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Information and Majorization Theory for Fermionic Phase-Space Distributions.

Nicolas J Cerf1, Tobias Haas1

  • 1Université libre de Bruxelles, Centre for Quantum Information and Communication, École polytechnique de Bruxelles, CP 165, 1050 Brussels, Belgium.

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|September 26, 2025
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Summary
This summary is machine-generated.

This study introduces information-theoretic measures for fermionic phase-space distributions, revealing all physical states are Gaussian. These measures, expressed as real-valued Berezin integrals, are physically relevant for fermionic uncertainty relations.

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Area of Science:

  • Quantum mechanics
  • Information theory
  • Mathematical physics

Background:

  • Fermionic systems possess unique anticommuting properties distinct from bosonic systems.
  • Analyzing uncertainty in fermionic phase-space distributions is complex due to Grassmann variables.

Purpose of the Study:

  • To develop information-theoretic measures for fermionic phase-space distributions.
  • To analyze the uncertainty relations of fermionic systems.
  • To establish physical relevance for Grassmann-valued distributions.

Main Methods:

  • Utilizing the theory of supernumbers.
  • Deriving simple expressions for Glauber P, Wigner W, and Husimi Q distributions for single fermionic modes.
  • Employing Berezin integration to evaluate uncertainty measures.

Main Results:

  • All physical states of a single fermionic mode are shown to be Gaussian.
  • Simple expressions for fermionic phase-space distributions were obtained.
  • Fermionic analogs of majorization and entropy conjectures, Lieb-Solovej theorem, and Wehrl-Lieb inequality were proven.
  • Berezin integrals of Grassmann-valued distributions yield physically relevant, real-valued uncertainty measures.

Conclusions:

  • Information-theoretic measures provide a robust framework for understanding fermionic uncertainty.
  • Despite their abstract nature, fermionic phase-space distributions yield concrete, physical insights through Berezin integration.
  • The study establishes new fermionic uncertainty relations with implications for quantum information and condensed matter physics.