Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review

Eigenvalues of Density Matrices

B. C. Carlson, Joseph M. Keller

Published Feb 1, 1961• 134 citations
Publisher

Abstract

For a system of $N$ identical particles in a pure bound state, the density matrices of positive orders $p$ and $N\ensuremath{-}p$ have the same nonzero eigenvalues with the same multiplicities. If the number of nonzero eigenvalues is finite, these density matrices are unitarily equivalent.

DOI
10.1103/physrev.121.659

Related papers

Variational Theorem for Reduced Density Matrices

Hsiao S. Kiang

This paper discusses a theorem concerning the variational description of the eigenfunctions and eigenvalues of the two complementary reduced density matrices for a many-particle system in a bound state.

Journal of Mathematical Physics•Mar 1, 1967•4 citations

Eigenvalues of Fermion Density Matrices

Fukashi Sasaki

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 Review•Jun 7, 1965•143 citations

Eigenvalue estimates for the one-particle density matrix

Alexander V. Sobolev

It is shown that the eigenvalues $λ_k, k=1, 2, \dots,$ of the one-particle density matrix satisfy the bound $λ_k\le C k^{-8/3}$ with a positive constant $C$.

arXiv:2008.10935•Aug 25, 2020•4 citations

Density Matrix in Quantum Mechanics and Distinctness of Ensembles Having the Same Compressed Density Matrix

Gui Lu Long, Yi-Fan Zhou, Jia-Qi Jin +2 more

arXiv:quant-ph/0508207•Foundations of Physics•May 8, 2006•25 citations

Equivalence of transfer matrices

M. A. Novotny

Three theorems dealing with transfer matrices in statistical mechanical systems are proved. The theorems state that the nonzero eigenvalues of transfer matrices formed through various prescriptions are identical. Hence it is possible to ascribe a physical meaning to all the eigenvalues of a transfer matrix, not just to the few largest eigenvalues. The first theorem states that the transfer matrix formed by building a system M layers at a time has as its only nonzero eigenvalues the eigenvalues of the transfer matrix formed by building the M layers of the system one at a time. This theorem relates the product of two nM×nM M-layer transfer matrices to the product of M one-layer M×M transfer matrices. The second theorem states that one of the nM×nM M-layer transfer matrices (for M>1) has only one nonzero eigenvalue. A procedure for finding this eigenvalue and all eigenvectors is given. The third theorem generalizes the first to the case where the chosen layering is not an integer multiple of the interaction length.

Journal of Mathematical Physics•Oct 1, 1988•12 citations

Reduced density matrices of a system of N coupled oscillators 2. eigenstructure of the 1-particle matrix for the canonical ensemble

S. Pruski, Jan F. Maćkowiak, O. Missuno

Reports on Mathematical Physics•Sep 1, 1972•17 citations
For a system of NNN identical particles in a pure bound state, the density matrices of positive orders ppp and N\ensuremath−pN\ensuremath{-}pN\ensuremath−p have the same nonzero eigenvalues with the same multiplicities. If the number of nonzero eigenvalues is finite, these density matrices are unitarily equivalent.
The least upper bound of the eigenvalues of second-order reduced density matrices for a system of fermions is proved to be nnn for a system of 2n2n2n or 2n+12n+12n+1 identical fermions. It is also shown that this limiting state may be interpreted as a system of identical pairs behaving as quasibosons.
It is shown that the eigenvalues λk,k=1,2,…,λ_k, k=1, 2, \dots,λk​,k=1,2,…, of the one-particle density matrix satisfy the bound λk≤Ck−8/3λ_k\le C k^{-8/3}λk​≤Ck−8/3 with a positive constant CCC.
Three theorems dealing with transfer matrices in statistical mechanical systems are proved. The theorems state that the nonzero eigenvalues of transfer matrices formed through various prescriptions are identical. Hence it is possible to ascribe a physical meaning to all the eigenvalues of a transfer matrix, not just to the few largest eigenvalues. The first theorem states that the transfer matrix formed by building a system M layers at a time has as its only nonzero eigenvalues the eigenvalues of the transfer matrix formed by building the M layers of the system one at a time. This theorem relates the product of two nM×nM M-layer transfer matrices to the product of M one-layer M×M transfer matrices. The second theorem states that one of the nM×nM M-layer transfer matrices (for M>1) has only one nonzero eigenvalue. A procedure for finding this eigenvalue and all eigenvectors is given. The third theorem generalizes the first to the case where the chosen layering is not an integer multiple of the interaction length.