Parametrization of quantum states and the quantum state discrimination problem

Abstract

A discrimination problem consists of $N$ linearly independent pure quantum states $Φ=\{\ket{ϕ_i}\}$ and the corresponding occurrence probabilities $η=\{η_i\}$. To any such problem we associate, up to a permutation over the probabilities $\{η_i\}$, a unique pair of density matrices $\boldsymbol{ρ_{_{T}}}$ and $\boldsymbol{η_{p}}$ defined on the $N$-dimensional Hilbert space $\mathcal{H}_N$. The first one, $\boldsymbol{ρ_{_{T}}}$, 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 $γ_{ij}=\bra{ϕ_i}ϕ_j\rangle$ and the occurrence probabilities $\{η_i\}$. The second one is defined as a diagonal density matrix $\boldsymbol{η_p}$ with the diagonal entries given by the probabilities $\{η_i\}$ with the ordering induced by the permutation $p$ of the probabilities. $\boldsymbol{ρ_{_{T}}}$ and $\boldsymbol{η_{p}}$ capture information about the quantum and classical versions of the discrimination problem, respectively. In this sense, when the set $Φ$ 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 $\boldsymbol{ρ_{_{T}}}\rightarrow \boldsymbol{η_{p}}$. Moreover, if the set lacks its independency and cannot be discriminated anymore the distinguishability of the pair, measured by the fidelity $F(\boldsymbol{ρ_{_{T}}}, \boldsymbol{η_{p}})$, becomes minimum. This enables one to associate to each discrimination problem a measure of discriminability defined by the fidelity $F(\boldsymbol{ρ_{_{T}}}, \boldsymbol{η_{p}})$. 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.