Information Theory interpretation of Quantum probabilities

This blog is named The Information Age for a purpose. It is nothing to do with the common sense view of what is information. Instead it aims to provide insights about information in the way it is interpreted in scientific and technological contexts. In these contexts, information and information theory is normally associated with statistics, signal processing, electrical and low voltages electronics engineering, computer science and no least physics, quantum physics and mathematical physics.

In all of those fields  of study, information is sometimes viewed with two mindsets. One is slightly pessimistic as to what it means and if it really is something really fundamental; others, on the optimistic side, tend to accept information as the crucial bit about understanding ultimate reality. I for my part settle somewhat in the middle of the pack, and try to cultivate healthy skepticism about any definite conclusion about the underlying  scientific validity of this topic. It is also a way to maintain the wise caution not to rush to erroneous concepts and conclusions; it might keep researchers  busy with more work to do for a great while…

Anyway this is all to introduce the paper I would like to present and review here today. It concerns precisely another application of information-theoretic interpretation of the phenomena of quantum physics and quantum probabilities. This blog has done already some early article on Quantum Physics, in the context of computing and computer science. But this time the choice dwell in a paper that is more theory and explanatory oriented. As such there is a more mathematical and physics specific content, that may not be easy for all readership. Anyway, it is worth a read, and important to feature in this site, given what it is all about.

Quantum physics, as we said before, is a quite unusual scientific theory. But its success is not only undeniable. It is at the base of many of our time and age life realities and familiar objects. There would not be the possibility of the existence of the computers, televisions, radios, fMRI imaging, mobile phones, smartphones, Playstation, the Internet, uff.. and more, if there weren’t quantum physics. Do we ever pause and think about that for a while…? But there is still a need to further understand this theory. For instance its mathematical foundations are different from the ‘other’ physics, named Classical Physics. This theory has at its mathematical framework the fundamental notions of randomness and probability. But even here we need to pause and think. Randomness and probability is also of a special kind in the quantum realm as opposed the classical realm. This is so that, as our paper title underline, there is quantum probabilities and then there is common sense classical probabilities; we are talking of different conceptual frameworks that we need to distinguish. This said we come full circle, as the principal motif of the paper is providing an interpretion to quantum probabilities based on the hallmarks of information theory to help us make sense of what is, if it is possible to understand, the transition from the classical realm to the quantum realm. Did Jeffrey Bub succeed in this?

The paper is from 2010. Six years and a little may be a lot of time in the current pace of research for a subject such as this. I do not know if there were deeper or more convincing developments from then until now. Also this may not have been a highly referenced and standard paper in the subject. But the way it is written, its content and the background of the author appeared to me worthy of a further look and a review here, even if only as matter of reminding me and the relevant readership of every perspective that we should take on this matter. Reviewing cross-referenced papers may have this desirable effect of a proper check on accepted theory and a way to a possible renewed inspiration to better and enhance existent frameworks.

Reminding myself that something in need of an interpretation is something not really well understood, here it is nevertheless the main highlights and abstract of the paper:

Quantum probabilities: an information-theoretic interpretation

 

Abstract

This Chapter develops a realist information-theoretic interpretation of the nonclassical features of quantum probabilities. On this view, what is fundamental in the transition from classical to quantum physics is the recognition that information in the physical sense has new structural features, just as the transition from classical to relativistic physics rests on the recognition that space-time is structurally different than we thought. Hilbert space, the event space of quantum systems, is interpreted as a kinematic (i.e., pre-dynamic) framework for an indeterministic physics, in the sense that the geometric structure of Hilbert space imposes objective probabilistic or information-theoretic constraints on correlations between events, just as the geometric structure of Minkowski space in special relativity imposes spatio-temporal kinematic constraints on events. The interpretation of quantum probabilities is more subjectivist in spirit than other discussions in this book (e.g., the chapter by Timpson), insofar as the quantum state is interpreted as a credence function—a bookkeeping device for keeping track of probabilities—but it is also objective (or intersubjective), insofar as the credences specified by the quantum state are understood as uniquely determined, via Gleason’s theorem, by objective correlational constraints on events in the nonclassical quantum event space defined by the subspace structure of Hilbert space.

