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Detecting Magic States via Characteristic Functions

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Abstract

Magic (non-stabilizerness) is a crucial resource for universal fault-tolerant quantum computation in the stabilizer formalism, without which the computation can be classically emulated, as demonstrated by the celebrated Gottesman-Knill theorem. Characterizations of magic, including its detection and quantification, are of interest in quantum computation and have attracted much attention. However, most quantifiers of magic are generally quite difficult to calculate even numerically, or are restricted to systems of special dimensions. The discrete Wigner formalism has been exploited to quantify magic, although it is computable, its application is limited since its formalism is quite different in odd prime power dimensions and other dimensions. In this work, we propose a quantifier of magic in terms of characteristic functions (Fourier transforms) of quantum states, and reveal its basic properties. This quantifier, apart from its nice properties, is well defined in all dimensions and is straightforward to compute. We illustrate the effectiveness of this quantifier in detecting the magic of several representative quantum states. Employing this quantifier, we show that the group covariant symmetric informationally complete (SIC) states possess the maximal magic. Finally, we compare the quantifier of magic with some existing ones including the well known sum negativity and the so called thauma, and show that they yield different orderings of magic.

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Acknowledgements

This work was supported by the National Key R&D Program of China, Grant Nos. 2020YFA0712700, 2019QY0702 and 2017YFA0303903, the Fundamental Research Funds for the Central Universities, Grant No. FRF-TP-19-012A3, and the National Natural Science Foundation of China, Grant No. 11875173.

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Authors and Affiliations

  1. Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, 100084, China

    Hao Dai

  2. School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China

    Shuangshuang Fu

  3. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Shunlong Luo

  4. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Shunlong Luo

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  1. Hao Dai
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Correspondence to Shunlong Luo.

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Dai, H., Fu, S. & Luo, S. Detecting Magic States via Characteristic Functions. Int J Theor Phys 61, 35 (2022). https://doi.org/10.1007/s10773-022-05027-8

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