[Read before The Royal Statistical Society at a meeting organized by the Research Section

Variance and Fisher information are ingredients of the Cramer-Rao inequality. We regard Fisher information as a Riemannian metric on a quantum statistical manifold and choose monotonicity under coarse graining as the fundamental property of variance and Fisher information. In this approach we show that there is a kind of dual one-to-one correspondence between the candidates of the two concepts. We emphasis that Fisher informations are obtained from relative entropies as contrast functions on the state space and argue that the scalar curvature might be interpreted as an uncertainty density on a statistical manifold.

Abstract not found

We derive essential elements of quantum mechanics from a parametric structure extending that of traditional mathematical statistics. The basic setting is a set A of incompatible experiments, and a transformation group G on the cartesian product Π of the parameter spaces of these experiments. The set of possible parameters is constrained to lie in a subspace of Π, an orbit or a set of orbits of G. Each possible model is then connected to a parametric Hilbert space. The spaces of different experiments are linked unitarily, thus defining a common Hilbert space H. A state is equivalent to a question together with an answer: the choice of an experiment a ∈ A plus a value for the corresponding parameter. Finally, probabilities are introduced through Born’s formula, which is derived from a recent version of Gleason’s theorem. This then leads to the usual formalism of elementary quantum mechanics in important special cases. The theory is illustrated by the example of a quantum particle with spin. 1. Introduction. Both

We deal with the maximization of classical Fisher information in a quantum system depending on an unknown parameter. This problem has been raised by physicists, who defined [Helstrom (1967) Phys. Lett. A 25 101–102] a quantum counterpart of classical Fisher information, which has been found to constitute an upper bound for classical information itself [Braunstein and Caves (1994) Phys. Rev. Lett. 72 3439–3443]. It has then become of relevant interest among statisticians, who investigated the relations between classical and quantum information and derived a condition for equality in the particular case of two-dimensional pure state systems [Barndorff-Nielsen and Gill (2000) J. Phys. A 33 4481–4490]. In this paper we show that this condition holds even in the more general setting of two-dimensional mixed state systems. We also derive the expression of the maximum Fisher information achievable and its relation with that attainable in pure states. 1. Introduction. Quantum

We propose an adaptive, two steps strategy, for the estimation of mixed qubit states. We show that the strategy is optimal in a local minimax sense for the norm one distance as well as other locally quadratic figures of merit. Local minimax optimality means that given n identical qubits, there exists no estimator which can perform better than the proposed estimator on a neighborhood of size n −1/2 of an arbitrary state. In particular, it is asymptotically Bayesian optimal for a large class of prior distributions. We present a physical implementation of the optimal measurement based on continuous time measurements in a field that couples with the qubits. The crucial ingredient of the result is the concept of local asymptotic normality (or LAN) for qubits. This means that, for large n, the statistical model described by n identically prepared qubits is locally equivalent to a model with only a classical Gaussian distribution and a Gaussian state of a quantum harmonic oscillator. The term ‘local ’ refers to a shrinking neighborhood around a fixed state ρ0. An essential result is that the neighborhood radius can be chosen arbitrarily close to n −1/4. This allows us to use a two steps procedure by which we first localize the state within a smaller neighborhood of radius n −1/2+ǫ, and then use LAN to perform optimal estimation. 1 1

Abstract not found

This paper concerns the design problem of choosing the measurement that provides the maximum Fisher information for the unknown parameter of a quantum system. We show that when the system under investigation is described by a one-parameter n-dimensional pure state model an optimal measurement exists, such that Fisher information attains the upper bound constituted by Hel-strom information. A characterisation theorem and two strategies of im-plementations are derived and discussed. These results generalise to n-dimensional spaces those obtained for n = 2 by Barndorff-Nielsen and Gill (2000). AMS (2000) subject classification. Primary 62B05; secondary 62F10.

Optimal estimation of one parameter quantum channels

Abstract. Variance and Fisher information are ingredients of the Cramer-Rao inequality. Fisher information is regarded as a Riemannian metric on a quantum statistical manifold and we choose monotonicity under coarse graining as the fundamental property. The quadratic cost functions are in a dual relation with the Fisher information quantities and they reduce to the variance in the commuting case. The scalar curvature in a certain geometry might be interpreted as an uncertainty on a statistical manifold. Information geometry has a surprising application to the theory of geometric mean of matrices. 1. The Cramer-Rao Inequality The Cramer-Rao inequality belongs to the basics of estimation theory in mathematical statistics. Its quantum analog was discovered immediately after the foundation of mathematical quantum estimation theory in the 1960's, see the book [10] of Helstrom, or the book [11] of Holevo for a rigorous summary of the subject. Although both the classical Cramer-Rao inequality and its quantum analog are as trivial as the Schwarz inequality, the subject takes a lot of attention because it is located on the highly exciting boundary of statistics, information and quantum theory. As a starting point we give a very general form of the quantum Cramer-Rao inequality in the simple setting of nite dimensional quantum mechanics. For 2 ( "; ") R a statistical operator is given and the aim is to estimate the value of the parameter close to 0. Formally is an n n positive semide nite matrix of trace 1 which describes a mixed state of a quantum mechanical system and we assume that is smooth (in). Assume that an estimation is performed by the measurement of a self-adjoint matrix A playing the role of an observable. A is called locally unbiased estimator if