 

 

Introduction

 

The paper begins with the realistic assumption of what was said above about the difference between quantum probabilities and classical probabilities:

Quantum probabilities are puzzling because quantum correlations are puzzling, and quantum correlations are puzzling in the way they differ from classical correlations. The aim of this Chapter is to argue for a realist information-theoretic interpretation of the nonclassical features of quantum probabilities. On this view, the transition from classical to quantum physics rests on the recognition that physical information, in Shannon’s sense [37], is structurally different than we thought, just as the transition from classical to relativistic physics rests on the recognition that space-time is structurally different than we thought. Hilbert space, the event space of quantum systems, is interpreted as a kinematic (i.e., pre-dynamic) framework for an indeterministic physics, in the sense that the geometric structure of Hilbert space imposes objective probabilistic or information-theoretic constraints on correlations between events, just as in special relativity the geometric structure of Minkowski space imposes spatio-temporal kinematic constraints on events.

Already here we see the author directly equating objective probabilistic Hilbert spaces with a information-theoretic frameworks. Is this already a kind of heretic boldness or unjustified intellectual courage ? Maybe not…

To deepen our wits:

The difference between classical correlations and nonclassical correlations can be brought out simply in terms of 2-person games. I discuss such games in §2, and in §3 I focus on quantum probabilities. It turns out that the irreversible loss of information in quantum conditionalization—the ‘irreducible and uncontrollable disturbance’ involved in a quantum measurement process, to use Bohr’s terminology—is a generic feature of nonclassical probabilistic theories that satisfy a ‘no signaling’ constraint. ‘No signaling’ is the requirement that no information should be available in the marginal probabilities of measurement outcomes in a region A about alternative choices made by an agent in region B. For example, an observer, Alice, in region A should not be able to tell what observable Bob measured in region B, or whether Bob performed any measurement at all, by looking at the statistics of her measurement outcomes, and conversely. Formally, if Alice measures the observable A with outcomes a in some set and Bob measures the observable B with outcomes b in some set, the constraint is:

quantum-prob1

Here p(a, b|A, B) is the probability of obtaining the pair of outcomes a, b in a joint measurement of the observables A on system A and B on system B, p(a|A, B) is the marginal probability of obtaining the outcome a for A when B is measured in region B, and p(b|A, B) is the marginal probability of obtaining the outcome b for B when A is measured in region A. The ‘no signaling’ constraint requires the marginal probability p(a|A, B) to be independent of the choice of measurement performed on system B (and independent of whether system B is measured at all), i.e., p(a|A, B) = p(a|A), and similarly for the marginal p(b|A, B) with respect to measurements on system A, p(b|A, B) = p(b|B).

(…)

In §3, I show that the quantum ‘measurement disturbance’ is an unavoidable consequence of the non-existence of a universal cloning device for nonclassical extremal states representing multipartite probability distributions. Such a device would allow signaling and so is excluded by the ‘no signaling’ constraint. In §4, following Pitowsky [33], I distinguish two measurement problems, a ‘big’ measurement problem and a ‘small’ measurement problem. I sketch a solution to the ‘small’ measurement problem as a consistency problem, exploiting the phenomenon of decoherence, and I argue that the ‘big’ measurement problem is a pseudo-problem that arises if we take the quantum pure state as the analogue of the classical pure state, i.e., as a representation of physical reality, in the sense that the quantum pure state is the ‘truthmaker’ for propositions about the occurrence and non-occurrence of events.

