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Peer-reviewedCC BYAdvances in Mathematics

Discrete quantum Gaussians and central limit theorem

  • Kaifeng Bu1,2
  • Weichen Gu3,1
  • Arthur Jaffe2
  1. 1Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA
  2. 2Departments of Mathematics and Physics, Harvard University, Cambridge, MA 02138, USA
  3. 3Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA
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Abstract

We introduce a quantum convolution and a conceptual framework to study states in discrete-variable (DV) quantum systems. All our results suggest that stabilizer states play a role in DV quantum systems similar to the role that Gaussian states play in continuous-variable systems. Hence we suggest the name “discrete quantum Gaussians” for stabilizer states. For example, we prove that the convolution of two stabilizer states is another stabilizer state, and that stabilizer states extremize both quantum entropy and Fisher information. We establish a “maximal entropy principle,” a “second law of thermodynamics for quantum convolution,” and a “quantum central limit theorem.” The latter is based on iterating the convolution of a zero-mean quantum state, which we prove converges to a stabilizer state. We bound the exponential rate of convergence of the quantum central limit theorem by the “magic gap,” defined in terms of the support of the characteristic function of the state. We elaborate on our general results with a discussion of some examples, as well as extending many of them to quantum channels.

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Cite

BibTeXRIS
BibTeX
@article{bu2026discrete,
  title = {{Discrete quantum Gaussians and central limit theorem}},
  author = {Kaifeng Bu and Weichen Gu and Arthur Jaffe},
  journal = {Advances in Mathematics},
  year = {2026},
  month = sep,
  doi = {10.1016/j.aim.2026.111204},
  url = {https://doi.org/10.1016/j.aim.2026.111204},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Discrete quantum Gaussians and central limit theorem
AU  - Kaifeng Bu
AU  - Weichen Gu
AU  - Arthur Jaffe
PY  - 2026
DA  - 2026/09/04
JO  - Advances in Mathematics
DO  - 10.1016/j.aim.2026.111204
UR  - https://doi.org/10.1016/j.aim.2026.111204
AB  - We introduce a quantum convolution and a conceptual framework to study states in discrete-variable (DV) quantum systems. All our results suggest that stabilizer states play a role in DV quantum systems similar to the role that Gaussian states play in continuous-variable systems. Hence we suggest the name “discrete quantum Gaussians” for stabilizer states. For example, we prove that the convolution of two stabilizer states is another stabilizer state, and that stabilizer states extremize both quantum entropy and Fisher information. We establish a “maximal entropy principle,” a “second law of thermodynamics for quantum convolution,” and a “quantum central limit theorem.” The latter is based on iterating the convolution of a zero-mean quantum state, which we prove converges to a stabilizer state. We bound the exponential rate of convergence of the quantum central limit theorem by the “magic gap,” defined in terms of the support of the characteristic function of the state. We elaborate on our general results with a discussion of some examples, as well as extending many of them to quantum channels.
SN  - 0001-8708
SN  - 1090-2082
ER  -