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Sufficiency in quantum statistical inference
"... This paper attempts to develop a theory of sufficiency in the setting of noncommutative algebras parallel to the ideas in classical mathematical statistics. Sufficiency of a coarsegraining means that all information is extracted about the mutual relation of a given family of states. In the paper s ..."
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This paper attempts to develop a theory of sufficiency in the setting of noncommutative algebras parallel to the ideas in classical mathematical statistics. Sufficiency of a coarsegraining means that all information is extracted about the mutual relation of a given family of states. In the paper su cient coarsegrainings are characterized in several equivalent ways and the noncommutative analogue of the factorization theorem is obtained. As an application we discuss exponential families. Our factorization theorem also implies two further important results, previously known only infinite Hilbert space dimension, but proved here in generality: the KoashiImoto theorem on maps leaving a family of states invariant, and the characterization of the general form of states in the equality case of strong subadditivity.
A quantum logic of down below
 Handbook of Quantum Logic, Quantum Structure, and Quantum Computation
"... The logic that was purposebuilt to accommodate the hopedfor reduction of arithmetic gave to language a dominant and pivotal place. Flowing from the founding efforts of Frege, Peirce, and Whitehead and Russell, this was a logic that incorporated proof theory into syntax, and in so doing made of gra ..."
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Cited by 14 (11 self)
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The logic that was purposebuilt to accommodate the hopedfor reduction of arithmetic gave to language a dominant and pivotal place. Flowing from the founding efforts of Frege, Peirce, and Whitehead and Russell, this was a logic that incorporated proof theory into syntax, and in so doing made of grammar
Accardi Contra Bell (cum Mundi): The Impossible Coupling
"... this paper allow the author to determine a protocol which will be acceptable for him ..."
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this paper allow the author to determine a protocol which will be acceptable for him
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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Cited by 6 (0 self)
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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
General probabilistic framework of randomness. Research Report No 8
, 2003
"... We introduce a new mathematical framework for the probabilistic description of an experiment upon a system of any type in terms of initial information representing this system. Based on the notions of an information state, an information state space and a generalized observable, this general framewo ..."
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We introduce a new mathematical framework for the probabilistic description of an experiment upon a system of any type in terms of initial information representing this system. Based on the notions of an information state, an information state space and a generalized observable, this general framework covers the description of a wide range of experimental situations including those where, with respect to a system, an experiment is perturbing. We prove that, to any experiment upon a system, there corresponds a unique generalized observable on a system initial information state space, which defines the probability distribution of outcomes under this experiment. We specify the case where initial information on a system provides ”no knowledge ” for the description of an experiment. Incorporating in a uniform way the basic notions of conventional probability theory and the noncommutativity aspects and the basic notions of quantum measurement theory, our framework clarifies the principle difference between Kolmogorov’s model in probability theory and the statistical model of quantum theory. Both models are included into our framework as particular cases. We show that the phenomenon of ”reduction ” of a system initial information
An Invitation to Quantum Tomography
 Journal of the Royal Statistical Society (B
"... this paper. This is indeed a classical tomography problem: we take observations from all possible onedimensional projections of a twodimensional probability density. The nonclassical feature is that though all these onedimensional projections are indeed bonafide probability densities, the under ..."
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this paper. This is indeed a classical tomography problem: we take observations from all possible onedimensional projections of a twodimensional probability density. The nonclassical feature is that though all these onedimensional projections are indeed bonafide probability densities, the underlying twodimensional "joint density" need not itself be a bonafide joint density, but can have small patches of "negative probability density"
Nonparametric estimation of the purity of a quantum state in quantum homodyne tomography with noisy data
 Mathematical Methods of Statistics
"... The aim of this paper is to answer an important issue in quantum mechanics, namely to determine if a quantum state of a light beam is pure or mixed. The estimation of the purity is done from measurements by Quantum Homodyne Tomography performed on identically prepared quantum systems. The quantum s ..."
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The aim of this paper is to answer an important issue in quantum mechanics, namely to determine if a quantum state of a light beam is pure or mixed. The estimation of the purity is done from measurements by Quantum Homodyne Tomography performed on identically prepared quantum systems. The quantum state of the light is entirely characterized by the Wigner function, a density of generalized joint probability which can take negative values and which must respect certain constraints of positivity imposed by quantum physics. We propose to estimate a quadratic functional of the Wigner function by a kernel method as the physical measure of the purity of the state. We give also an adaptive estimator that does not depend on the smoothness parameters and we establish upper bound on the minimax risk over a class of infinitely differentiable functions. 1 ha l0
Quantum theory as a statistical theory under symmetry
 In Foundations of Probability and Physics 3
, 2005
"... The aim of the paper is to derive essential elements of quantum mechanics from a parametric structure extending that of traditional mathematical statistics. The main extensions, which also can be motivated from an applied statistics point of view, relate to symmetry, the choice between complementary ..."
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The aim of the paper is to derive essential elements of quantum mechanics from a parametric structure extending that of traditional mathematical statistics. The main extensions, which also can be motivated from an applied statistics point of view, relate to symmetry, the choice between complementary experiments and hence complementary parametric models, and use of the fact that there for simple systems always is a limited experimental basis that is common to all potential experiments. Concepts related to transformation groups together with the statistical concept of sufficiency are used in the construction of the quantummechanical Hilbert space. The Born formula is motivated through recent analysis by Deutsch and Gill, and is shown to imply the formulae of elementary quantum probability / quantum inference theory in the simple case. Planck’s constant, and the Schrödinger equation are also derived from this conceptual framework. The theory is illustrated by one and
Maximum likelihood versus likelihoodfree quantum system identification in the atom maser. arXiv:1311.4091
, 2013
"... the atom maser ..."
Posterior distribution for negative binomial parameter p using a group invariant
, 2007
"... We obtain a noninformative prior measure for the p parameter of the negative binomial distribution by use of a group theoretic method. Heretofore, group theoretic inference methods have not been applicable in the case of discrete distributions. A linear representation of a group leads to quantities ..."
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We obtain a noninformative prior measure for the p parameter of the negative binomial distribution by use of a group theoretic method. Heretofore, group theoretic inference methods have not been applicable in the case of discrete distributions. A linear representation of a group leads to quantities whose squared moduli constitute the probability distribution. The group invariant measure yields prior measure dp/p2.