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Lecture Notes on Quantum Statistics

Richard D. Gill Part of book project with O.E. Barndorff-Nielsen and P.E. Jupp

Version: 30 January 2000

Preface These notes are meant to form the material for an introductory course on quantum statistics at the graduate level, aimed at mathematical statisticians and probabilists. No background in physics (quantum or otherwise) is required. They are still far from complete.

Quantum statistics as we mean it here is statistical inference based on data obained from measurement of a quantum system. The reader is probably aware that quantum physics makes stochastic predictions about reality. The actual outcome of an experiment involving measurements on some small number of elementary particles cannot be predicted. Rather, quantum mechanics allows one to compute the probability distribution of the outcomes. This probability distribution will depend on a specification of the system under study and on the chosen measurement apparatus. Often such a specification depends on parameters which are not known in advance and then the data could be used to gain information about them. These parameters could correspond to classical, macroscopic features of the system producing these elementary particles. Examples could be the orientation or position of apparatus producing a stream of photons in a quantum communication system, or properties of some distant star or other astronomical object so distant or weak that we can only detect a finite number of photons from this source during a finite observational period.

In the past, physical predictions made on quantum systems typically involved huge numbers of identical particles and focussed on their collective behaviour. The random nature of the outcome was submerged by the law of large numbers, and the aim was to compute expectated values and correlations. Thus quantum physics provided exact predictions of non-random, aggregate properties. Statistical questions might have been important but the underlying stochastic nature of the phenomenon under study at the level of individual particles did not play a role. Nowadays however experimentation and theory are focussing more and more on manipulating truly small quantum systems consisting of just one or a really small number of atoms, electrons, photons or whatever. Technology will surely follow. Theory and

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iii speculation on such topics as quantum computing, quantum cryptography, and quantum communication channels, far outrun experiment and technology but these are also developing fast and will in the near future, we believe, supply real life statistical problems prototypical forms of which are studied in these notes.

Does the reader need a background in physics in order to study quantum statistics? We believe not. A beautiful feature of quantum mechanics is that it can be described abstractly on the basis of a fairly straightforward mathematical apparatus: the mathematics of linear operators on a complex Hilbert space (the so-called state-space) together with some elementary probability theory. Moreover, when we are concerned purely with `discrete' properties of a quantum system such as spin of a finite number of electrons, polarization of a finite number of photons, the energy level (ground state or excited) of a finite array of atoms at very low temperature, and can ignore other aspects of the system under study, finite-dimensional Hilbert spaces suffice. Thus the mathematical model can be formulated in the language of finite dimensional complex vectors and matrices. Already in such a finite-dimensional set-up one can pose and try to solve important and non-trivial problems of quantum statistical inference. And these problems could turn up in quite diverse applications since the basic model could apply to different aspects of quite different systems. We do not yet know if the particular `toy' problems we consider will turn up in practice in the coming years, but we are confident that what we have learnt from their study will be of value in future applications.

The notes concentrate on the natural statistical problems arising from the simplest possible quantum settings. Moreover we study whether various key notions in mathematical statistics--such as exponential families or transformation models--have extensions or analogues in quantum statistics. In particular the statistical notion of information, and information bounds based on the Fisher information matrix, have already turned out to be very useful in quantum statistical design and inference problems. As we have said, the results of measurement of a quantum system are random, and quantum theory tells us how to compute the resulting probability distributions. Consequently quantum statistical inference is `ordinary' statistics applied to the kind of models which turn up in quantum mechanics.

Quantum theory places fundamental limits to the amount of information which can be obtained from a single quantum system. We have a very precise description of the class of all possible measurements on a given quantum system and this generates fundamental limits to the precision with which the state of the system can be determined through measurements on the system. Moroever one can distinguish subclasses of measurements which

iv might be more easy to implement in practice. Thus `experimental design' problems with a clean mathematical description arise very naturally.

Despite the enormous success of quantum mechanics, its foundations are still the subject of much controversy. This could be either an appealing or an off-putting feature of this problem area for readers, depending on their personal inclinations. Fortunately we do not need to take sides concerning foundational aspects of quantum mechanics. We simply take the model of quantum mechanics as a description of empirical reality. However the reader who is interested in foundational aspects--which involve what we mean by randomness and determinism, and hence could be of interest to philosophically inclined probabilists and statisticians--can meet all the paradoxical features of quantum mechanics arising from wave-particle duality such as entanglement, non-locality, and the measurement problem, in the context of simple finite-dimensional models. The controversies concern the proper interpretation and consistency of this mathematical picture. We plan to include some introductory material on these matters but emphasize that it is not necessary to take sides or even become aquainted with these problems in order to solve concrete quantum statistical problems, just as quantum physicists manage very well to make perfect predictions about the world without any agreement on the interpretation of their theory.

In recent years a large body of theory has been developed under the name of quantum probability. This sophisticated mathematical theory, which certainly has much to say about quantum physics itself, is largely inspired by a powerful analogy between the mathematics, in quantum theory, of so-called states and observables, and the mathematics of probability measures and random variables as studied in ordinary probability theory. One can consider quantum probability as a generalisation of ordinary probability obtained by dropping the requirement of commutativity of certain objects. This reflects the physical fact that measurement of a quantum system disturbs that system; its state changes as a result of the measurement and the sequence in which one performs certain measurements influences strongly the outcomes one will obtain. Quantum probability theory has however not traditionally been much interested in the `ordinary' probability theory of the outcomes of measurements, but rather has studied the internal evolution of a quantum system, and exploiting mathematical analogy has brought into the quantum world generalisations of classical notions from stochastic processes such as Markov processes, diffusions, and stochastic differential equations. Consequently we will not have much to say on this kind of quantum probability theory, though we hope to give some introduction to some current developments in that field.

Contents 1 Introduction 1

1.1 The Stern-Gerlach experiment . . . . . . . . . . . . . . . . . . 2 1.2 The basic model . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 States . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Spin half . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 The geometry of spin half . . . . . . . . . . . . . . . . 10 1.3.2 Measurement for spin half . . . . . . . . . . . . . . . . 13 1.3.3 Polarization. . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Problems and further results . . . . . . . . . . . . . . . . . . . 15 1.5 Bibliographic comments . . . . . . . . . . . . . . . . . . . . . 16

2 Observables and wave functions 17