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 $\mathrm{\Phi }(\varrho )={\sum }_i{a}_i\varrho a_i^{\mathrm{†}}$. We conjecture that spectra of ${\mathrm{\Phi }}^p(\varrho )$ and conventional $p$-particle reduced density matrix ${\varrho }_p$ coincide. Nontrivial generalized Pauli constraints are derived for states satisfying the parity superselection rule.