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Theoretical and Mathematical Physics

Quantifying non-Gaussianity via the Hellinger distance

Yue Zhang, Shunlong Luo

Published Aug 1, 2020• 6 citations
Publisher

Abstract

Abstract Non-Gaussianity is an important resource for quantum information processing with continuous variables. We introduce a measure of the non-Gaussianity of bosonic field states based on the Hellinger distance and present its basic features. This measure has some natural properties and is easy to compute. We illustrate this measure with typical examples of bosonic field states and compare it with various measures of non-Gaussianity. In particular, we highlight its similarity to and difference from the measure based on the Bures distance (or, equivalently, fidelity).

DOI
10.1134/s0040577920080061

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Abstract Non-Gaussianity is an important resource for quantum information processing with continuous variables. We introduce a measure of the non-Gaussianity of bosonic field states based on the Hellinger distance and present its basic features. This measure has some natural properties and is easy to compute. We illustrate this measure with typical examples of bosonic field states and compare it with various measures of non-Gaussianity. In particular, we highlight its similarity to and difference from the measure based on the Bures distance (or, equivalently, fidelity).
While Gaussian states and associated Gaussian operations are basic ingredients and convenient objects for continuous-variable quantum information, it is also realized that non-Gaussianity is an important resource for quantum information processing. The characterization and quantification of non-Gaussianity have been widely studied in the past decade, with several significant measures for non-Gaussianity introduced. In this work, by exploiting an information-theoretic refinement of the conventional Heisenberg uncertainty relation and a physical characterization of Gaussian states as minimum uncertainty states, we introduce an easily computable measure for non-Gaussianity of bosonic field states in terms of the Wigner-Yanase skew information. Fundamental properties, as well as intuitive meaning, of this measure are unveiled. The concept is illustrated by prototypical non-Gaussian states, and compared with various existent measures for non-Gaussianity. Its merit and physical significance are elucidated.
Abstract Two prominent classification schemes for Bosonic field states are (i) classical versus nonclassical, and (ii) Gaussian versus non-Gaussian. These two paradigms are closely related yet fundamentally different. In particular, classical states may be Gaussian or non-Gaussian, and Gaussian states may be classical or nonclassical. Both non-Gaussianity and nonclassicality are important resources for quantum information processing. It is desirable to study the interplay between non-Gaussianity and nonclassicality, and seek effective methods of detecting them. In this work, we introduce a quantifier for non-Gaussianity by exploiting and optimizing an information-theoretic quantifier for nonclassicality, and exhibit its basic properties. A criterion for simultaneously detecting non-Gaussianity and nonclassicality follows, and its applications are illustrated through several examples. This unveils some intrinsic relations between non-Gaussianity and nonclassicality.
We propose three alternative measures for non-Gaussianity of quantum states: sine distance, Bures angle, and Bures distance, which are based on quantum fidelity introduced by Wang [Phys. Lett. A 373 58 (2008)]. Using them, we evaluate the non-Gaussianity of some relevant single-mode and two-mode non-Gaussian states and find a good consistency of the three examined measures. In addition, we show that such metrics can exactly quantify the degree of Gaussianity of even Schrödinger-cat-like states of small amplitudes that can not be measured by other known non-Gaussianity measures such as the Hilbert–Schmidt metric and the relative entropy metric. We make a comparative study between all existing non-Gaussianity measures according to the metric axioms and point out that the sine distance is the best candidate among them.
Quantum coherence and nonclassical correlation are key features of a quantum world. Quantifying coherence and nonclassical correlation are two key tasks in quantum information theory. First, we present a bona fide measure of quantum coherence by utilizing the Hellinger distance. This coherence measure is proven to fulfill all the criteria of a well-defined coherence measure, including the strong monotonicity in the resource theories of quantum coherence. In terms of this coherence measure, the distribution of quantum coherence in multipartite systems is studied and a corresponding polygamy relation is proposed. Its operational meanings and the relations between the generation of quantum correlations and the coherence are also investigated. Moreover, we present the Hellinger distance-based measure of nonclassical correlation, which not only inherits the nice properties of the Hellinger distance including contractivity, but also shows a powerful analytic computability for a large class of quantum states. We show that there is an explicit tradeoff relation satisfied by the quantum coherence and this nonclassical correlation.
Non-Gaussian states are essential resources in quantum information processing. In this work, we investigate methods for quantifying bosonic non-Gaussianity in many-body systems. Building on recent theoretical insights into the self-convolution properties of bosonic pure states, we introduce non-Gaussian entropy as a new measure to characterize non-Gaussianity in bosonic pure states. We further propose a practical protocol for measuring non-Gaussian entropy using three beam splitters and four copies of the input state. In addition, we extend this framework to mixed states, providing a general approach to quantifying non-Gaussianity. Our results offer a convenient and efficient method for characterizing bosonic non-Gaussianity, paving the way for its implementation on near-term experimental platforms.
We consider how to quantify non-Gaussianity for the correlation of a bipartite quantum state by using various measures such as relative entropy and geometric distances. We first show that an intuitive approach, i.e., subtracting the correlation of a reference Gaussian state from that of a target non-Gaussian state, fails to yield a non-negative measure with monotonicity under local Gaussian channels. Our finding clearly manifests that quantum-state correlations generally have no Gaussian extremality. We therefore propose a different approach by introducing relevantly averaged states to address correlation. This enables us to define a non-Gaussianity measure based on, e.g., the trace-distance and the fidelity, fulfilling all requirements as a measure of non-Gaussian correlation. For the case of the fidelity-based measure, we also present readily computable lower bounds of non-Gaussian correlation.