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Register a new userVerification of joint measurability using phase-space quasiprobability distributions
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Added by
Saleh Rahimi-Keshari
Publication date
2020
fields
Physics
and research's language is
English
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Measurement incompatibility is a distinguishing property of quantum physics and an essential resource for many quantum information processing tasks. We introduce an approach to verify the joint measurability of measurements based on phase-space quasiprobability distributions. Our results therefore establish a connection between two notions of non-classicality, namely the negativity of quasiprobability distributions and measurement incompatibility. We show how our approach can be applied to the study of incompatibility-breaking channels and derive incompatibility-breaking sufficient conditions for bosonic systems and Gaussian channels. In particular, these conditions provide useful tools for investigating the effects of errors and imperfections on the incompatibility of measurements in practice. To illustrate our method, we consider all classes of single-mode Gaussian channels. We show that pure lossy channels with 50% or more losses break the incompatibility of all measurements that can be represented by non-negative Wigner functions, which includes the set of Gaussian measurements.
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Occupying a position between entanglement and Bell nonlocality, Einstein-Podolsky-Rosen (EPR) steering has attracted increasing attention in recent years. Many criteria have been proposed and experimentally implemented to characterize EPR-steering. Nevertheless, only a few results are available to quantify steerability using analytical results. In this work, we propose a method for quantifying the steerability in two-qubit quantum states in the two-setting EPR-steering scenario, using the connection between joint measurability and steerability. We derive an analytical formula for the steerability of a class of X-states. The sufficient and necessary conditions for two-setting EPRsteering are presented. Based on these results, a class of asymmetric states, namely, one-way steerable states, are obtained.
A mapping between operators on the Hilbert space of $N$-dimensional quantum system and the Wigner quasiprobability distributions defined on the symplectic flag manifold is discussed. The Wigner quasiprobability distribution is constructed as a dual pairing between the density matrix and the Stratonovich-Weyl kernel. It is shown that the moduli space of the Stratonovich-Weyl kernel is given by an intersection of the coadjoint orbit space of the $SU(N)$ group and a unit $(N-2)$-dimensional sphere. The general consideration is exemplified by a detailed description of the moduli space of 2, 3 and 4-dimensional systems.
In order to analyze joint measurability of given measurements, we introduce a Hermitian operator-valued measure, called $W$-measure, such that it has marginals of positive operator-valued measures (POVMs). We prove that ${W}$-measure is a POVM {em if and only if} its marginal POVMs are jointly measurable. The proof suggests to employ the negatives of ${W}$-measure as an indicator for non-joint measurability. By applying triangle inequalities to the negativity, we derive joint measurability criteria for dichotomic and trichotomic variables. Also, we propose an operational test for the joint measurability in sequential measurement scenario.
Quantum measurements can be interpreted as a generalisation of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalisation of doubly stochastic matrices that we call doubly normalised tensors (DNTs), and formulate a corresponding version of Birkhoff-von Neumanns theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability arises as a mathematical feature of DNTs in this context, needed to establish a characterisation similar to Birkhoff-von Neumanns. Conversely, we also show that DNTs emerge naturally from a particular instance of a joint measurability problem, remarking its relevance in general operator theory.
This talk is a survey of the question of joint measurability of coexistent observables and its is based on the monograph Operational Quantum Physics [1] and on the papers [2,3,4].
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