A fermionic de Finetti theorem

TL;DR

This paper establishes a fermionic de Finetti theorem for finite Majorana systems, linking symmetry to local mode separability and improving understanding of fermionic quantum states and approximations.

Contribution

It derives a new de Finetti theorem for fermionic systems, connecting symmetry properties to mode separability and extending Hudson's fermionic central limit theorem.

Findings

States invariant under certain permutations are well approximated by mode separable states

The theorem provides a quantitative link to Hartree-Fock approximation quality

Extension of Hudson's fermionic central limit theorem based on the new de Finetti result

Abstract

Quantum versions of de Finetti's theorem are powerful tools, yielding conceptually important insights into the security of key distribution protocols or tomography schemes and allowing to bound the error made by mean-field approaches. Such theorems link the symmetry of a quantum state under the exchange of subsystems to negligible quantum correlations and are well understood and established in the context of distinguishable particles. In this work, we derive a de Finetti theorem for finite sized Majorana fermionic systems. It is shown, much reflecting the spirit of other quantum de Finetti theorems, that a state which is invariant under certain permutations of modes loses most of its anti-symmetric character and is locally well described by a mode separable state. We discuss the structure of the resulting mode separable states and establish in specific instances a quantitative link to the quality of Hartree-Fock approximation of quantum systems. We hint at a link to generalized Pauli principles for one-body reduced density operators. Finally, building upon the obtained de Finetti theorem, we generalize and extend the applicability of Hudson's fermionic central limit theorem.