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Constructing Fermionic Dynamics with Closed Moment Hierarchies

A. E. Teretenkov

Abstract

We construct a broad class of completely positive maps and Go\-rini--Kossakowski--Sudarshan-Lindblad generators for fermionic systems induced by linear transformations of system and environment modes. For these maps, we derive explicit Heisenberg-picture formulas for arbitrary normally ordered monomials in terms of minors of the underlying mode-transformation matrices and environment correlation tensors. We show that for even environment states the linear span of monomials up to any fixed order is invariant, which yields closed equations for low-order moments and makes their computation efficient. We also discuss the relation of this construction to second quantization of non-Hermitian one-particle contractions and extend the formalism to completely positive maps arising from post-selection.

Constructing Fermionic Dynamics with Closed Moment Hierarchies

Abstract

We construct a broad class of completely positive maps and Go\-rini--Kossakowski--Sudarshan-Lindblad generators for fermionic systems induced by linear transformations of system and environment modes. For these maps, we derive explicit Heisenberg-picture formulas for arbitrary normally ordered monomials in terms of minors of the underlying mode-transformation matrices and environment correlation tensors. We show that for even environment states the linear span of monomials up to any fixed order is invariant, which yields closed equations for low-order moments and makes their computation efficient. We also discuss the relation of this construction to second quantization of non-Hermitian one-particle contractions and extend the formalism to completely positive maps arising from post-selection.

Paper Structure

This paper contains 8 sections, 7 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Fermionic channels induced by linear mode transformations
  3. Second quantization candidates for non-Hermitian systems
  4. Dynamics averaged over a Poisson process
  5. Post-selected linear fermionic maps
  6. Conclusions
  7. Acknowledgments
  8. Funding

Key Result

Lemma 1

Let $A,B\in\mathbb{C}^{m\times m}$ and fix an ordered multi-index $I=(i_1<\dots<i_p)$. Then where the sum runs over all ordered subsets $L=(l_1<\dots<l_{p-r})$ and $\Lambda=(\lambda_1<\dots<\lambda_r)$, such that $|I|= |L|+|\Lambda|$, we denote by $A_{I\times L}$ (resp. $B_{I\times \Lambda}$) the submatrix of $A$ (resp. $B$) formed by rows $I$ and columns $L$ (resp. $\Lambda$), and by their hor

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • Lemma 3
  • Theorem 2
  • Lemma 4
  • proof
  • Theorem 3