Spectral properties of reduced fermionic density operators and parity superselection rule

  • Grigori G. Amosov
  • Sergey N. Filippov


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 \(\varPhi (\varrho ) = \sum _i a_i \varrho a_i^{\dag }\). We conjecture that spectra of \(\varPhi ^p(\varrho )\) and conventional p-particle reduced density matrix \(\varrho _p\) coincide. Non-trivial generalized Pauli constraints are derived for states satisfying the parity superselection rule.


Fermionic state Reduced density matrix Tracing out modes Tracing out particles Spectrum Equispectrality Superselection rule Generalized Pauli constraints 



The authors are delighted to thank A.S. Holevo for a motivation of this work and illuminating discussions. The authors are grateful to Christian Schilling for fruitful discussions and bringing Ref. [64] to our attention. Propositions 1 and 2, Corollary 1, Theorems 1 and 2 are proved by G.G. Amosov. Corollary 2, Proposition 3, Examples 1 and 3, Sect. 5 are due to S.N. Filippov. Both authors discussed the results and commented on the manuscript. The work of G.G. Amosov is supported by Russian Science Foundation under Grant No. 14-21-00162 and performed in Steklov Mathematical Institute of Russian Academy of Sciences. S.N. Filippov’s work on Sects. 3 and 4 is supported by the Russian Foundation for Basic Research under Project No. 16-37-60070 mol_a_dk and performed in Institute of Physics and Technology, Russian Academy of Sciences. S.N. Filippov’s work on Sects. 5 is supported by Russian Science Foundation under Project No. 16-11-00084 and performed in Moscow Institute of Physics and Technology.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.V. A. Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Institute of Physics and TechnologyRussian Academy of SciencesMoscowRussia
  3. 3.Moscow Institute of Physics and TechnologyMoscow RegionRussia
  4. 4.P. N. Lebedev Physical InstituteRussian Academy of SciencesMoscowRussia

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