scaling properties, the efficiency of different classes of local quantum measurement detecting typical bipartite entanglement is investigated quantitatively. For this purpose Euclidean volume ratios between locally detectable bipartite entangled states and all bipartite quantum states are determined numerically with the help of a Monte Carlo algorithm. Our results demonstrate that physically realizable (N,M)(N,M)(N,M) POVMs are sufficient for optimal local entanglement detection. In particular, this implies that for this purpose the construction of optimal (N,M)(N,M)(N,M) POVMs is not necessary. As questions concerning the existence and construction of optimal (N,M)(N,M)(N,M) POVMs are still largely open, this may offer interesting perspectives for practical applications in quantum information processing.
Abstract Recently proposed correlation-matrix-based sufficient conditions for bipartite steerability from Alice to Bob for arbitrary measurements are applied to local informationally complete positive operator valued measures (POVMs) of the (N,M)-type. These POVMs include a large class of local generalized measurements of current interest [Siudzińska K 2022 Phys. Rev. A 105 042209]. It is shown that the trace norm of correlation matrices with local (N,M)-POVMs is proportional to that of local orthonormal hermitian operator bases (LOOs). This implies that all types of informationally complete (N,M)-POVMs are equally powerful in detecting bipartite steerability from Alice to Bob and, in addition, they are as powerful as LOOs. In order to explore the typicality of steering numerical calculations of lower bounds on Euclidean volume ratios between steerable bipartite quantum states from Alice to Bob and all quantum states are determined with the help of a hit-and-run Monte-Carlo algorithm. These results demonstrate that with the single exception of two qubits this correlation-matrix-based sufficient condition significantly underestimates these volume ratios. These results are also compared to a recently proposed method that determines bipartite steerability from Alice’s qubit to Bob’s arbitrary dimensional quantum system by bipartite entanglement detection. It is demonstrated that in general this method is significantly more effective in detecting typical steerability provided non local entanglement detection methods are used which transcend local measurements.
Abstract Highly symmetric quantum measurements, such as mutually unbiased measurements (MUMs) and general symmetric informationally complete positive-operator-valued measures (GSIC-POVMs), play an important role in both foundational and practical aspects of quantum information theory. Recently, a broad class of symmetric measurements were introduced [K Siudzińska, (2022) Phys. Rev. A 105, 042209], which can be viewed as a common generalization of MUMs and GSIC-POVMs. In this work, the role of these symmetric measurements in entanglement detection is studied. More specifically, based on the correlation matrices defined via (informationally complete) symmetric measurements, a separability criterion for arbitrary dimensional bipartite systems is proposed. It is shown that the criterion is stronger than the method provided by Siudzińska, meanwhile, it can unify several popular separability criteria based on MUMs or GSIC-POVMs. Furthermore, using these (informationally complete) symmetric measurements, two efficient criteria are presented to detect the entanglement of tripartite quantum states. The detection power and advantages of these criteria are illustrated through several examples.
Existing methods for deciding whether a bipartite quantum state is separable or entangled typically fall into one of two categories: they are either complete but require access to an explicit density matrix followed by numerical optimization, or they can be evaluated directly by measuring the quantum system but are incomplete, in the sense that they cannot detect all forms of entanglement. In this work, we overcome both limitations in a unified framework. First, we bypass numerical optimization by deriving separability criteria in the form of universal bounds on tensor powers of separable states. We prove that these bounds are complete: every entangled state violates them for sufficiently large tensor powers. Second, we explicitly construct a corresponding complete family of nonlinear entanglement witnesses, which can detect all forms of entanglement without requiring an explicit density matrix. The witnesses we construct are moreover basis-independent, in the sense that they are invariant under conjugation by local unitaries. Altogether, our results expand the toolbox for entanglement detection in arbitrary local dimensions in a manifestly invariant way.
We study multipartite entanglement and genuine tripartite entanglement based on general symmetric informationally complete positive operator valued measurements (GSIC-POVMs). By representing the bipartite density matrices in terms of GSIC-POVMs, we obtain the lower bound of the sum of squares of the corresponding probability. We then construct a special matrix with the correlation probability of GSIC-POVMs to derive useful and operational criteria to detect genuine tripartite entanglement. We also generalize the results to obtain a sufficient criterion to detect entanglement for multipartite quantum states in arbitrary dimensions. Detailed examples show that the new method can detect more entangled and genuine entangled states than previous criteria.