Parametrization of quantum states and the quantum state discrimination problem

Abstract

A discrimination problem consists of $N$ linearly independent pure quantum states $\Phi=\{\ket{\phi_i}\}$ and the corresponding occurrence probabilities $\eta=\{\eta_i\}$. To any such problem we associate, up to a permutation over the probabilities $\{\eta_i\}$, a unique pair of density matrices $\boldsymbol{\rho_{_{T}}}$ and $\boldsymbol{\eta_{{p}}}$ defined on the $N$-dimensional Hilbert space $\mathcal{H}_N$. The first one, $\boldsymbol{\rho_{_{T}}}$, 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 $\gamma_{ij}=\bra{\phi_i}\phi_j\rangle$ and the occurrence probabilities $\{\eta_i\}$. The second one is defined as a diagonal density matrix $\boldsymbol{\eta_p}$ with the diagonal entries given by the probabilities $\{\eta_i\}$ with the ordering induced by the permutation $p$ of the probabilities. $\boldsymbol{\rho_{_{T}}}$ and $\boldsymbol{\eta_{{p}}}$ 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 $\boldsymbol{\rho_{_{T}}}\rightarrow \boldsymbol{\eta_{{p}}}$. Moreover, if the set lacks its independency and cannot be discriminated anymore the distinguishability of the pair, measured by the fidelity $F(\boldsymbol{\rho_{_{T}}}, \boldsymbol{\eta_{{p}}})$, becomes minimum. This enables one to associate to each discrimination problem a measure of discriminability defined by the fidelity $F(\boldsymbol{\rho_{_{T}}}, \boldsymbol{\eta_{{p}}})$. This quantity, 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.