Diagonal elements linear inequalities

LINEAR INEQUALITIES FOR DIAGONAL ELEMENTS OF DENSITY MATRICES... 

G. Constraints on Off-Diagonal Elements from Other Positive-Definite Hamiltonians Linear Inequalities from the Spatial Representation Linking the Orbital and Spatial Representations... 

The preceding is a rather comprehensive—but not exhaustive— review of N-representability constraints for diagonal elements of reduced density matrices. The most general and most powerful V-representability conditions seem to take the form of linear inequalities, wherein one states that the expectation value of some positive semidefinite linear Hermitian operator is greater than or equal to zero, Tr [PnTn] > 0. If Pn depends only on 2-body operators, then it can be reduced into a g-electron reduced operator, Pq, and Tr[Pg vrg] > 0 provides a constraint for the V-representability of the g-electron reduced density matrix, or 2-matrix. Requiring that Tr[Pg Arrg] > 0 for every 2-body positive semidefinite linear operator is necessary and sufficient for the V-representability of the 2-matrix [22]. 

Ayers, P W. Davidson, E. R. Linear inequalities for diagonal elements of density matrices. Adv. Chem. Phys. 2007,134, 443-483. 

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