Quantum Foundations

In this talk we quickly review the basics of the modal "toy model" of quantum theory described by Schumacher in his September 22 colloquium at PI. We then consider how the theory addresses more general open systems. Because the modal theory has a more primitive mathematical structure than actual quantum mechanics, it lacks density operators, positive operator measurements, and completely positive maps. As we will show, however, modal quantum theory has an elegant description of the states, effects and operations of open modal systems -- a description with close analogies to actual quantum mechanics.

The question of the existence of gravitational stress-energy in general relativity has exercised investigators in the field since the very inception of the theory. Folklore has it that no adequate definition of a localized gravitational stress-energetic quantity can be given. Most arguments to that effect invoke one version or another of the Principle of Equivalence. I argue that not only are such arguments of necessity vague and hand-waving but, worse, are beside the point and do not address the heart of the issue. Based on an analysis of what it may mean for one tensor to depend in the proper way on another, I prove that, under certain natural conditions, there can be no tensor whose interpretation could be that it represents gravitational stress-energy in general relativity. It follows that gravitational energy, such as it is in general relativity, is necessarily non-local. Along the way, I prove a result of some interest in own right about the structure of the associated jet bundles of the bundle of Lorentz metrics over spacetime.

The uncertainty principle bounds the uncertainties about the outcomes of two incompatible measurements, such as position and momentum, on a particle. It implies that one cannot predict the outcomes for both possible choices of measurement to arbitrary precision, even if information about the preparation of the particle is available in a classical memory. However, if the particle is prepared entangled with a quantum memory, it is possible to predict the outcomes for both measurement choices precisely. I will explain a recent extension of the uncertainty principle to incorporate this case. The new relation gives a lower bound on the uncertainties, which depends on the amount of entanglement between the particle and the quantum memory. If time permits, I will also outline a couple of applications.

A brief review of some recent work on the causal set approach to quantum gravity. Causal sets are a discretisation of spacetime that allow the symmetries of GR to be preserved in the continuum approximation. One proposed application of causal sets is to use them as the histories in a quantum sum-over-histories, i.e. to construct a quantum theory of spacetime. It is expected by many that quantum gravity will introduce some kind of fuzziness uncertainty and perhaps discreteness into spacetime, and generic effects of this fuzziness are currently being sought. Applied as a model of discrete spacetime, causal sets can be used to construct simple phenomenological models which allow us to understand some of the consequences of this general expectation.

Quantum states are not observables like in any wave mechanics but co-observables describing the reality as a possible knowledge about the statistics of all quantum events, like quantum jumps, quantum decays, quantum diffusions, quantum trajectories, etc. However, as we show, the probabilistic interpretation of the traditional quantum mechanics is inconsistent with the probbilistic causality and leads to the infamous quantum measurement problem. Moreover, we prove that all attempts to solve this problem as suggested by Bohr are doomed in the traditional framework of the reversible interactions. We explore the only possibility left to resolve the quantum causality problem while keeping the reversibility of Schroedinger mechanics. This is to break the time symmetry of the Heisenberg mechanics using the nonequivalence of the Schroedinger and Heisenberg quantum mechanics on nonsimple operator algebras in infinite dimensional Hilbert spaces. This is the main idea of Eventum Mechanics, which enhances the quantum world of the future by classical events of the past and constructs the reversible Schroedinger evolutions compatible with observable quantm trajectories by irreversible quantum to classicl interfaces in terms of the reversible unitary scatterings. It puts the idea of hidden variables upside down by declaring that what is visible (in the past by now) is not quantum but classical and what is visible (by now in the future) is quantum but not classical. More on the philosophy of Eventum Mechanics can be found in [1]. We demonstrate these ideas on the toy model of the nontrivial quantum - classical bit interface. The application of these ideas in the continuous time leads to derivation of the quantum stochastic master equations reviewed in [1] and of my research pages [3]. V. P. Belavkin: Quantum Causality, Stochastics, Trajectories and Information. Reports on Progress in Physics 65 (3): 353-420 (2002). quant-ph/0208087, PDF. http://www.maths.nott.ac.uk/personal/vpb/vpb_research.html http://www.maths.nott.ac.uk/personal/vpb/research/cau_idy.html

Many results have been recently obtained regarding the power of hypothetical closed time-like curves (CTC’s) in quantum computation. Most of them have been derived using Deutsch’s influential model for quantum CTCs [D. Deutsch, Phys. Rev. D 44, 3197 (1991)]. Deutsch’s model demands self-consistency for the time-travelling system, but in the absence of (hypothetical) physical CTCs, it cannot be tested experimentally. In this paper we show how the one-way model of measurement-based quantum computation (MBQC) can be used to test Deutsch’s model for CTCs. Using the stabilizer formalism, we identify predictions that MBQC makes about a specific class of CTCs involving travel in time of quantum systems. Using a simple example we show that Deutsch’s formalism leads to predictions conflicting with those of the one-way model. There exists an alternative, little-discussed model for quantum time-travel due to Bennett and Schumacher (in unpublished work, see http://bit.ly/cjWUT2), which was rediscovered recently by Svetlichny [arXiv:0902.4898v1]. This model uses quantum teleportation to simulate (probabilistically) what would happen if one sends quantum states back in time. We show how the Bennett/ Schumacher/ Svetlichny (BSS) model for CTCs fits in naturally within the formalism of MBQC. We identify a class of CTC’s in this model that can be simulated deterministically using techniques associated with the stabilizer formalism. We also identify the fundamental limitation of Deutsch's model that accounts for its conflict with the predictions of MBQC and the BSS model. This work was done in collaboration with Raphael Dias da Silva and Elham Kashefi, and has appeared in preprint format (see website). Website: http://arxiv.org/abs/1003.4971

