Parametrization of quantum states and the quantum state discrimination problem

TL;DR

This paper introduces a new parametrization of quantum states based on discrimination problem parameters, linking quantum and classical distinguishability, and proposes a fidelity measure to quantify discriminability.

Contribution

It defines a unique pair of density matrices for each discrimination problem, connecting quantum overlaps and classical probabilities, and introduces a fidelity-based measure of discriminability.

Findings

The pair of density matrices captures quantum and classical discrimination information.

Fidelity between the matrices quantifies the set's discriminability.

The measure is computationally simple and applicable to estimate discriminability extent.

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.