Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Infinite Dimensional Analysis Quantum Probability and Related Topics

EXTREMAL DECISION RULES IN QUANTUM HYPOTHESIS TESTING

Kalyanapuram Rangachari Parthasarathy

Published Dec 1, 1999• 47 citations
Publisher

Abstract

We present a complete description of the extreme points of the convex set of all decision rules in testing multiple hypotheses concerning a quantum system whose states are described by density matrices in a finite-dimensional Hilbert space. Holevo's equations for an optimal decision are derived by methods of elementary calculus and a simple example is given in order to illustrate the nonuniqueness of optimal decision rules.

DOI
10.1142/s0219025799000321

Related papers

Optimal multiple quantum statistical hypothesis testing

V. P. Belavkin

This paper is concerned with the problem of optimal M-alternative determination of quantum statistical states. A review of newest achievement of solving this problem is given. A notion of an effective decision Hilbert space is introduced and necessary and sufficient condkions for optimality of multiple quantum hypothesis testing in this space are formulated. The general solution is found for the case of a two-dimensional decision space. Another problem solved is that of discrimination of quantum pure non-orthogonal states. The result is represented in explicit analytical form for an "equidiagonal" case, which is quite general. In particular, we find explicit solutions of optimal discrimination problem of homogeneous and equiangle sets of pure states. These results are used for the M-ary detection problem in solving for the quantum coherent non-orthogonal signals. It is proved that the simplex signals are optimal elso in quantum case. The optimal estimatesof phaseandamplitude of quantum coherent signals are found. For decision operators a notion of IT-representation is introduced to get a general quasi-classical (optimal in quasi-classical limit) M-ary detection procedure of stochastic fields and particles, which submits to Bose-Einstein statistics. An optimal solution of problem of non-coherent detection of quantum stochastic (including optical) signals are found in the extreme quantum limit (weaknoise and signals with unknown phase).

Stochastics•Jan 1, 1973•114 citations

Extreme points of the set of density matrices with positive partial transpose

Jon Magne Leinaas, Jan Myrheim, Eirik Ovrum

We present a necessary and sufficient condition for a finite-dimensional density matrix to be an extreme point of the convex set of density matrices with positive partial transpose with respect to a subsystem. We also give an algorithm for finding such extreme points and illustrate this by some examples.

arXiv:0704.3348•Physical Review A•Sep 18, 2007•38 citations

EXTREMAL COVARIANT MEASUREMENTS

Giulio Chiribella, Giacomo

Abstract. We characterize the extremal points of the convex set of quantum measurements that are covariant under a finite-dimensional projective representation of a compact group, with action of the group on the measurement probability space which is generally non-transitive. In this case the POVM density is made of multiple orbits of positive operators, and, in the case of extremal measurements, we provide a bound for the number of orbits and for the rank of POVM elements. Two relevant applications are considered, concerning state discrimination with mutually unbiased bases and the maximization of the mutual information. 1. introduction A fundamental issue in the theory of quantum information [1] is the investigation of the ultimate precision limits for extracting classical information from a quantum system. Indeed, when the information is encoded on quantum states, its read-out suffers the intrinsically quantum limitation of discriminating among nonorthogonal states. One then need to optimize the discrimination with respect to a given optimality

•Jan 1, 2006•5 citations

On Consistency of the Maximum Likelihood Method in Testing Multiple Quantum Hypotheses

Kalyanapuram Rangachari Parthasarathy

We define the maximum likelihood method in testing a finite number of quantum hypotheses for quantum systems in a finite dimensional Hilbert space and prove its asymptotic consistency when the number of trials increases to infinity.

