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Cambridge University Press eBooks

Geometry of probability distributions

Ingemar Bengtsson, Karol Życzkowski

Published May 11, 2006• 0 citations
Publisher

Abstract

Some people hate the very name of statistics, but I find them full of beauty and interest. Sir Francis Galton In quantum mechanics one often encounters sets of non-negative numbers that sum to unity, having a more or less direct interpretation as probabilities. This includes the squared moduli of the coefficients when a pure state is expanded in an orthonormal basis, the eigenvalues of density matrices, and more. Continuous distributions also play a role, even when the Hilbert space is finite dimensional. From a purely mathematical point of view a probability distribution is simply a measure on a sample space, constrained so that the total measure is one. Whatever the point of view one takes on this, the space of states will turn into a convex set when we allow probabilistic mixtures of its pure states. In classical mechanics the sample space is phase space, which is typically a continuous space. This leads to technical complications but the space of states in classical mechanics does share a major simplifying feature with the discrete case, namely that every state can be expressed as a mixture of pure states in a unique way. This was not so in the case of colour space, nor will it be true for the convex set of all states in quantum mechanics. Majorization and partial order Our first aim is to find ways of describing probability distributions; we want to be able to tell when a probability distribution is ‘more chaotic’ or ‘more uniform’ than another. One way of doing this is provided by the theory of majorization .

DOI
10.1017/cbo9780511535048.003

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Some people hate the very name of statistics, but I find them full of beauty and interest. Sir Francis Galton In quantum mechanics one often encounters sets of non-negative numbers that sum to unity, having a more or less direct interpretation as probabilities. This includes the squared moduli of the coefficients when a pure state is expanded in an orthonormal basis, the eigenvalues of density matrices, and more. Continuous distributions also play a role, even when the Hilbert space is finite dimensional. From a purely mathematical point of view a probability distribution is simply a measure on a sample space, constrained so that the total measure is one. Whatever the point of view one takes on this, the space of states will turn into a convex set when we allow probabilistic mixtures of its pure states. In classical mechanics the sample space is phase space, which is typically a continuous space. This leads to technical complications but the space of states in classical mechanics does share a major simplifying feature with the discrete case, namely that every state can be expressed as a mixture of pure states in a unique way. This was not so in the case of colour space, nor will it be true for the convex set of all states in quantum mechanics. Majorization and partial order Our first aim is to find ways of describing probability distributions; we want to be able to tell when a probability distribution is ‘more chaotic’ or ‘more uniform’ than another. One way of doing this is provided by the theory of majorization .
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It is usually taken for granted that the natural mathematical framework for quantum mechanics is the theory of Hilbert spaces, where pure states of a quantum system correspond to complex vectors of unit length. These vectors can be combined to create more general states expressed in terms of positive semidefinite matrices of unit trace called density operators. A density operator tells us everything we know about a quantum system. In particular, it specifies a unique probability for any measurement outcome. Thus, to fully appreciate quantum mechanics as a statistical model for physical phenomena, it is necessary to understand the basic properties of its set of states. Studying the convex geometry of quantum states provides important clues as to why the theory is expressed most naturally in terms of complex amplitudes. At the very least, it gives us a new perspective into thinking about structure of quantum mechanics. \n \n \nThis thesis is concerned with the structure of quantum state space obtained from the geometry of the convex set of probability distributions for a special class of measurements called symmetric informationally complete (SIC) measurements. In this context, quantum mechanics is seen as a particular restriction of a regular simplex, where the state space is postulated to carry a symmetric set of states called SICs, which are associated with equiangular lines in a complex vector space. The analysis applies specifically to 3-dimensional quantum systems or qutrits, which is the simplest nontrivial case to consider according to Gleason's theorem. It includes a full characterization of qutrit SICs and includes specific proposals for implementing them using linear optics. The infinitely many qutrit SICs are classified into inequivalent families according to the Clifford group, where equivalence is defined by geometrically invariant numbers called triple products. The multiplication of SIC projectors is also used to define structure coefficients, which are convenient for elucidating some additional structure possessed by SICs, such as the Lie algebra associated with the operator basis defined by SICs, and a linear dependency structure inherited from the Weyl-Heisenberg symmetry. After describing the general one-to-one correspondence between density operators and SIC probabilities, many interesting features of the set of qutrits are described, including an elegant formula for its pure states, which reveals a permutation symmetry related to the structure of a finite affine plane, the exact rotational equivalence of different SIC probability spaces, the shape of qutrit state space defined by the radial distance of the boundary from the maximally mixed state, and a comparison of the 2-dimensional cross-sections of SIC probabilities to known results. Towards the end, the representation of quantum states in terms of SICs is used to develop a method for reconstructing quantum theory from the postulate of maximal consistency, and a procedure for building up qutrit state space from a finite set of points corresponding to a Hesse configuration in Hilbert space is sketched briefly.
In classical probability theory one assumes that all the events concerning a statistical experiment constitute a Boolean σ-algebra and defines a probability measure as a completely additive non-negative function which assigns the value unity for the identity element of the σ-algebra. Invariably, the σ-algebra is the Borel σ-algebra B \mathcal{B} B X (i.e., the smallest σ-algebra generated by the open subsets) of a nice topological space X. Under very general conditions it turns out that all probability measures on B \mathcal{B} B X constitute a convex set whose extreme points are degenerate probability measures.
Continuous degrees of freedom The quantum systems we have discussed so far have been described by finite-dimensional Hilbert spaces. Basic measurements on such systems have a finite number of possible outcomes, and a quantum state predicts a discrete probability distribution over these. Now we wish to extend our theory to handle systems with one or more continuous degrees of freedom, such as the position of a particle that can move in one dimension. This will require an extension of our theory to Hilbert spaces of infinite dimension, and to systems with continuous observables. There is a philosophical issue here. How do we know that there really are infinitely many distinct locations for a particle? The short answer is, we don't. It might be that space itself is both discrete (at the tiniest scales) and bounded (at the largest), so that the number of possible locations of a particle is some very large but finite number. If this is the case, then the continuum model for space is nothing more than a convenient approximation. Infinity is just a simplified way of describing a quantity that is immense, but still finite. In this section, we will adopt this view of infinity. We will imagine that any continuous variable is really an approximation of a “true” discrete variable. This idea will motivate the continuous quantities and operations that we need.
The source of inspiration for doing quantum probability lies in the ingenious methods adopted by physicists to compute probabilities of events concerning the subatomic world of elementary particles where the laws of classical mechanics break down and the distinction between a particle and a wave becomes vague. These methods lead to a generalisation of classical probability which may be described as a study of observable quantities concerning any system subject to chance and exhibiting the following basic features: (i) each observable quantity behaves like a classical real valued random variable on a probability space and hence has a probability distribution on the real line; (ii) it may be impossible to measure simultaneously two individually observable quantities in the sense that measurement of one may interfere with and destroy already existing information about the value of the other; (iii) bounded observable quantities constitute a real linear space in which the vectorial sum of two individually observable quantities interfering with each other is different from the arithmetic sum of their actual values. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.