Tasks and premises in quantum state determination

Abstract

The purpose of quantum tomography is to determine an unknown quantum state\nfrom measurement outcome statistics. There are two obvious ways to generalize\nthis setting. First, our task need not be the determination of any possible\ninput state but only some input states, for instance pure states. Second, we\nmay have some prior information, or premise, which guarantees that the input\nstate belongs to some subset of states, for instance the set of states with\nrank less than half of the dimension of the Hilbert space. We investigate state\ndetermination under these two supplemental features, concentrating on the cases\nwhere the task and the premise are statements about the rank of the unknown\nstate. We characterize the structure of quantum observables (POVMs) that are\ncapable of fulfilling these type of determination tasks. After the general\ntreatment we focus on the class of covariant phase space observables, thus\nproviding physically relevant examples of observables both capable and\nincapable of performing these tasks. In this context, the effect of noise is\ndiscussed.\n