Spectral properties of reduced fermionic density operators and parity superselection rule

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(rho) = Sigma(i)a(i)rho a(i)(dagger). We conjecture that spectra of Phi(p)(rho) and conventional p-particle reduced density matrix rho(p) coincide. Non-trivial generalized Pauli constraints are derived for states satisfying the parity superselection rule.