Quantifying non-Gaussianity via the Hellinger distance
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QUANTIFYING NON-GAUSSIANITY VIA THE HELLINGER DISTANCE Yue Zhang∗ and Shunlong Luo∗
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).
Keywords: bosonic field, Gaussian state, non-Gaussianity, Hellinger distance, Bures distance
DOI: 10.1134/S0040577920080061
1. Introduction Gaussian states and the associated Gaussian operations play a crucial role in quantum information processing involving quantum optics [1]–[7]. Although Gaussian states belong to infinite-dimensional spaces in the mathematical sense, they are easy to handle both theoretically and experimentally. This is due to the following remarkable features: 1. Gaussian states can be fully and neatly characterized by their first and second moments [8], [9]. 2. The evolution of Gaussian states under quadratic Hamiltonians can be conveniently described in the language of symplectic groups [10], [11]. 3. Gaussian states have a physical interpretation as minimum uncertainty states [12]. 4. Gaussian states constitute a rather wide family of important states including coherent states and thermal states, among others. 5. Gaussian states can be easily prepared and manipulated in experiments [13]–[15]. ∗
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China; School of Mathematical Sciences, University of the Chinese Academy of Sciences, Beijing, China, e-mail: [email protected]. This research was supported by the National Natural Science Foundation of China (Grant No. 11875317), the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences (Grant No. Y029152K51), and the Key Laboratory of Random Complex Structures and Data Science, Chinese Academy of Sciences (Grant No. 2008DP173182). Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 204, No. 2, pp. 242–257, August, 2020. Received March 2, 2020. Revised March 19, 2020. Accepted March 30, 2020. 1046
c 2020 Pleiades Publishing, Ltd. 0040-5779/20/2042-1046
6. Gaussian states and Gaussian operations are an available, important resource for quantum information protocols [16]–[20]. Given the significance and ubiquity of Gaussian states, it can be asked what non-Gaussian states are good for. Since the last decade, it has been recognized that non-Gaussian states and operations can be used to improve the efficiency of quantum protocols [21]–[26], and they play an increasingly important role in quantum information [27]–[29]. While Gaussian states are usually defined by Gau
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