Quantum Operations in an Information Theory for Fermions

TL;DR

This paper develops a quantum information framework for fermions that respects the parity super-selection rule, defining allowed quantum operations and exploring their implications for fermionic correlations.

Contribution

It introduces a formalism for fermionic quantum operations consistent with super-selection rules, including unitary, measurement, and general maps, and proves their equivalence.

Findings

Defined physically allowed fermionic quantum operations

Extended formalism to general quantum operations for fermions

Discussed implications for fermionic correlation characterization

Abstract

A reasonable quantum information theory for fermions must respect the parity super-selection rule to comply with the special theory of relativity and the no-signaling principle. This rule restricts the possibility of any quantum state to have a superposition between even and odd parity fermionic states. It thereby characterizes the set of physically allowed fermionic quantum states. Here we introduce the physically allowed quantum operations, in congruence with the parity super-selection rule, that map the set of allowed fermionic states onto itself. We first introduce unitary and projective measurement operations of the fermionic states. We further extend the formalism to general quantum operations in the forms of Stinespring dilation, operator-sum representation, and axiomatic completely-positive-trace-preserving maps. We explicitly show the equivalence between these three representations of fermionic quantum operations. We discuss the possible implications of our results in characterization of correlations in fermionic systems.