Universal Bound and Phase Transition in Many-Body Fermionic Non-Gaussianity

Abstract

Fermionic non-Gaussianity is a resource for universal quantum computation that can be generated by interactions in quantum many-body systems. Using the magic Rényi entropy (MRE) as a measure of non-Gaussianity, we derive its universal upper bound and Haar mean, proving that typical Haar-random states attain the maximal MRE density of ln(4/3) per Majorana in the large-system limit. We also show that the MRE can differ extensively between states with identical covariance matrices. To investigate how non-Gaussianity grows toward this maximal density, we evolve a Gaussian state in imaginary and real time under the Sachdev-Ye-Kitaev Hamiltonian. By varying the imaginary- and real-time durations, we identify a first-order transition marked by a kink in the MRE density and spontaneous breaking of the permutation symmetry among the four copies used to evaluate the MRE. This transition represents a qualitative change in the non-Gaussianity of the state that is not reflected in the thermal free energy.

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BibTeX RIS

BibTeX

@misc{hoshino2026universal,
  title = {{Universal Bound and Phase Transition in Many-Body Fermionic Non-Gaussianity}},
  author = {Masahiro Hoshino and Ryota Matsuda and Yuto Ashida},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02075},
  eprint = {2610.02075},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02075},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Universal Bound and Phase Transition in Many-Body Fermionic Non-Gaussianity
AU  - Masahiro Hoshino
AU  - Ryota Matsuda
AU  - Yuto Ashida
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.02075
UR  - https://arxiv.org/abs/2610.02075
AB  - Fermionic non-Gaussianity is a resource for universal quantum computation that can be generated by interactions in quantum many-body systems. Using the magic Rényi entropy (MRE) as a measure of non-Gaussianity, we derive its universal upper bound and Haar mean, proving that typical Haar-random states attain the maximal MRE density of ln(4/3) per Majorana in the large-system limit. We also show that the MRE can differ extensively between states with identical covariance matrices. To investigate how non-Gaussianity grows toward this maximal density, we evolve a Gaussian state in imaginary and real time under the Sachdev-Ye-Kitaev Hamiltonian. By varying the imaginary- and real-time durations, we identify a first-order transition marked by a kink in the MRE density and spontaneous breaking of the permutation symmetry among the four copies used to evaluate the MRE. This transition represents a qualitative change in the non-Gaussianity of the state that is not reflected in the thermal free energy.
ER  -

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