Niels Bohr supposedly said that if quantum mechanics did not make you dizzy then you did not understand it. I think that the same can be said about statistical inference. Robert D. Cousins In this chapter we quantify how easy it may be too distinguish probability distributions from each other (a discussion that was started in Chapter 2). The issue is a very practical one and arises whenever one is called upon to make a decision based on imperfect data. There is no unique answer because everything depends on the data – the l 1 -distance appears if there has been just one sampling of the distributions, the relative entropy governs the approach to the ‘true’ distribution as the number of samplings goes to infinity, and so on. The quantum case is even subtler. A quantum state always stands ready to produce a large variety of classical probability distributions, depending on the choice of measurement procedure. It is no longer possible to distinguish pure states from each other with certainty, unless they are orthogonal. The basic idea behind the quantum distinguishability measures is the same as that which allowed us, in Section 5.3, to relate the Fubini–Study metric to the Fisher–Rao metric. We will optimize over all possible measurements. Classical distinguishability measures If a distance function has an operational significance as a measure of statistical distinguishability, then we expect it to be monotone (and decreasing) under general stochastic maps. Coarse graining means that information is being discarded, and this cannot increase distinguishability.
Niels Bohr supposedly said that if quantum mechanics did not make you dizzy then you did not understand it. I think that the same can be said about statistical inference. Robert D. Cousins In this chapter we quantify how easy it may be too distinguish probability distributions from each other (a discussion that was started in Chapter 2). The issue is a very practical one and arises whenever one is called upon to make a decision based on imperfect data. There is no unique answer because everything depends on the data – the l 1 -distance appears if there has been just one sampling of the distributions, the relative entropy governs the approach to the ‘true’ distribution as the number of samplings goes to infinity, and so on. The quantum case is even subtler. A quantum state always stands ready to produce a large variety of classical probability distributions, depending on the choice of measurement procedure. It is no longer possible to distinguish pure states from each other with certainty, unless they are orthogonal. The basic idea behind the quantum distinguishability measures is the same as that which allowed us, in Section 5.3, to relate the Fubini–Study metric to the Fisher–Rao metric. We will optimize over all possible measurements. Classical distinguishability measures If a distance function has an operational significance as a measure of statistical distinguishability, then we expect it to be monotone (and decreasing) under general stochastic maps. Coarse graining means that information is being discarded, and this cannot increase distinguishability.
Probability theory is a measure theory – with a soul. Mark Kac Section 2.6 was devoted to classical ensembles , that is to say ensembles defined by probability measures on the set of classical probability distributions over N events. In this chapter quantum ensembles are defined by choosing probability measures on the set of density matrices of size N . A warning should be issued first: there is no single, naturally distinguished measure in M ( N ) , so we have to analyse several measures, each of them with different physical motivations, advantages and drawbacks. This is in contrast to the set of pure quantum states, where the Fubini–Study measure is the only natural choice for a measure that defines ‘random states’. A simple way to define a probability measure goes through a metric. Hence we will start this chapter with a review of the metrics defined on the set of mixed quantum states. Monotone metrics In Section 2.5 we explained how the Fisher metric holds its distinguished position due to the theorem of Čencov, which states that the Fisher metric is the unique monotone metric on the probability simplex Δ N –1 . Now that the latter has been replaced by the space of quantum states M ( N ) we must look again at the question of metrics. Since the uniqueness in the classical case came from the behaviour under stochastic maps, we turn our attention to stochastic quantum maps – the completely positive, trace preserving maps discussed in Chapter 10. A distance D in the space of quantum states M ( N ) is called monotone if it does not grow under the action of a stochastic map Φ, If a monotone distance is geodesic the corresponding metric on M ( N ) is called monotone.
