Spectral properties of reduced fermionic density operators and parity superselection rule

TL;DR

This paper investigates the spectral properties of reduced fermionic states, characterizes states with equal spectra under mode reduction, and explores implications of the parity superselection rule for fermionic systems.

Contribution

It introduces a detailed analysis of mode- and particle-reduced states in fermionic systems, characterizes equispectral states, and proposes a new operation for particle reduction with conjectured spectral equivalence.

Findings

Spectra of mode-reduced states are generally not identical.

States with equal spectra are related via local unitaries and satisfy the parity superselection rule.

Derived generalized Pauli constraints for parity superselection states.

Abstract

We consider pure fermionic states with a varying number of quasiparticles and analyze two types of reduced density operators: one is obtained via tracing out modes, the other is obtained via tracing out particles. We demonstrate that spectra of mode-reduced states are not identical in general and fully characterize pure states with equispectral mode-reduced states. Such states are related via local unitary operations with states satisfying the parity superselection rule. Thus, valid purifications for fermionic density operators are found. To get particle-reduced operators for a general system, we introduce the operation $\Phi(\varrho) = \sum_i a_i \varrho a_i^{\dag}$. We conjecture that spectra of $\Phi^p(\varrho)$ and conventional $p$-particle reduced density matrix $\varrho_p$ coincide. Nontrivial generalized Pauli constraints are derived for states satisfying the parity superselection rule.