And, as a possible riposte to criticisms about ungrounded or unsubstantiated claims by the author about what is meant by the assertion that an information-theoretic interpretation is a valid one for quantum probabilities, the paper has some concluding remarks:

Finally, in §5, I clarify the sense in which the information-theoretic interpretation here is proposed as a realist interpretation of quantum mechanics. The interpretation of quantum probabilities is more subjectivist in spirit than other discussions in this book (e.g., the chapter by Timpson), insofar as the quantum state is interpreted as a credence function—a bookkeeping device for keeping track of probabilities. Nevertheless, the interpretation is objective (or intersubjective), because the credences specified by the quantum state are understood as uniquely determined, via Gleason’s theorem [23], by objective correlational constraints on events in the nonclassical quantum event space defined by the subspace structure of Hilbert space. On this view, in the sense of Lewis’s Principal Principle, Gleason’s theorem relates an objective feature of the world, the nonclassical structure of objective chances, to the credence function of a rational agent. The notion of objective chance can be understood in the metaphysically ‘thin’ Humean or Lewisian sense outlined by Hoefer [25], and Frigg and Hoefer [16], for whom chances are not irreducible modalities, or propensities, or necessary connections, but simply features of the pattern of actual events: numbers satisfying probability rules that are part of the best system of such rules, in the sense of simplicity, strength, and fit, characterizing the ‘Humean mosaic,’ the collection of everything that actually happens at all times.

Classical vs Non-Classical correlations

 

This section is the heart of the argument in this paper. I will outline the broad issues without much detail, but as always I highly recommend the full readership of the whole paper to all interested.

This argument hinges on the distinction between what the author calls classical and non-classical correlations. It is presented as a game-theoretic framework, well-known by the economic subject of Game Theory, nevertheless that isn’t the subject of study here:

 

 

To bring out the difference between classical and nonclassical correlations, consider the following game between two players, Alice and Bob, and a moderator. The moderator supplies Alice and Bob with a prompt, or an input, at each round of the game, where these inputs are selected randomly from a set of possible inputs, and Alice and Bob are supposed to respond with an output, either 0 or 1, depending on the input. They win the round if their outputs for the given inputs are correlated in a certain way. They win the game if they have a winning strategy that guarantees a win on each round.2 Alice and Bob are told what the inputs will be (i.e., the set of possible inputs for Alice, and the set of possible inputs for Bob), and what the required correlations are, i.e., what counts as winning a round. They are allowed to confer on a joint strategy before the game starts, but once the game starts they are separated and not allowed further communication

quantum-prob2

 

(…)

 

Denote the marginal probability of output 1 by p. The winning correlations are summed up in Table 1.

quantum-corrl1

Going further deep on the classical versus non-classical correlations distinctions:

 

From the perspective of nonlocal PR boxes and other nonclassical correlations, we see that quantum correlations are not particularly special. Indeed, classical correlations appear to be rather special. The convex set of classical probability distributions has the structure of a simplex. An n-simplex is a particular sort of convex set: a convex polytope4 generated by n + 1 vertices that are not confined to any (n − 1)-dimensional subspace (e.g., a triangle as opposed to a rectangle). The simplest classical state space in this sense (where the points of the space represent probability distributions) is the 1-bit space (the 1-simplex), consisting of two pure or extremal deterministic states, 0 = (1 0) [binomial distribution] and 1 = (0 1) [binomial distribution] , represented by the vertices of the simplex, with mixed states—convex combinations of pure states—represented by the line segment between the two vertices: p = p 0 + (1 − p) 1, for 0 ≤ p ≤ 1. A simplex has the property that a mixed state can be represented in one and only one way as a mixture of extremal states, the vertices of the simplex. No other state space has this feature: if the state space is not a simplex, the representation of mixed states as convex combinations of extremal states is not unique.

