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

Birkhäuser Boston eBooks

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

Kalyanapuram Rangachari Parthasarathy

Published Jan 1, 2001• 8 citations
Publisher

Abstract

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.
DOI
10.1007/978-1-4612-0167-0_19

Related papers

A scalable maximum likelihood method for quantum state tomography

T Baumgratz, A Nüßeler, M. Cramer +1 more

The principle of maximum likelihood reconstruction has proven to yield satisfactory results in the context of quantum state tomography for many-body systems of moderate system sizes. Until recently, however, quantum state tomography has been considered to be infeasible for systems consisting of a large number of subsystems due to the exponential growth of the Hilbert space dimension with the number of constituents. Several reconstruction schemes have been proposed since then to overcome the two main obstacles in quantum many-body tomography: experiment time and post-processing resources. Here we discuss one strategy to address these limitations for the maximum likelihood principle by adopting a particular state representation to merge a well established reconstruction algorithm maximizing the likelihood with techniques known from quantum many-body theory.

arXiv:1308.2395•New Journal of Physics•Dec 4, 2013•60 citations

An asymptotic error bound for testing multiple quantum hypotheses

Michael Nussbaum, Arleta Szkoła

We consider the problem of detecting the true quantum state among r possible ones, based of measurements performed on n copies of a finite-dimensional quantum system. A special case is the problem of discriminating between r probability measures on a finite sample space, using n i.i.d. observations. In this classical setting, it is known that the averaged error probability decreases exponentially with exponent given by the worst case binary Chernoff bound between any possible pair of the r probability measures. Define analogously the multiple quantum Chernoff bound, considering all possible pairs of states. Recently, it has been shown that this asymptotic error bound is attainable in the case of r pure states, and that it is unimprovable in general. Here we extend the attainability result to a larger class of r-tuples of states which are possibly mixed, but pairwise linearly independent. We also construct a quantum detector which universally attains the multiple quantum Chernoff bound up to a factor 1/3.

arXiv:1112.1529•The Annals of Statistics•Dec 1, 2011•20 citations

An asymptotic error bound for testing multiple quantum hypotheses

Michael Nussbaum, Arleta Szko La

We consider the problem of detecting the true quantum state among r possible ones, based of measurements performed on n copies of a finite-dimensional quantum system. A special case is the problem of discriminating between r probability measures on a finite sample space, using n i.i.d. observations. In this classical setting, it is known that the averaged error probability decreases exponentially with exponent given by the worst case binary Chernoff bound between any possible pair of the r probability measures. Define analogously the multiple quantum Chernoff bound, considering all possible pairs of states. Recently, it has been shown that this asymptotic error bound is attainable in the case of r pure states, and that it is unimprovable in general. Here we extend the attainability result to a larger class of r-tuples of states which are possibly mixed, but pairwise linearly independent. We also construct a quantum detector which universally attains the multiple quantum Chernoff bound up to a factor 1/3.

•Jan 1, 2011•17 citations

Strong consistency and asymptotic efficiency for adaptive quantum estimation problems

Akio Fujiwara

It is shown that for an adaptive quantum estimation scheme based on locally unbiased measurements, the sequence of maximum likelihood estimators is strongly consistent and asymptotically efficient.

Journal of Physics A Mathematical and General•Sep 19, 2006•112 citations

EXTREMAL DECISION RULES IN QUANTUM HYPOTHESIS TESTING

Kalyanapuram Rangachari Parthasarathy

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.

Infinite Dimensional Analysis Quantum Probability and Related Topics•Dec 1, 1999•47 citations

10 Maximum-Likelihood Estimation in Experimental Quantum Physics

Gerald Badurek, Er Lvovsky, Alipasha Vaziri +1 more

Abstract. The theory of quantum state reconstruction is illustrated here on sev-eral examples taken from modern experimental praxis. Maximum-likelihood estima-tion is applied to experiments on physical systems of increasing complexity, starting with a simple one-dimensional problem of quantum phase estimation, continuing with the absorption and phase neutron tomographies, further discussing quantum tomography of higher-dimensional discrete quantum systems, and closing with the homodyne tomography of an infinite dimensional system- a mode of light. All these experiments nicely demonstrate the utility of present state-of-art techniques for ma-nipulating states of a neutron and internal as well as external states of a photon. Over the past hundred years, quantum theory proved to be an extremely useful tool for describing and understanding nature on a microscopic level. Its usual applications consist in predicting the outcomes of future experimental results given a quantum state describing the physical system.

•Nov 14, 2015•0 citations
The principle of maximum likelihood reconstruction has proven to yield satisfactory results in the context of quantum state tomography for many-body systems of moderate system sizes. Until recently, however, quantum state tomography has been considered to be infeasible for systems consisting of a large number of subsystems due to the exponential growth of the Hilbert space dimension with the number of constituents. Several reconstruction schemes have been proposed since then to overcome the two main obstacles in quantum many-body tomography: experiment time and post-processing resources. Here we discuss one strategy to address these limitations for the maximum likelihood principle by adopting a particular state representation to merge a well established reconstruction algorithm maximizing the likelihood with techniques known from quantum many-body theory.
We consider the problem of detecting the true quantum state among r possible ones, based of measurements performed on n copies of a finite-dimensional quantum system. A special case is the problem of discriminating between r probability measures on a finite sample space, using n i.i.d. observations. In this classical setting, it is known that the averaged error probability decreases exponentially with exponent given by the worst case binary Chernoff bound between any possible pair of the r probability measures. Define analogously the multiple quantum Chernoff bound, considering all possible pairs of states. Recently, it has been shown that this asymptotic error bound is attainable in the case of r pure states, and that it is unimprovable in general. Here we extend the attainability result to a larger class of r-tuples of states which are possibly mixed, but pairwise linearly independent. We also construct a quantum detector which universally attains the multiple quantum Chernoff bound up to a factor 1/3.
We consider the problem of detecting the true quantum state among r possible ones, based of measurements performed on n copies of a finite-dimensional quantum system. A special case is the problem of discriminating between r probability measures on a finite sample space, using n i.i.d. observations. In this classical setting, it is known that the averaged error probability decreases exponentially with exponent given by the worst case binary Chernoff bound between any possible pair of the r probability measures. Define analogously the multiple quantum Chernoff bound, considering all possible pairs of states. Recently, it has been shown that this asymptotic error bound is attainable in the case of r pure states, and that it is unimprovable in general. Here we extend the attainability result to a larger class of r-tuples of states which are possibly mixed, but pairwise linearly independent. We also construct a quantum detector which universally attains the multiple quantum Chernoff bound up to a factor 1/3.
Abstract. The theory of quantum state reconstruction is illustrated here on sev-eral examples taken from modern experimental praxis. Maximum-likelihood estima-tion is applied to experiments on physical systems of increasing complexity, starting with a simple one-dimensional problem of quantum phase estimation, continuing with the absorption and phase neutron tomographies, further discussing quantum tomography of higher-dimensional discrete quantum systems, and closing with the homodyne tomography of an infinite dimensional system- a mode of light. All these experiments nicely demonstrate the utility of present state-of-art techniques for ma-nipulating states of a neutron and internal as well as external states of a photon. Over the past hundred years, quantum theory proved to be an extremely useful tool for describing and understanding nature on a microscopic level. Its usual applications consist in predicting the outcomes of future experimental results given a quantum state describing the physical system.