Ensemble and purification

Let $H$ be a nonzero finite-dimensional complex-inner-product space. A finite ensemble of density operators on $H$ is a positive integer $n$, weights $p_1,\dots ,p_n\ge 0$ with ${\sum }_{a=1}^n{p}_a=1$, and density operators ${\rho }_1,\dots ,{\rho }_n$ on $H$. Its barycenter is $\bar{\rho }={\sum }_{a=1}^n{p}_a{\rho }_a$. A zero-weight component still names a density operator but contributes zero to the barycenter.

Let $K$ be any finite-dimensional complex-inner-product space. A purification of a density operator $\rho$ on $H$ is a unit vector $\mathrm{\Psi }\in H\otimes K$ such that

${\mathrm{Tr}}_K(\mathrm{\mid }\mathrm{\Psi }⟩⟨\mathrm{\Psi }\mathrm{\mid })=\rho \mathrm{.}$

The tensor factors are ordered $H$ then $K$. Tensor products, outer products and partial trace have the conventions of Ordered tensor registers, their inner product, and wire permutations, Dirac Notation Conversion and Partial trace over a finite tensor factor. The auxiliary space may be any finite dimension that supports such a unit vector.