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On quantum statistical inference
 J. Roy. Statist. Soc. B
, 2001
"... [Read before The Royal Statistical Society at a meeting organized by the Research Section ..."
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[Read before The Royal Statistical Society at a meeting organized by the Research Section
Point estimation of states of finite quantum systems
 J. Phys. A
, 2007
"... The estimation of the density matrix of a klevel quantum system is studied when the parametrization is given by the real and imaginary part of the entries, and they are estimated by independent measurements. It is established that the properties of the estimation procedure depend very much on the i ..."
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The estimation of the density matrix of a klevel quantum system is studied when the parametrization is given by the real and imaginary part of the entries, and they are estimated by independent measurements. It is established that the properties of the estimation procedure depend very much on the invertibility of the true state. In particular, in the case of a pure state, the estimation should be constrained to ensure the positive definiteness of the estimate. An efficient constraining algorithm is proposed and it yields an asymptotically unbiased estimate. Moreover, several estimation schemes are compared for the unknown state of a qubit when one copy is measured at a time. It is shown that the average mean quadratic error matrix is the smallest if the applied observables are complementary. All the results are illustrated by computer simulations. PACS numbers: 03.67.−a, 03.65.Wj, 03.65.Fd (Some figures in this article are in colour only in the electronic version) 1.
Asymptotic information bounds in quantum statistics
, 2009
"... We derive an asymptotic lower bound on the Bayes risk when N identical quantum systems whose state depends on a vector of unknown parameters are jointly measured in an arbitrary way and the parameters of interest estimated on the basis of the resulting data. The bound is an integrated version of a q ..."
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We derive an asymptotic lower bound on the Bayes risk when N identical quantum systems whose state depends on a vector of unknown parameters are jointly measured in an arbitrary way and the parameters of interest estimated on the basis of the resulting data. The bound is an integrated version of a quantum CramérRao bound due to Holevo (1982), and it thereby links the fixed N exact Bayesian optimality usually pursued in the physics literature with the pointwise asymptotic optimality favoured in classical mathematical statistics. By heuristic arguments the bound can be expected to be sharp. This does turn out to be the case in various important examples, where it can be used to prove asymptotic optimality of interesting and useful measurementandestimation schemes. On the way we obtain a new family of “dual Holevo bounds ” of independent interest. 1
Conciliation of Bayes and Pointwise Quantum State Estimation:
, 2009
"... asymptotic information bounds in quantum statistics ∗ ..."
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asymptotic information bounds in quantum statistics ∗
Quantum state estimation and large deviations
, 2004
"... In this paper we propose a method to estimate the density matrix ρ of a dlevel quantum system by measurements on the Nfold system in the joint state ρ ⊗N. The scheme is based on covariant observables and representation theory of unitary groups and it extends previous results concerning the estimat ..."
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In this paper we propose a method to estimate the density matrix ρ of a dlevel quantum system by measurements on the Nfold system in the joint state ρ ⊗N. The scheme is based on covariant observables and representation theory of unitary groups and it extends previous results concerning the estimation of the spectrum of ρ. We show that it is consistent (i.e. the original input state ρ is recovered with certainty if N → ∞), analyze its large deviation behavior, and calculate explicitly the corresponding rate function which describes the exponential decrease of error probabilities in the limit N → ∞. Finally we discuss the question whether the proposed scheme provides the fastest possible decay of error probabilities. 1
Quantum Information
"... . Recent developments in the mathematical foundations of quantum mechanics have brought the theory closer to that of classical stochastics (probability and statistics). On the other hand, the unique character of quantum physics sets many of the questions addressed apart from those met classically in ..."
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. Recent developments in the mathematical foundations of quantum mechanics have brought the theory closer to that of classical stochastics (probability and statistics). On the other hand, the unique character of quantum physics sets many of the questions addressed apart from those met classically in stochastics. Furthermore, concurrent advances in experimental techniques have led to a strong interest in questions of quantum information, in particular in the sense of the amount of information about unknown parameters in given observational data or accessible through various possible types of measurements. This scenery is outlined. 1 Introduction In the last two decades, developments of an axiomatic type in the mathematical foundations of quantum mechanics have brought the theory closer to that of classical stochastics 1 . On the other hand, the unique character of quantum physics sets many of the questions addressed apart from those met classically in stochastics. The key mathematic...
Complementarity and state estimation
 REP. MATH. PHYSICS, 65(2010), 203{214.
, 2010
"... The concept of complementarity (or quasiorthogonality) is extended to POVM's. It is shown in the setting of unconstrained state estimation that the determinant of the mean quadratic error matrix is minimal if the POVM's are complementary (and informationally complete). Several examples of ..."
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The concept of complementarity (or quasiorthogonality) is extended to POVM's. It is shown in the setting of unconstrained state estimation that the determinant of the mean quadratic error matrix is minimal if the POVM's are complementary (and informationally complete). Several examples of the scheme are given.