Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fermionic optimal transport

Rocco Duvenhage, Dylan van Zyl, Paola Zurlo

TL;DR

This work extends quantum optimal transport to fermionic, ${\mathbb Z}_2$-graded settings by formulating fermionic transport plans on graded tensor products and translating them to usual tensor products via cyclic representations and twisted commutants. It defines quadratic fermionic Wasserstein distances $W^{\mathrm{F}}, W^{\mathrm{F}}_{\sigma}, W^{\mathrm{F}}_{\sigma\sigma}$ through a cost functional built from the associated even channels $E_{\omega}$, and shows these distances inherit key metric properties from the non-graded theory by translation. A central result is a one-to-one correspondence between fermionic transport plans and usual plans, enabling a transfer of metric results and symmetry properties, including KMS duality and Klein-type gradings, to the fermionic context. The framework also develops a detailed balance theory for indistinguishable fermions, connecting fermionic detailed balance to standard quantum detailed balance via copying and reversing operations, and derives symmetry-based bounds on deviations from fermionic detailed balance. Collectively, the paper provides a rigorous canonical approach to measuring dynamical distance and non-equilibrium behavior in fermionic quantum systems, with potential implications for quantum statistical mechanics and noncommutative geometry.

Abstract

Quadratic Wasserstein distances are obtained between dynamical systems (with states as special case), on $\mathbb{Z}_2$-graded von Neumann algebras. This is achieved through a systematic translation from non-graded to $\mathbb{Z}_2$-graded transport plans, on usual and fermionic (or $\mathbb{Z}_2$-graded) tensor products respectively. The metric properties of these fermionic Wasserstein distances are shown, and their symmetries relevant to deviation of a system from quantum detailed balance are investigated. The latter is done in conjunction with the development of a complete mathematical framework for detailed balance in systems involving indistinguishable fermions.

Fermionic optimal transport

TL;DR

This work extends quantum optimal transport to fermionic, -graded settings by formulating fermionic transport plans on graded tensor products and translating them to usual tensor products via cyclic representations and twisted commutants. It defines quadratic fermionic Wasserstein distances through a cost functional built from the associated even channels , and shows these distances inherit key metric properties from the non-graded theory by translation. A central result is a one-to-one correspondence between fermionic transport plans and usual plans, enabling a transfer of metric results and symmetry properties, including KMS duality and Klein-type gradings, to the fermionic context. The framework also develops a detailed balance theory for indistinguishable fermions, connecting fermionic detailed balance to standard quantum detailed balance via copying and reversing operations, and derives symmetry-based bounds on deviations from fermionic detailed balance. Collectively, the paper provides a rigorous canonical approach to measuring dynamical distance and non-equilibrium behavior in fermionic quantum systems, with potential implications for quantum statistical mechanics and noncommutative geometry.

Abstract

Quadratic Wasserstein distances are obtained between dynamical systems (with states as special case), on -graded von Neumann algebras. This is achieved through a systematic translation from non-graded to -graded transport plans, on usual and fermionic (or -graded) tensor products respectively. The metric properties of these fermionic Wasserstein distances are shown, and their symmetries relevant to deviation of a system from quantum detailed balance are investigated. The latter is done in conjunction with the development of a complete mathematical framework for detailed balance in systems involving indistinguishable fermions.

Paper Structure

This paper contains 17 sections, 28 theorems, 292 equations.

Table of Contents

  1. Introduction
  2. Duality
  3. Transport plans and dynamical systems
  4. Cyclic representations
  5. Representations from fermionic transport Plans
  6. Gradings and Klein isomorphisms
  7. Representations from usual transport plans
  8. Fermionic versus usual transport plans
  9. Supercommutation
  10. Translating representations
  11. Fermionic transport plans as channels
  12. The one-to-one correspondence between fermionic and usual transport plans
  13. Fermionic Wasserstein distances
  14. Deviation from fermionic detailed balance
  15. Fermionic detailed balance
  16. ...and 2 more sections

Key Result

Proposition 2.4

For $E$ as given in Definition verwDuaalDef, the twisted dual $E^\wr$ exists if and only if the dual $E'$ does, in which case Moreover, the two possibilities verwrDuaal and verwrDuaal' in defining $E^\wr$ are in fact equivalent.

Theorems & Definitions (85)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • Proposition 2.6
  • Remark 2.7
  • Definition 3.2
  • Definition 3.3
  • ...and 75 more