Entropy of Reduced Density Matrices

Abstract

The entropy of a positive trace class operator, P, is defined as S(P) = -Tr P log P. We will review known inequalities relating the entropy of the density matrix representing the state of an N-particle system to the entropy of the corresponding reduced density matrices. One important inequality of this type is $$ s\left( {{\rho _N}} \right)s\left( {{\rho _1}} \right) \pm s\left( {I \mp {\rho _1}} \right) $$ where ρ1 is the 1-particle density matrix of the system. The density matrices are normalized so that TrρN = 1 and Trρ1 = N; the upper sign refers to fermions, the lower to bosons, and I denotes the identity operator. We will discuss a conjectured generalization of this inequality to the case of two-particle density matrices. We also conjecture that S(ρ2) ≥ 0 in the case of fermions when the two-particle density matrix, ρ2, is normalized so that \(Tr{\rho _2} = \left( {\begin{array}{*{20}{c}} N \\ 2 \end{array}} \right)\). Evidence for the validity of both conjectures will be presented.