It has long been recognized that there are two distinct laws that go by the name of the Second Law of Thermodynamics. The original says that there can be no process resulting in a net decrease in the total entropy of all bodies involved. A consequence of the kinetic theory of heat is that this law will not be strictly true; statistical fluctuations will result in small spontaneous transfers of heat from a cooler to a warmer body. The currently accepted version of the Second Law is probabilistic: tiny spontaneous transfers of heat from a cooler to a warmer body will be occurring all the time, while a larger transfer is not impossible, merely improbable. There can be no process whose expected result is a net decrease in total entropy. According to Maxwell, the Second Law has only statistical validity, and this statement is easily read as an endorsement of the probabilistic version. I argue that a close reading of Maxwell, with attention to his use of "statistical," shows that the version of the second law endorsed by Maxwell is strictly weaker than our probabilistic version. According to Maxwell, even the probable truth of the second law is limited to situations in which we deal with matter only in bulk and are unable to observe or manipulate individual molecules. Maxwell's version does not rule out a device that could, predictably and reliably, transfer heat from a cooler to a warmer body without a compensating increase in entropy. I will discuss the evidence we have for these two laws, Maxwell's and ours.

We first discuss quantum measure and integration theory. We then consider various anhomomorphic logics. Finally, we present some connections between the two theories. One connection is transferring a quantum measure to a measure on an anhomomorphic logic. Another is the creation of a reality filter that is stronger than Sorkin's preclusivity. This is accomplished by generating a preclusive coevent from a quantum measure. No prior knowledge of quantum measure theory or anhomomorphic logics will be assumed.

Non-relativistic quantum mechanics is derived as an example of entropic inference. The basic assumption is that the position of a particle is subject to an irreducible uncertainty of unspecified origin. The corresponding probability distributions constitute a curved statistical manifold. The probability for infinitesimally small changes is obtained from the method of maximum entropy and the concept of time is introduced as a book-keeping device to keep track of how they accumulate. This requires introducing appropriate notions of instant and of duration. A welcome feature is that this entropic notion of time incorporates a natural distinction between past and future. The Schroedinger equation is recovered when the statistical manifold participates in the dynamics in such a way that there is a conserved energy: its curved geometry guides the motion of the particles while they, in their turn, react back and determine its evolving geometry. The phase of the wave function—not just its magnitude—is explained as a feature of purely statistical origin. Finally, the model is extended to include external electromagnetic fields and gauge transformations.

Many putative explanations in physics rely on idealized models of physical systems. These explanations are inconsistent with standard philosophical accounts of explanation. A common view holds that idealizations can underwrite explanation nonetheless, but only when they are what have variously been called Galilean, approximative, traditional or controllable. Controllability is the least vague of these categories, and this paper focuses on the relation between controllability and explanation. Specifically, it argues that the common view is an untenable half-measure. It gives the example of a simple pendulum with quadratic damping, an uncontrollable idealization that makes use of singular limits and for which the behaviour at the limit is qualitatively new—but a system whose behaviour is fully explained in terms of the idealization. It shows that uncontrollable idealizations can have explanatory capacities (and in a way distinct from Batterman’s “asymptotic explanation”).

One might have hoped that philosophers had sorted out what ‘truth’ is supposed to be by now. After all, Aristotle offered what seems to be a clear and simple characterization in his Metaphysics. So perhaps it is surprising (and then again perhaps it isn’t), that contemporary philosophers have not settled on a consensus regarding the nature of truth to this day. In fact, the most obvious theory of truth, that truth consists in correspondence to the facts, seems to be steadily waning in popularity in technical circles, replaced instead by a perhaps puzzlingly austere minimalist theory that restricts its characterization of truth to the familiar equivalence schema:

is true if and only if p. The differences between such deflationary theories and the ‘traditional’ correspondence theory of truth, and perhaps even more strikingly between these theories and epistemic theories of truth, call to mind counterpart features in different attitudes about the proper interpretation of quantum mechanics. By reviewing the most striking features of different theories of truth, as well as some of their most difficult objections, we can start to see where different interpretations seem to be reliant on (or at least quite congenial to) particular theories of truth and also where these theories begin to reveal themselves as variously helping and hindering the smooth functioning of different interpretations.

Our starting point is a particular `canvas' aimed to `draw' theories of physics, which has symmetric monoidal categories as its mathematical backbone. With very little structural effort (i.e. in very abstract terms) and in a very short time this categorical quantum mechanics research program has reproduced a surprisingly large fragment of quantum theory. Philosophically speaking, this framework shifts the conceptual focus from `material carriers' such as particles, fields, or other `material stuff', to `logical flows of information', by mainly encoding how things stand in relation to each other. These relations could, for example, be induced by operations. Composition of these relations is the carrier of all structure. Thus far the causal structure has been treated somewhat informally within this approach. In joint work with my student Raymond Lal, by restricting the capabilities to compose, we were able to formally encode causal connections. We call the resulting mathematical structure a CauCat, since it combines the symmetric monoidal stricture with Sorkin's CauSets within a single mathematical concept. The relations which now respect causal structure are referred to as processes, which make up the actual `happenings'. As a proof of concept, we show that if in a quantum teleportation protocol one omits classical communication, no information is transfered. We also characterize Galilean theories. Classicality is an attribute of certain processes, and measurements are special kinds of processes, defined in terms of their capabilities to correlate other processes to these classical attributes. So rather than quantization, what we do is classicization within our universe of processes. We show how classicality and the causal structure are tightly intertwined. All of this is still very much work in progress!