Linear, generally Hermitian operators

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]. 

It was von Neumann [26] who first dealt with the problem of the quantum statistical ensemble. The density operator is the statistical operator of a quantum statistical ensemble. In our case, the statistical ensemble is a set of linearly damped oscillators of several quantum states in contact with a heat bath with temperature T. The density operator is an operator whose eigenvalues are the classical statistical probability of the chosen microstates denoted by p(. If the chosen microstates are denoted by I i), which are eigenstates of a Hermitian operator but not necessarily the eigenstate of a Bohlinian or Hamiltonian, the general density operator is written as... 

In general ViAfi is not equal to MiVi but is its Hermitian conjugate, since (p7r)t = Trtpt. Therefore, it should be reasonably obvious that the ViAfi operators are also linearly independent. We note that an alternative, but very similar, proof that all a, = 0 in Eq. (5.42) could be constructed by multiplying on the left by Vj, j = 1,2,...,/ sequentially. 

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