Braunstein and Caves (Braunstein S L and Caves C M 1994 Phys. Rev. Lett. 72 3439-43) proposed to use Helstrom's quantum information number to define, meaningfully, a metric on the set of all possible states of a given quantum system. They showed that the quantum information is nothing other than the maximal Fisher information in a measurement of the quantum system, maximized over all possible measurements. Combining this fact with classical statistical results, they argued that the quantum information determines the asymptotically optimal rate at which neighbouring states on some smooth curve can be distinguished, based on arbitrary measurements on n identical copies of the given quantum system. We show that the measurement which maximizes the Fisher information typically depends on the true, unknown, state of the quantum system. We close the resulting loophole in the argument by showing that one can still achieve the same, optimal, rate of distinguishability, by a two-stage adaptive measurement procedure. When we consider states lying not on a smooth curve, but on a manifold of higher dimension, the situation becomes much more complex. We show that the notion of `distinguishability of close-by states' depends strongly on the measurement resources one allows oneself, and on a further specification of the task at hand. The quantum information matrix no longer seems to play a central role.
Abstract Our purpose here is to draw attention to an interesting relationship that can be shown to hold between (a) information geometry, and (b) quantum geometry. By ‘information geometry’, we mean the natural geometry associated with families of probability distributions. This is a subject that goes back more than half a century to the pioneering work of Rao (1945), who showed that the Fisher information matrix associated with a parametrised family of probability dis tributions gives rise to a natural Riemannian metric on the parameter space. This then allows one to speak in a precise way of the ‘distance’ between two probability distributions, an idea that turns out to be useful in problems of statistical inference. An extensive literature has thus developed, following this line of enquiry, on applications of differential geometry to statistics (see, e.g., Amari (1985), Barndorff-Nielsen et al. (1986), Murray and Rice (1993)). By ‘quantum geometry’, on the other hand, we mean the geometry of the manifold of states associated with a given quantum mechanical system. In particular, we consider the manifold of ‘pure’ states associated with a given complex Hilbert space. This is the space of ‘rays’ through the origin in the Hilbert space, which has the structure of a complex projective space. It is known that the usual rules of quantum mechanics allow one to construct a natural metric on this space, namely the Fubini-Study metric, the specification of which is equivalent to all the familiar structure associated with ordinary quantum mechanics.
The astute reader might have formed the impression that quantum in formation science is a rather qualitative discipline because we have not, as yet, explained how to quantify quantum information. There are three good reasons for leaving this important question until the final chapter. Firstly, quantum information theory is technically demanding and to treat it at an earlier stage might have suggested that our subject was more complicated than it is. Secondly, there is the fact that many of the ideas in the field, such as teleportation and quantum circuits, are unfamiliar and it was important to present these as simply as possible. Finally, and most importantly, the theory of quantum information is not yet fully developed. It has not yet reached, in particular, the level of completeness of its classical counterpart. For this reason we can answer only some of the many questions we would like a quantum theory of information to address. Having said this, we can say that however, there are beautiful and useful mathematical results and it seems certain that these will continue to form an important part of the theory as it develops. We noted in the introduction to Chapter 1 that ‘quantum mechanics is a probabilistic theory and so it was inevitable that a quantum information theory would be developed’. A presentation of at least the beginnings of a quantitative theory is the objective of this final chapter. The entropy or information derived from a given probability distribution is, as we have seen, a convenient measure of the uncertainty associated with the distribution. If many of the probabilities are large, so that many of the possible events are comparably likely, then the entropy will be large. If one probability is close to unity, however, then the entropy will be small. It is convenient to introduce entropy in quantum mechanics as a measure of the uncertainty, or lack of knowledge, of the form of the state vector. If we know that our system is in a particular pure state then the associated uncertainty or entropy should be zero. For mixed states, however, it will take a non-zero value.
In the previous chapters it was stated that the entropy of a system can be determined by statistical means by counting the number of micro-states in which the system may occur under the given conditions. By failing to enlarge upon this point, we have falsely given the impression that the way in which the count should be made was definitely established. Even in the simple case of the various configurations of 50 white and 50 red billiard balls (Fig. 11), counting is more ambiguous than was suggested in Chapter 2. If two white balls are interchanged, a state is reached which, to all appearances, is indistinguishable from the original state and which was thus not counted separately. However, a more thorough investigation with the help of scales and measuring instruments would certainly make it possible to distinguish two similarly coloured balls from one another. The number of visually-distinguishable configurations is thus much smaller than the number — 100! — which can be distinguished on closer investigation.