  The minimum error discrimination problem for ensembles of linearly
independent pure states are known to have an interesting structure; for such a
given ensemble the optimal POVM is given by the pretty good measurment of
another ensemble which can be related to the former ensemble by a bijective
mapping $\mathscr{R}$ on the "space of ensembles". In this paper we generalize
this result to ensembles of general linearly independent states (not
necessarily pure) and also give an analytic expression for the inverse of the
map, i.e., for $\mathscr{R}^{-1}$. In the process of proving this we also
simplify the necessary and sufficient conditions that a POVM needs to satisfy
to maximize the probability of success for the MED of an LI ensemble of states.
This simplification is then employed to arrive at a rotationally invariant
necessary and sufficient conditions of optimality. Using these rotationally
invariant conditions it is established that every state of a LI mixed state
ensemble can be resolved to a pure state decomposition so that the
corresponding pure state ensemble (corresponding to pure states of all mixed
states together) has as its optimal POVM a pure state decomposition of the
optimal POVM of mixed state ensemble. This gives the necessary and sufficient
conditions for the PGM of a LI ensemble to be its optimal POVM; another
generalization for the pure state case. Also, these rotationally invariant
conditions suggest a technique to give the optimal POVM for an ensemble of LI
states. This technique is polynomial in time and outpeforms standard
barrier-type interior point SDP in terms of computational complexity.

Quantum Physics