Birkhäuser Boston eBooks•Jan 1, 2001•8 citations

Quantum hypothesis testing and state discrimination

J. Prabhu Tej, Syed Raunaq Ahmed, A. R. Usha Devi +1 more

This expository article gives an overview of the theory of hypothesis testing of quantum states in finite dimensional Hilbert spaces. Optimal measurement strategy for testing binary quantum hypotheses, which result in minimum error probability, is discussed. Collective and individual adaptive measurement strategies in testing hypotheses in the multiple copy scenario, with various upper and lower bounds on error probability, are outlined. A brief account on quantum channel discrimination and the role of entangled states in achieving enhanced precision in the task of channel discrimination is given.

arXiv:1803.04944•Mar 13, 2018•5 citations

Extremal covariant measurements

Giulio Chiribella, Giacomo Mauro D’Ariano

We characterize the extremal points of the convex set of quantum measurements that are covariant under a finite-dimensional projective representation of a compact group, with action of the group on the measurement probability space which is generally nontransitive. In this case the POVM density is made of multiple orbits of positive operators, and, in the case of extremal measurements, we provide a bound for the number of orbits and for the rank of POVM elements. Two relevant applications are considered, concerning state discrimination with mutually unbiased bases and the maximization of the mutual information.

arXiv:quant-ph/0603168•Journal of Mathematical Physics•Sep 1, 2006•14 citations
We present a complete description of the extreme points of the convex set of all decision rules in testing multiple hypotheses concerning a quantum system whose states are described by density matrices in a finite-dimensional Hilbert space. Holevo's equations for an optimal decision are derived by methods of elementary calculus and a simple example is given in order to illustrate the nonuniqueness of optimal decision rules.
This paper is concerned with the problem of optimal M-alternative determination of quantum statistical states. A review of newest achievement of solving this problem is given. A notion of an effective decision Hilbert space is introduced and necessary and sufficient condkions for optimality of multiple quantum hypothesis testing in this space are formulated. The general solution is found for the case of a two-dimensional decision space. Another problem solved is that of discrimination of quantum pure non-orthogonal states. The result is represented in explicit analytical form for an "equidiagonal" case, which is quite general. In particular, we find explicit solutions of optimal discrimination problem of homogeneous and equiangle sets of pure states. These results are used for the M-ary detection problem in solving for the quantum coherent non-orthogonal signals. It is proved that the simplex signals are optimal elso in quantum case. The optimal estimatesof phaseandamplitude of quantum coherent signals are found. For decision operators a notion of IT-representation is introduced to get a general quasi-classical (optimal in quasi-classical limit) M-ary detection procedure of stochastic fields and particles, which submits to Bose-Einstein statistics. An optimal solution of problem of non-coherent detection of quantum stochastic (including optical) signals are found in the extreme quantum limit (weaknoise and signals with unknown phase).
Abstract. We characterize the extremal points of the convex set of quantum measurements that are covariant under a finite-dimensional projective representation of a compact group, with action of the group on the measurement probability space which is generally non-transitive. In this case the POVM density is made of multiple orbits of positive operators, and, in the case of extremal measurements, we provide a bound for the number of orbits and for the rank of POVM elements. Two relevant applications are considered, concerning state discrimination with mutually unbiased bases and the maximization of the mutual information. 1. introduction A fundamental issue in the theory of quantum information [1] is the investigation of the ultimate precision limits for extracting classical information from a quantum system. Indeed, when the information is encoded on quantum states, its read-out suffers the intrinsically quantum limitation of discriminating among nonorthogonal states. One then need to optimize the discrimination with respect to a given optimality
This expository article gives an overview of the theory of hypothesis testing of quantum states in finite dimensional Hilbert spaces. Optimal measurement strategy for testing binary quantum hypotheses, which result in minimum error probability, is discussed. Collective and individual adaptive measurement strategies in testing hypotheses in the multiple copy scenario, with various upper and lower bounds on error probability, are outlined. A brief account on quantum channel discrimination and the role of entangled states in achieving enhanced precision in the task of channel discrimination is given.
We characterize the extremal points of the convex set of quantum measurements that are covariant under a finite-dimensional projective representation of a compact group, with action of the group on the measurement probability space which is generally nontransitive. In this case the POVM density is made of multiple orbits of positive operators, and, in the case of extremal measurements, we provide a bound for the number of orbits and for the rank of POVM elements. Two relevant applications are considered, concerning state discrimination with mutually unbiased bases and the maximization of the mutual information.