Optimal quantum state identification with qudit-encoded unknown states

Abstract

We consider the problem of optimally identifying the state of a probe qudit, prepared with given prior probability in a pure state belonging to a finite set of possible states which together span a $D$-dimensional subspace of the $d$-dimensional Hilbert space the qudit is defined in. It is assumed that we do not know some or all of the states in the set, but for each unknown state we are given a reference qudit into which this state is encoded. We show that from the measurement for optimal state identification with $d=D$ one can readily determine the optimal figure of merit for qudits with $d>D$, without solving a new optimization problem. This result is applied to the minimum-error identification and to the optimal unambiguous identification of two qudit states with $d\ensuremath{\ge}2$, where either one or both of the states are unknown, and also to the optimal unambiguous identification of $N$ equiprobable linearly independent unknown pure qudit states with $d\ensuremath{\ge}N$. In all cases the optimal figure of merit, averaged over the unknown states, increases with growing dimensionality $d$ of the qudits.