Asymptotic Theory of Quantum Statistical Inference

The following sections are included:

• PREFACE

• CONTENTS

• FIRST APPEARANCE

• LIST OF CONTRIBUTORS

The following sections are included:

• Classical/Quantum Statistical Inference
  • Buildup of Classical Statistical Inference
  • Research Direction of Quantum Statistical Inference

• Significance of Classical/Quantum Statistical Inference

• Organization of This Book

• Mathematical Formulation

• References

The following sections are included:

• Quantum Hypothesis Testing
  • Setting of Quantum Hypothesis Testing
  • Quantum Stein's Lemma
  • Further Analysis of Quantum Stein's Lemma
  • Further Topics in Quantum Hypothesis Testing

• Multi-Hypotheses Testing and Distinguishability

• Mathematical Treatment of Classical Hypothesis Testing

• Further Reading

The hypothesis testing problem for two quantum states is treated. We show a new inequality between the errors of the first kind and the second kind, which complements the result of Hiai and Petz to establish the quantum version of Stein's lemma. The inequality is also used to show a bound on the probability of errors of the first kind when the power exponent for the probability of errors of the second kind exceeds the quantum relative entropy, which yields the strong converse in quantum hypothesis testing. Finally, we discuss the relation between the bound and the power exponent derived by Han and Kobayashi in classical hypothesis testing.

Umegaki's relative entropy S(ω, φ) = Tr D_ω(log D_ω − log D_φ) (of states ω and φ with density operators D_ω and D_φ, respectively) is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint. It is also proved that some other versions of the relative entropy give rise to the same asymptotics as Umegaki's one. As a byproduct, the inequality Tr A log AB ≥ Tr A(log A + log B) is obtained for positive definite matrices A and B.

There are many situations in information theory where it is required to find the threshold value of some relevant quantity for which a kind of error probability can asymptotically be made arbitrarily small. The strong converse property means that when the quantity exceeds the threshold value the error probability inevitably goes to one asymptotically. In quantum information theory, the strong converse property is known for (classical) capacity of quantum memoryless channel ([1, 4]) and for hypothesis testing problem of two quantum i.i.d. states ([2]). In this article we provide new proofs, which are based on a simple and unified argument, with these results.

In this paper it is proved that the quantum relative entropy D(ρ ‖ σ) can be asymptotically attained by the relative entropy of probabilities given by a certain sequence of positive-operator-valued measurements (POVMs). The sequence of POVMs depends on σ, but is independent of the choice of ρ.

This correspondence explores asymptotics of randomly generated vectors on extended Hilbert spaces. In particular, we are interested to know how “orthogonal” these vectors are. We investigate two types of asymptotic orthogonality, the weak orthogonality and the strong orthogonality, that are regarded as quantum analogs of the classical birthday problem and its variant. As regards the weak orthogonality, a new characterization of the von Neumann entropy is derived, and a mechanism behind the noiseless quantum channel coding theorem is clarified. As regards the strong orthogonality, on the other hand, a characterization of the quantum Réenyi entropy of degree 2 is derived.

The following sections are included:

• Main Issue in Quantum Estimation

• Asymptotic Theory of Classical Estimation

• Attainable Quantum Cramér-Rao Type Bound

• Attainable Quantum Cramér-Rao Type Bound with Quantum Correlation

• Global Attainability of CR Bound

• Mathematical Formulation in Classical Estimation
  • Cramér-Rao Inequality
  • Asymptotic Setting
  • Multi-Parametric Case

• Further Reading

The following sections are included:

• Introduction

• Quantum Measurement Theory

• Parameter Estimation and Lower Bounds of Cramér-Rao Type (CR Bounds)

• Some Examples of CR Bounds

• The Most Informative CR Bound

• A New CR Bound for the 2-Parameter Case

• References

• Additional Note by Author

The following sections are included:

• Introduction

• Statistical Operator and Non-Commutative Probability Theory

• Quantum Cramér-Rao Inequalities

• Quantum Exponential Family

• Measurements of the Quantum System and Fisher Information

• Quasi-Classical Model

• Sufficiency of Measurements and Fisher Information

• Conclusion

• References

The parameter estimation problem for a quantum statistical model , where {S_θ} are quantum states (i.e. density operators) and Θ is an open set in Rⁿ, is considered. The problem is to estimate the true state which is unknown but is assumed to lie in . The quantum analogues of various notions developed in the classical statistical estimation theory are studied; e.g., unbiased estimator, lower bound of Craméer-Rao type, efficient estimator, asymptotic efficiency, differential geometrical structures, etc. It is shown that the quantum theory for the one-parameter case (n = 1) is constructed almost in parallel with the classical theory, while several difficulties lie ahead of us to construct the general theory of quantum estimation for the multi-parameter case.

We study the problem of minimizing a quadratic quantity defined for given two Hermitian matrices X, Y and a positive-definite Hermitian matrix. This problem is reduced to the simultaneous diagonalization of X, Y when XY = YX. We derive a lower bound for the quantity, and in some special cases solve the problem by showing that the lower bound is achievable. This problem is closely related to a simultaneous measurement of quantum mechanical observables which are not commuting and has an application in the theory of quantum state estimation.

