Parametrization of quantum states and the quantum state discrimination problem

Abstract

A discrimination problem consists of $N$ linearly independent pure quantum states $\Phi =\{|{\varphi }_i⟩\}$ 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 ${\mathbf{\rho }}_{{}_{\mathbf{T}}}$ and ${\mathbf{\eta }}_{\mathbf{p}}$ defined on the $N$-dimensional Hilbert space ${\mathcal{H}}_N$. The first one, ${\mathbf{\rho }}_{{}_{\mathbf{T}}}$, 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 ${\gamma }_{ij}=⟨{\varphi }_i|{\varphi }_j⟩$ and the occurrence probabilities $\{{\eta }_i\}$. The second one is defined as a diagonal density matrix ${\mathbf{\eta }}_{\mathbf{p}}$ with the diagonal entries given by the probabilities $\{{\eta }_i\}$ with the ordering induced by the permutation $p$ of the probabilities. ${\mathbf{\rho }}_{{}_{\mathbf{T}}}$ and ${\mathbf{\eta }}_{\mathbf{p}}$ 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 ${\mathbf{\rho }}_{{}_{\mathbf{T}}}\to {\mathbf{\eta }}_{\mathbf{p}}$. Moreover, if the set lacks its independency and cannot be discriminated anymore the distinguishability of the pair, measured by the fidelity $F({\mathbf{\rho }}_{{}_{\mathbf{T}}},{\mathbf{\eta }}_{\mathbf{p}})$, becomes minimum. This enables one to associate to each discrimination problem a measure of discriminability defined by the fidelity $F({\mathbf{\rho }}_{{}_{\mathbf{T}}},{\mathbf{\eta }}_{\mathbf{p}})$. 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.

7 pages, 3 figures, Title, introduction and presentation are changed

References in corpus (8)

• No-local-broadcasting theorem for quantum correlations
• On the conditions for discrimination between quantum states with minimum error
• Quantum state discrimination: a geometric approach
• Optimal Unambiguous Discrimination of Quantum States
• Complete solution for unambiguous discrimination of three pure states with real inner products
• Unambiguous discrimination of linearly independent pure quantum states: Optimal average probability of success
• Squaring parametrization of constrained and unconstrained sets of quantum states
• An Explicit Computation of the Bures Metric Over the Space of $N$-Dimensional Density Matrices

Cited by in corpus (2)

• Geometric Bloch Vector Solution to Minimum Error Discriminations of Mixed Qubit States
• Minimum-error discrimination of thermal states