Eigenvalues of fermion density matrices

The least upper bound of the eigenvalues of second-order reduced density matrices for a system of fermions is proved to be $n$ for a system of $2n$ or $2n+1$ identical fermions. It is also shown that this limiting state may be interpreted as a system of identical pairs behaving as quasibosons.The least upper bound of the eigenvalues of second-order reduced density matrices for a system of fermions is proved to be $n$ for a system of $2n$ or $2n+1$ identical fermions. It is also shown that this limiting state may be interpreted as a system of identical pairs behaving as quasibosons.

Physical ReviewJun 7, 1965143 citations

STRUCTURE OF THE FERMION DENSITY MATRIX.

Louis Witten

Jan 1, 19670 citations

THE DENSITY MATRIX FOR THE SYSTEM OF NONINTERACTING FERMIONS

Yu. N. Demkov, Igor V. Komarov

Jan 1, 19650 citations

Entanglement criteria for the bosonic and fermionic induced ensembles

Stéphane Dartois, Ion Nechita, Adrian Tanasă

Abstract We introduce the bosonic and fermionic ensembles of density matrices and study their entanglement. In the fermionic case, we show that random bipartite fermionic density matrices have non-positive partial transposition; hence, they are typically entangled. The similar analysis in the bosonic case is more delicate, due to a large positive outlier eigenvalue. We compute the asymptotic ratio between the size of the environment and the size of the system Hilbert space for which random bipartite bosonic density matrices fail the PPT criterion, being thus entangled. We also relate moment computations for tensor-symmetric random matrices to evaluations of the circuit counting and interlace graph polynomials for directed graphs.Abstract We introduce the bosonic and fermionic ensembles of density matrices and study their entanglement. In the fermionic case, we show that random bipartite fermionic density matrices have non-positive partial transposition; hence, they are typically entangled. The similar analysis in the bosonic case is more delicate, due to a large positive outlier eigenvalue. We compute the asymptotic ratio between the size of the environment and the size of the system Hilbert space for which random bipartite bosonic density matrices fail the PPT criterion, being thus entangled. We also relate moment computations for tensor-symmetric random matrices to evaluations of the circuit counting and interlace graph polynomials for directed graphs.

arXiv:2111.05638Quantum Information ProcessingNov 11, 20220 citations

The least upper bound of the eigenvalues of second-order reduced density matrices for a system of fermions is proved to be

n

for a system of

2n

2n+1

identical fermions. It is also shown that this limiting state may be interpreted as a system of identical pairs behaving as quasibosons.