The author studies the relation between the attainable Cramér-Rao type bound and the duality theorem in the infinite dimensional linear programming. By this approach, the attainable Cramér-Rao type bound for a 3-parameter spin-1/2 model is explicitly derived.

The asymptotic efficiency of statistical estimate of unknown quantum states is discussed, both in adaptive and collective settings. Adaptive bounds are written in single-letterized form, and collective bounds are written in limiting expression. Our arguments clarify mathematical regularity conditions.

Concerning state estimation, we will compare two cases. In one case we cannot use the quantum correlations between samples. In the other case, we can use them. In addition, under the later case, we will propose a method which simultaneously measures the complex amplitude and the expected photon number for the displaced thermal states.

We consider the problem of estimating the state of a large but finite number N of identical quantum systems. As N becomes large the problem simplifies dramatically. The only relevant measure of the quality of estimation becomes the mean quadratic error matrix. Here we present a bound on this quantity: a quantum Cramér-Rao inequality. This bound expresses succinctly how in the quantum case one can trade information about one parameter for information about another. The bound holds for arbitrary measurements on pure states, but only for separable measurements on mixed states—a striking example of nonlocality without entanglement for mixed but not for pure states. Cramér-Rao bounds are generally only derived for unbiased estimators. Here we give a version of our bound for biased estimators, and a simple asymptotic version for large N. Finally we prove that when the unknown state belongs to a two-dimensional Hilbert space our quantum Cramér-Rao bound can always be attained and we provide an explicit measurement strategy that attains it. Thus we have a complete solution to the problem of estimating as efficiently as possible the unknown state of a large ensemble of qubits in the same pure state. The same is true for qubits in the same mixed state if one restricts oneself to separable measurements, but non-separable measurements allow dramatic increase of efficiency. Exactly how much increase is possible is a major open problem.

The following sections are included:

• Quantum Cramér-Rao Bound in Pure States Model

• Geometrical Study of Quantum Estimation

• Further Reading

A statistical parameter estimation theory for quantum pure state models is presented. We first investigate the basic framework of pure state estimation theory and derive quantum counterparts of the Fisher metric. We then formulate a one-parameter estimation theory, based on the symmetric logarithmic derivatives, and clarify the differences between pure state models and strictly positive models.

The following sections are included:

• Introduction

• Dualistic Geometry Based on the Logarithmic Derivatives
  • Dualistic structure on quantum states
  • Autoparallelity in quantum estimation theory
  • Autoparallelity in unitary models
  • Conclusions of Section 2

• Pure State Estimation Theory
  • Preliminaries
  • Quantum Fisher metric
  • One parameter pure state estimation
    • Time–energy uncertainty relation
    • Efficient estimator
  • Multi-parameter pure state estimation
    • Canonical coherent states
    • Spin coherent states
  • Linear random measurement
    • Gaussian model
    • Pure coherent models
  • Geometry of pure state estimation
    • Dualistic structure of pure state space
    • Curvature and torsion
  • Conclusions

• Appendix
  • Parameter Estimation of a Quantum Gaussian Model
  • Spin Coherent State
  • Proof of Inequality (7.33)
  • Photon Squeezed State

• References

We introduce a class of quantum pure state models called the coherent models. A coherent model is an even dimensional manifold of pure states whose tangent space is characterized by a symplectic structure. In a rigorous framework of noncommutative statistics, it is shown that a coherent model inherits and expands the original spirit of the minimum uncertainty property of coherent states.

The following sections are included:

• Introduction
  • The purposes of the thesis
  • Organization of the thesis

• Preliminaries
  • Classical estimation theory
  • Quantum mechanics and measurement theory
  • Unbiased estimator in quantum estimation theory

• Conceptual Framework
  • Horizontal lift and SLD
  • Definition of Uhlmann's parallelism
  • RPF for infinitesimal loop
  • The SLD Cramér-Rao inequality
  • The attainable Cramér-Rao type bound
  • The locally quasi-classical model and the quasi-classical model

• Uhlmann Connection and the Estimation Theory
  • Some new facts about RPF
  • Uhlmann's parallelism in the quantum estimation theory

• The Pure State Estimation Theory
  • Historical review of the theory and the purpose of the section
  • Notations
  • The commuting theorem and the locally quasi-classical model
  • The reduction theorem and the direct approach
  • Lagrange's method of indeterminate coefficients in the pure state estimation theory
  • The model with two parameters
  • Multiplication of the imaginary unit
  • The coherent model
  • Informationally exclusive, and independent parameters
  • Direct sum of models

• Berry's Phase in Quantum Estimation Theory
  • Berry's phase
  • Berry's phase in quantum estimation theory
  • Berry's phase in the global theory of quantum estimation
  • Antiunitary operators
  • Time reversal symmetry

• References

The following sections are included:

• Optimal Performance in the Covariant Model

• Optimization among Realizable POVMs

• Application to Other Problems

• Further Reading

Given only a finite ensemble of identically prepared particles, how precisely can one determine their states? We describe optimal measurement procedures in the case of spin-1/2 particles. Furthermore, we prove that optimal measurement procedures must necessarily view the ensemble as a single composite system rather than as the sum of its components, i.e., optimal measurements cannot be realized by measurements on each particle.

