Correct interpretation of trace normalized density matrices as ensembles

Abstract

A density operator, $\rho = {P}_{\alpha } |\alpha > <\alpha | + {P}_{\beta } |\beta > <\beta |$, with ${P}_{\alpha }$ and ${P}_{\beta }$ linearly independent normalized wave functions, must be traced normalized, so ${P}_{\beta } = 1 - {P}_{\alpha }$. However, unless $<\alpha |\beta > = 0$, ${P}_{\alpha }$ and ${P}_{\beta }$ cannot be interpreted as probabilities of finding $|\alpha >$ and $|\beta >$ respectively. We show that a density matrix comprised of two (${P}_{\alpha }$ and ${P}_{\beta }$ nonzero) non-orthogonal projectors have unique spectral decomposition into diagonal form with orthogonal projectors. Only then, according to axioms of Von Neumann and Fock, can we have probability interpretation of that density matrix, only then can the diagonal elements be interpreted as probabilities of an ensemble. Those probabilities on the diagonal are not ${P}_{\alpha }$ and ${P}_{\beta}$. Further, only in the case of orthogonal projectors can we have the degenerate situation in which multiple ensembles are permitted.