Parametrization of quantum states based on the quantum state discrimination problem

A discrimination problem consists of N linearly independent pure quantum states \(\Phi =\{|\phi _i\rangle \}\) and the corresponding occurrence probabilities \(\eta =\{\eta _i\}\). With any such problem, we associate, up to a permutation over the probabilities \(\{\eta _i\}\), a unique pair of density matrices \(\varvec{\rho _{_{T}}}\) and \(\varvec{\eta _{{p}}}\) defined on the N-dimensional Hilbert space \(\mathcal {H}_N\). The first one, \(\varvec{\rho _{_{T}}}\), provides a new parametrization of a generic full-rank density matrix in terms of the parameters of the discrimination problem, i.e., the mutual overlaps \(\gamma _{ij}=\langle \phi _i|\phi _j\rangle \) and the occurrence probabilities \(\{\eta _i\}\). The second one, on the other hand, is defined as a diagonal density matrix \(\varvec{\eta _p}\) with the diagonal entries given by the probabilities \(\{\eta _i\}\) with the ordering induced by the permutation p of the probabilities. \(\varvec{\rho _{_{T}}}\) and \(\varvec{\eta _{{p}}}\) capture information about the quantum and classical versions of the discrimination problem, respectively. In this sense, when the set \(\Phi \) can be discriminated unambiguously with probability one, i.e., when the states to be discriminated are mutually orthogonal and can be distinguished by a classical observer, then \(\varvec{\rho _{_{T}}} \rightarrow \varvec{\eta _{{p}}}\). Moreover, if the set lacks its independency and cannot be discriminated anymore, the distinguishability of the pair, measured by the fidelity \(F(\varvec{\rho _{_{T}}}, \varvec{\eta _{{p}}})\), becomes minimum. This enables one to associate with each discrimination problem a measure of discriminability defined by the fidelity \(F(\varvec{\rho _{_{T}}}, \varvec{\eta _{{p}}})\). This quantity, though distinct from the maximum probability of success, has the advantage of being easy to calculate, and in this respect, it can find useful applications in estimating the extent to which the set is discriminable.