2008/06/30 by B. F. Samsonov, Boris F. Samsonov
Computer Science · Mathematics · Physics and Astronomy · #Algorithm #Cartesian product #Combinatorics #Dimension (graph theory) #Extreme point #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Operator (biology) #POVM #Pure mathematics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum mechanics #Quantum operation #State (computer science) #Unit sphere #quant-ph
paper · pdf · doi:10.1103/physreva.79.042312
published as Phys. Rev. A 79, 042312 (2009) · Thanks to the referee the paper is essentially enlarged and corrected. To be published in Phys. Rev. A
arxiv created 2009/03/18 · openalex publication_date 2009/04/08 · arxiv updated 2013/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Optimization of the mean efficiency for unambiguous (or error-free) discrimination among N given linearly independent nonorthogonal states should be realized in a way to keep the probabilistic quantum-mechanical interpretation. This imposes a condition on a certain matrix to be positive semidefinite. We reformulated this condition in such a way that the conditioned optimization problem for the mean efficiency was reduced to finding an unconditioned maximum of a function defined on a unit N sphere for equiprobable states and on an N ellipsoid if the states are given with different probabilities. We established that for equiprobable states a point on the sphere with equal values of Cartesian coordinates, which we call symmetric point, plays a special role. Sufficient conditions for a vector set are formulated for which the mean efficiency for equiprobable states takes its maximal value at the symmetric point. This set, in particular, includes previously studied symmetric states. A subset of symmetric states, for which the optimal measurement corresponds to a positive-operator-valued measure (POVM) requiring a one-dimensional ancilla space is constructed. We presented our constructions of a POVM suitable for the ancilla space dimension varying from 1 to N and the Neumark extension differing from the existing schemes by the property that it is straightforwardly applicable to the case when it is desirable to present the whole space system+ancilla as the tensor product of a two-dimensional ancilla space and the N-dimensional system space.