 

The state space of classical mechanics is an infinitedimensional simplex, where the extremal states are all deterministic states, with enough structure to support transformations acting on the vertices that include the canonical transformations generated by Hamiltonians. The space of ‘no signaling’ bipartite probability distributions, with arbitrary inputs x ∈ {1, . . . , n}, y ∈ {1, . . . , n} and binary outputs, 0 or 1, is a convex polytope that is not a simplex—the ‘no signaling’ correlational polytope—with the vertices (in the case n = 2) representing generalized PR-boxes (which differ from the standard PR box only with respect to permutations of the inputs and/or outputs), or deterministic boxes (deterministic states), or (in the case n > 2) combinations of these (where the probabilities for some pairs of inputs are those of a generalized PR-box, while for other pairs of inputs they are the probabilities of a deterministic box; see [28], [2], [3]). Inside this polytope is the convex set of quantum correlations, which is not a polytope (the simplest quantum system is the qubit, whose state space as a convex set is a sphere: the Bloch sphere), and inside the quantum convex set is the convex set of classical correlations, which has the rather special structure of a simplex, where the extremal states are all deterministic states.

 

( Note: 6 Think of a PR-box as a nonlocal bipartite state, like a maximally entangled quantum state, e.g., the singlet state for spin-1/2 particles, or any of the Bell states. Copying one side of a PR-box would be like copying one half of a Bell state.)

 

(…)

 

It follows that:

quantum-prob3

So Bob could compute x, the value of Alice’s input, from the Boolean sum of his two outputs: if his outputs take the same value, then Alice’s input is 0; if they take opposite values, Alice’s input is 1. If such a cloning device were possible, Alice and Bob could use the combined PR-box and cloning device to signal instantaneously. Since we are assuming ‘no signaling,’ such a device must be impossible. An analogous argument applies not only to the hypothetical correlations of nonlocal boxes such as the PR-box, but to quantum correlations, i.e., there can be no device that will copy one half of an entangled quantum state without allowing the possibility of instantaneous signaling.

 

Similarly, nonclassical correlations are monogamous: the correlations of a PRbox, for example, can be shared by Alice and Bob, but not by Alice and Bob as well as Charles. If the correlations with Alice could be shared by Bob and Charles, then Bob and Charles could use their outputs to infer Alice’s input, allowing instantaneous signaling between Alice and Bob-Charles. By contrast, there is no such constraint on classical correlations: Alice can happily share any classical correlations with Bob and also with Charles, David, . . . without violating the ‘no signaling’ constraint.

 

(…)

 

 

quantum-corrl2
Figure 1: A schematic representation of the space of no-signaling correlations. The vertices are labelled L and NL for local and nonlocal. Bell inequalities characterize the facets represented by dashed lines. The set bounded by these is L. The region accessible to quantum mechanics is Q. Superquantum correlations lie in region P outside the quantum region. From a diagram in [2].

 

 

The classical simplex represents the classical state space regarded as a space of classical (multipartite) probability distributions; the associated Boolean algebra represents the classical event structure. The conceptually puzzling features of nonclassical ‘no signaling’ theories—quantum and superquantum—can all be associated with state spaces that have the structure of a polytope whose vertices include the local deterministic extremal states of the classical simplex, as well as nonlocal nondeterministic extremal states (like PR-boxes) that lie outside the classical simplex (see Fig. 1). Mixed states that lie outside the classical polytope decompose non-uniquely into convex combinations of these extremal states. The non-unique decomposition of mixed states into convex combinations of pure states is a characteristic feature of nonclassical ‘no signaling’ theories, including quantum theories.

