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arXiv:2304.14038

On Kirkwood--Dirac quasiprobabilities and unravelings of quantum channel assigned to a tight frame

Alexey E. Rastegin

Published Apr 27, 2023• 0 citations
arXivHTMLPDF

Abstract

An issue which has attracted increasing attention in contemporary researches are Kirkwood--Dirac quasiprobabilities. List of their use includes many questions of quantum physics. Applications of complex tight frames in quantum information science were recently demonstrated. It is shown in this paper that quasiprobabilities naturally appear in the context of unravelings of a quantum channel. Using vectors of the given tight frame to build principal Kraus operators generates quasiprobabilities with interesting properties. For an equiangular tight frame, we characterize the Hilbert--Schmidt and spectral norms of the matrix consisted of quasiprobabilities. Hence, novel uncertainty relations in terms of Rényi and Tsallis entropies are obtained. New inequalities for characterizing the location of eigenvalues are derived. They give an alternative to estimating on the base of Geršgorin's theorem. A utility of the presented inequalities is exemplified with symmetric informationally complete measurement in dimension two.

DOI
10.48550/arxiv.2304.14038

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An issue which has attracted increasing attention in contemporary researches are Kirkwood--Dirac quasiprobabilities. List of their use includes many questions of quantum physics. Applications of complex tight frames in quantum information science were recently demonstrated. It is shown in this paper that quasiprobabilities naturally appear in the context of unravelings of a quantum channel. Using vectors of the given tight frame to build principal Kraus operators generates quasiprobabilities with interesting properties. For an equiangular tight frame, we characterize the Hilbert--Schmidt and spectral norms of the matrix consisted of quasiprobabilities. Hence, novel uncertainty relations in terms of Rényi and Tsallis entropies are obtained. New inequalities for characterizing the location of eigenvalues are derived. They give an alternative to estimating on the base of Geršgorin's theorem. A utility of the presented inequalities is exemplified with symmetric informationally complete measurement in dimension two.
Protocols of quantum information science often realize in terms of specially selected states. In particular, such states are used to perform measurements at the final stage of a protocol. This study aims to explore measurements assigned to a mutually unbiased-equiangular tight frame. The utilized method deals with Kirkwood–Dirac quasiprobabilities, which are increasingly used in contemporary research. These quasiprobabilities constitute a matrix that can be linked to unravelings of certain quantum channels. Using states of the given frame to build principal Kraus operators leads to quasiprobabilities that represent the measured state. The structure of a mutually unbiased-equiangular tight frame allows one to characterize entropies associated with a particular unraveling. To do this, we estimate some of the Schatten and Ky Fan norms of the matrix consisted of quasiprobabilities. New uncertainty relations in terms of Rényi and Tsallis entropies follow from the obtained inequalities. A utility of the presented inequalities is exemplified with mutually unbiased bases of a qubit and equiangular tight frames of a ququart.
Finite tight frames are interesting in various topics including questions of quantum information. Each complex tight frame leads to a resolution of the identity in the Hilbert space. Symmetric informationally complete measurements are a special class of equiangular tight frames. Applications of such frames in quantum physics deserve more attention than they have obtained. We derive uncertainty relations for a quantum measurement assigned to an equiangular tight frame. Main results follow from estimation of the corresponding index of coincidence. State-dependent and state-independent formulations are both addressed. Also, we discuss applications of considered measurements to detect entanglement and other correlations.
Kirkwood-Dirac (KD) quasiprobability is a quantum analog of phase space probability of classical statistical mechanics, allowing negative or/and nonreal values. It gives an informationally complete representation of a quantum state. Recent works have revealed the important roles played by the KD quasiprobability in the broad fields of quantum science and quantum technology. In the present work, we use the KD quasiprobability to access the quantum coherence in a quantum state. We show that the l1{l}_{1}l1​ norm of the imaginary part of the KD quasiprobability over an incoherent reference basis and a second basis, maximized over all possible choices of the latter, can be used to quantify quantum coherence, satisfying certain desirable properties. It is upper bounded by the quantum uncertainty, i.e., the quantum standard deviation, of the incoherent basis in the state. It gives a lower bound to the l1{l}_{1}l1​ norm quantum coherence, and for a single qubit, they are identical. We discuss the measurement of the KD coherence based on the measurement of the KD quasiprobability and an optimization procedure in hybrid quantum-classical schemes, and suggest statistical interpretations. We also discuss its relevance in the physics of linear response regime.
Abstract The role of quantum coherence as an information resource has attracted increasing attention in recent years. Many protocols of quantum information science deal with specially selected states. Complete sets of mutually unbiased bases and symmetric informationally complete measurements are widely used in this regard. Other constructions were found to be useful, including projective designs and equiangular tight frames. As a rule, there are additional restrictions on the probabilities generated by quantum measurements assigned to such sets. Inequalities in terms of coherence quantifiers allow one to examine complementarity with respect to different sets. This study aims to examine uncertainty relations for coherence quantifiers averaged with respect to a set of mutually unbiased equiangular tight frames. To quantify the amount of coherence, quantum coherence quantifiers of the Tsallis type and the geometric coherence are used. The first case is induced by the Tsallis relative entropies. The derived inequalities are exemplified with equiangular tight frames of a ququart.