The optimization of measurement for n samples of pure sates are studied. The error of the optimal measurement for n samples is asymptotically compared with the one of the maximum likelihood estimators from n data given by the optimal measurement for one sample.

We derive a tight upper bound for the fidelity of a universal N → M qubit cloner, valid for any M ≥ N, where the output of the cloner is required to be supported on the symmetric subspace. Our proof is based on the concatenation of two cloners and the connection between quantum cloning and quantum state estimation. We generalize the operation of a quantum cloner to mixed and/or entangled input qubits described by a density matrix supported on the symmetric subspace of the constituent qubits. We also extend the validity of optimal state estimation methods to inputs of this kind.

The following sections are included:

• Introduction

• Quantum Covariant Family

• Pure State Case

• References

The following sections are included:

• Large Deviation Analysis in Classical Estimation

• First Large Deviation Analysis on Quantum Estimation

• Further Large Deviation Analysis on Quantum Estimation

• Estimation of Eigenvalues

• Further Reading

Kullback divergence D(p ∥ q) for a couple of probability distributions {p, q} and Fisher information J(θ) for a parametric family of probability distributions {p_θ} are very important notions in the classical theory of statistical inference. Quantum analogues of these notions are known as Umegaki's relative entropy D(ρ ‖ σ) for a couple of density operators {ρ, σ} and the quantum Fisher information J(θ) based on the pymmetric logarithmic derivatives for a parametric family of density operators {ρ_Θ}. In the classical case, the relation holds, which connects the asymptotic theory of hypothesis testing and that of parameter estimation. In the quantum case, the relation defines , which is not equal to J(θ) in general and is another quantum analogue of Fisher information. We show the inequality and elucidate its meaning in connection with the statistical inference for quantum states.

We discuss two quantum analogues of Fisher information, symmetric logarithmic derivative (SLD) Fisher information and Kubo-Mori-Bogoljubov (KMB) Fisher information from a large deviation viewpoint of quantum estimation and prove that the former gives the true bound and the latter gives the bound of consistent superefficient estimators. In another comparison, it is shown that the difference between them is characterized by the change of the order of limits.

Given N quantum systems prepared according to the same density operator ρ, we propose a measurement on the N-fold system that approximately yields the spectrum of ρ. The projections of the proposed observable decompose the Hilbert space according to the irreducible representations of the permutations on N points, and are labeled by Young frames, whose relative row lengths estimate the eigenvalues of ρ in decreasing order. We show convergence of these estimates in the limit N → ∞, and that the probability for errors decreases exponentially with a rate we compute explicitly.

The following sections are included:

• Estimation of Suantum State-Evolution Process

• Tomography

• Further Reading

A quantum clock must satisfy two basic constraints. The first is a bound on the time resolution of the clock given by the difference between its maximum and minimum energy eigenvalues. The second follows from Holevo's bound on how much classical information can be encoded in a quantum system. We show that asymptotically, as the dimension of the Hilbert space of the clock tends to infinity, both constraints can be satisfied simultaneously. The experimental realization of such an optimal quantum clock using trapped ions is discussed.

This paper explores an application of quantum entanglement. The problem treated here is the quantum channel identification problem: given a parametric family {Γ_θ}_θ of quantum channels, find the best strategy of estimating the true value of the parameter θ. As a simple example, we study the estimation problem of the isotropic depolarization parameter θ for a two-level quantum system . In the framework of noncommutative statistics, it is shown that the optimal input state on to the channel exhibits a transition-like behavior according to the value of the parameter θ.

Homodyne tomography — i.e., homodyning while scanning the local oscillator phase — is now a well assessed method for “measuring” the quantum state. In this paper I will show how it can be used as a kind of universal detection, for measuring generic field operators, however at expense of some additional noise. The general class of field operators that can be measured in this way is presented, and includes also operators that are inaccessible to heterodyne detection. The noise from tomographical homodyning is compared to that from heterodyning, for those operators that can be measured in both ways. It turns out that for some operators homodyning is better than heterodyning when the mean photon number is sufficiently small. Finally, the robustness of the method to additive phase-insensitive noise is analyzed. It is shown that just half photon of thermal noise would spoil the measurement completely.

We describe in detail the theory underpinning the measurement of density matrices of a pair of quantum two-level systems (“qubits”). Our particular emphasis is on qubits realized by the two polarization degrees of freedom of a pair of entangled photons generated in a down-conversion experiment; however, the discussion applies in general, regardless of the actual physical realization. Two techniques are discussed, namely, a tomographic reconstruction (in which the density matrix is linearly related to a set of measured quantities) and a maximum likelihood technique which requires numerical optimization (but has the advantage of producing density matrices which are always non-negative definite). In addition, a detailed error analysis is presented, allowing errors in quantities derived from the density matrix, such as the entropy or entanglement of formation, to be estimated. Examples based on down-conversion experiments are used to illustrate our results.

The following sections are included:

• INDEX