 

 

Concluding remarks

 

The interest, importance and significance of the issues highlighted in this paper (and in this ver review,  by the way…), are hardly an overstatement. This is required reading for the researcher in the topics of information theory, quantum physics and quantum computing. The paper is somewhat dense and long on mathematical knowledge, so the scope of a Blog post will never do full justice to the what is involved. Again it is recommended the readership of the whole paper. I will just share now here the main concluding remarks and hope it to be a source of inspiration to anyone in a position to further deepen  our understanding of this very important field of scientific and technological inquiry and study:

(…)

Dynamical solutions to the measurement problem amend quantum mechanics in such a way that the loss of information in quantum conditionalization is accounted for dynamically, and the quantum probabilities are reconstructed dynamically as measurement probabilities. The quantum probabilities are not regarded as a kinematic feature of the nonclassical event structure but are derived dynamically, as artifacts of the measurement process. Even on the Everett interpretation, where Hilbert space is interpreted as the representation space for a new sort of ontological entity, represented by the quantum state, and no definite outcome out of a range of alternative outcomes is selected in a quantum measurement process (so no explanation is required for such an event), probabilities arise as a feature of the branching structure that emerges in the dynamical process of decoherence.

If, instead, we look at the quantum theory as a member of a class of nonclassical ‘no signaling’ theories, in which the state space (considered as a space of multipartite probability distributions) does not have the structure of a simplex, then there is no unique decomposition of mixed states into a convex combination of extremal states, there is no general cloning procedure for an arbitrary extremal state, and there is no measurement in the nondisturbing sense that one has in classical theories, where it is in principle possible, via measurement, to extract enough information about an extremal state to produce a copy of the state without irreversibly changing the state. Hilbert space as a projective geometry (i.e., the subspace structure of Hilbert space) represents a nonBoolean event space, in which there are built-in, structural probabilistic constraints on correlations between events (associated with the angles between events)—just as in special relativity the geometry of Minkowski space represents spatio-temporal constraints on events. These are kinematic, i.e., pre-dynamic, objective probabilistic or information-theoretic constraints on events to which a quantum dynamics of matter and fields conforms, through its symmetries, just as the structure of Minkowski space imposes spatio-temporal kinematic constraints on events to which a relativistic dynamics conforms.

(…)

On the information-theoretic interpretation, the ‘big’ measurement problem is a pseudo-problem, a consequence of taking the quantum pure state as the analogue of the classical pure state, i.e., as the ‘truthmaker’ for propositions about the occurrence and non-occurrence of events, rather than as a credence function associated with the interpretation of Hilbert space as a new kinematic framework for the physics of an indeterministic universe, in the sense that Hilbert space defines objective probabilistic or information-theoretic constraints on correlations between events. The ‘small’ measurement problem is a consistency problem that can be resolved by considering the dynamics of the measurement process and the role of decoherence in the emergence of an effectively classical probability space of macro-events to which the Born probabilities refer (alternatively, by considering certain combinatorial features of the probabilistic structure: see Pitowsky [33, §4.3]

(…)

There are other information-theoretic interpretations of quantum mechanics (see [38, 40, 39] for a critical discussion), the most prominent of which is the information-theoretic interpretation of Fuchs [17, 18, 19, 20], in which quantum states represent subjective degrees of belief, and the loss of information on measurement is attributed to Bayesian conditionalization as the straighforward refinement of prior degrees of belief in the usual sense, together with a further readjustment of the observer’s beliefs, which is required roughly because, as Fuchs puts it [19, p.8]: ‘The world is sensitive to our touch.’ For Fuchs, as for de Finetti (see [21]), physics is an extension of common sense. What does the work in allowing Fuchs’ Bayesian analysis of measurement updating to avoid the measurement problem is, ultimately, an instrumentalist interpretation of quantum probabilities as the probabilities of measurement outcomes.

By contrast, the information-theoretic interpretation outlined here is proposed as a realist interpretation, in the context of an analysis of nonclassical probabilistic correlations in an indeterministic (non-Boolean) universe, analogous to the analysis of nonclassical spatio-temporal relations in a relativistic universe. A salient feature of this interpretation is the rejection of one aspect of the measurement problem, the ‘big’ measurement problem, as a pseudo-problem, and the recognition of the ‘small’ measurement problem as a legitimate consistency problem that requires resolution.

 

Body text and featured images:  Quantum probabilities: an information-theoretic interpretation

 

 

 

 

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