Fermionic systems for quantum information people

TL;DR

This paper explores the algebraic structure of fermionic systems in quantum information, introducing new tools and concepts to handle their unique features like superselection rules and mode embeddings, with pedagogical clarity and practical formulas.

Contribution

It develops new mathematical tools such as fermionic tensor products and embeddings, and clarifies the role of superselection rules in fermionic quantum information processing.

Findings

Introduces fermionic (quasi-)tensor product and embeddings.

Provides formulas for fermionic partial trace and reduced states.

Clarifies the necessity of superselection rules in fermionic systems.

Abstract

The operator algebra of fermionic modes is isomorphic to that of qubits, the difference between them is twofold: the embedding of subalgebras corresponding to mode subsets and multiqubit subsystems on the one hand, and the parity superselection in the fermionic case on the other. We discuss these two fundamental differences extensively, and illustrate these through the Jordan--Wigner representation in a coherent, self-contained, pedagogical way, from the point of view of quantum information theory. Our perspective leads us to develop useful new tools for the treatment of fermionic systems, such as the fermionic (quasi-)tensor product, fermionic canonical embedding, fermionic partial trace, fermionic products of maps and fermionic embeddings of maps. We formulate these by direct, easily applicable formulas, without mode permutations, for arbitrary partitionings of the modes. It is also shown that fermionic reduced states can be calculated by the fermionic partial trace, containing the proper phase factors. We also consider variants of the notions of fermionic mode correlation and entanglement, which can be endowed with the usual, local operation based motivation, if the parity superselection rule is imposed. We also elucidate some other fundamental points, related to joint map extensions, which make the parity superselection inevitable in the description of fermionic systems.