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Journal of Mathematical Physics

Linear Inequalities for Density Matrices

Ernest R. Davidson

Published Apr 1, 1969• 80 citations
Publisher

Abstract

A set of linear inequalities for the reduced two-body density matrix is presented. The general problem of finding all such inequalities is discussed. It is shown that all of the linear inequalities found so far for ensemble representability may be written as a single family of inequalities.
DOI
10.1063/1.1664899

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Abstract A partial trace over the occupation numbers of all but k states in the density matrix of an ensemble with an arbitrary number of single‐particle states is defined as the (reduced) k‐state density matrix. This matrix is used to obtain a complete, practical solution to the problem of determining the representability of the diagonal elements of the one‐ and two‐particle (reduced) density matrices. This solution is expressed as a series of linear inequalities involving the density‐matrix elements; the inequalities are identical with those derived previously by Davidson and McCrae by a different method. In addition, our method is used to obtain nonlinear, matrix inequalities on the off‐diagonal elements of the density matrices.
A general method is described for finding linear inequalities relating physical properties of an ensemble which can exist only in a finite number of states. This algorithm is used to solve a few examples of the fixed-N and variable-N Slater hull problem and the fixed-N Boson hull problem. Interpretations of the results are made in terms of boundary conditions for fermion and boson reduced density matrices, pair distributions of lattice vacancies, and pair distributions of spins in an Ising model.
Properties and NNN-representability conditions of the two-body density matrix
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It is a hard and important problem to find the criterion of a set of positive-definite matrices which can be written as reduced density operators of a multipartite global quantum state. This problem is closely related to the study of many-body quantum entanglement, which is one of the focuses of current quantum-information theory. We give several results on the necessary compatibility relations between a set of reduced and global density matrices, including (i) compatibility conditions for the one-party reduced density matrices of any $({N}_{A}\ifmmode\times\else\texttimes\fi{}{N}_{B})$-dimensional bipartite mixed quantum state, (ii) compatibility conditions for the one-party reduced matrices of any MMM-partite quantum states with the dimension ${N}^{\ensuremath{\bigotimes}M}$, and (iii) compatibility conditions for the two-party reduced density matrixes of $(2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2)$-dimensional tripartite mixed quantum state.
The reduction of density matrices defined with respect to a finite basis set is considered. A basis set can be introduced into the vector space of Hermitian matrices acting on functions of the coordinates of ppp particles for each value of ppp. An analogy with the construction of spin eigenfunctions is used to obtain a particular basis set called the reducing basis. Reduction of one of these basis elements is either one to one or maps into the origin. Any element in the preimage of a density matrix can thus be resolved into two components, one uniquely determined by the density matrix and the other arbitrary within a certin subset. A restatement of the NNN-representability problem is given, and two sufficient conditions and one necessary condition for NNN representability are given in terms of distances and norms.