Quantum Foundations

Classical constraints come in various forms: first and second class, irreducible and reducible, regular and irregular, all of which will be illustrated. They can lead to severe complications when classical constraints are quantized. An additional complication involves whether one should quantize first and reduce second or vice versa, which may conflict with the axiom that canonical quantization requires Cartesian coordinates. Most constraint quantization procedures (e.g., Dirac, BRST, Faddeev) run into difficulties with some of these issues and may lead to erroneous results. The Projection Operator Method involves no gauge fixing, no auxiliary variables of any kind, and can treat simultaneously any and all kinds of constraints. It also admits a phase space path integral formulation with similar features.

A family of probability distributions (i.e. a statistical model) is said to be sufficient for another, if there exists a transition matrix transforming the probability distributions in the former to the probability distributions in the latter. The so-called Blackwell-Sherman-Stein Theorem provides necessary and sufficient conditions for one statistical model to be sufficient for another, by comparing their "information values" in a game-theoretical framework. In this talk, I will extend some of these ideas to the quantum case. I will begin by considering the comparison of ensembles of quantum states in terms of their "information value" in quantum statistical decision problems. In this case, I will prove that one ensemble is "more informative" than another if and only if there exists a suitable processing of the former into the latter. I will then move on to the comparison of bipartite quantum states in terms of their "nonlocality value" in nonlocal games. In this case, I will prove that one bipartite state is "more nonlocal" than another if and only if the former can be transformed into the latter by local operations and shared randomness, arguing, moreover, that the framework provided by nonlocal games can be useful in understanding analogies and differences between the notions of quantum entanglement and nonlocality.

Wheeler's delayed choice (WDC) is one of the "standard experiments in foundations". It aims at the puzzle of a photon simultaneously behaving as wave and particle. Bohr-Einstein debate on wave-particle duality prompted the introduction of Bohr's principle of complementarity, ---`.. the study of complementary phenomena demands mutually exclusive experimental arrangements" . In WDC experiment the mutually exclusive setups correspond to the presence or absence of a second beamsplitter in a Mach-Zehnder interferometer (MZI). A choice of the setup determines the observed behaviour. The delay ensures that the behaviour cannot be adapted before the photon enters MZI. Using WDC as an example, we show how replacement of classical selectors by quantum gates streamlines experiments and impacts on foundational questions. We demonstrate measurements of complementary phenomena with a single setup, where observed behaviour of the photon is chosen after it has been already detected. Spacelike separation of the setup components becomes redundant. The complementarity principle has to be reformulated --- instead of complementarity of experimental setups we now have complementarity of measurement results. Finally we present a quantum-controlled scheme of Bell-type experiments. To reach any of these conclusions in either classical or quantum setting a (simple) hidden variable model that represents the "reality" of "particle" and "wave" should be analyzed. The model is never fully exorcised but just pushed to have more and more conspiratorial set of assumptions.

This is a very informal talk about some of the issues associated with the notion of "macroscopic realism" (MR) and its relation to quantum mechanics (QM). After a brief discussion of the motivation for attempts to modify QM at some point between the level of the atom and that of our own direct experience, and a survey of some candidate experimental systems, I will discuss attempts to quantify the notion of "macroscopic distinctness", the use of temporal Bell inequalities to discriminate between the predictions of QM and those of MR, the related concept of "noninvasive measurement" and existing and possible ways to implement it. I will also comment briefly on the relation between Bell-EPR experiments and those on MR.

This is a geometric tutorial about straight and twisted vectors and forms (ie, de Rham currents) leading to some wild thoughts about the EM field as a *thing*, ie with properties similar to a piece of matter; and to some even wilder thoughts about a metric-free GR.

In my talk I raise the question of the fundamental limits to the size of thermal machines - refrigerators, heat pumps and work producing engines - and I will present the smallest possible ones. I will also discuss the issue of a possible complementarity between size and efficiency and show that even the smallest machines could be maximally efficient. Finally I will present a new point of view over what is work and what do thermal machines actually do.

I provide a reformulation of finite dimensional quantum theory in the circuit framework in terms of mathematical axioms, and a reconstruction of quantum theory from operational postulates. The mathematical axioms for quantum theory are the following: [Axiom 1] Operations correspond to operators. [Axiom 2] Every complete set of positive operators corresponds to a complete set of operations. The following operational postulates are shown to be equivalent to these mathematical axioms: [P1] Definiteness. Associated with any given pure state is a unique maximal effect giving probability equal to one. This maximal effect does not give probability equal to one for any other pure state. [P2] Information locality. A maximal measurement on a composite system is effected if we perform maximal measurements on each of the components. [P3] Tomographic locality. The state of a composite system can be determined from the statistics collected by making measurements on the components. [P4] Compound permutatability. There exists a compound reversible transformation on any system effecting any given permutation of any given maximal set of distinguishable states for that system. [P5] Preparability. Filters are non-mixing and non-flattening. Hence, from these postulates we can reconstruct all the usual features of quantum theory: States are represented by positive operators, transformations by completely positive trace non-increasing maps, and effects by positive operators. The Born rule (i.e. the trace rule) for calculating probabilities also follows. See arXiv:1104.2066 for more details. These operational postulates are deeper than those I gave ten years ago in quant-ph/0101012.

Quantum theory can be thought of a noncommutative generalization of classical probability and, from this perspective, it is puzzling that no quantum generalization of conditional probability is in widespread use. In this talk, I discuss one such generalization and show how it can unify the description of ensemble preparations of quantum states, POVM measurements and the description of correlations between quantum systems. The conditional states formalism allows for a description of prepare-and-measure experiments that is neutral with respect to the direction of inference, such that both the retrodictive formalism and the more usual predictive formalism are consequences of a more fundamental description in terms of a conditionally independent tripartite state, and the two formalisms are related by a quantum generalization of Bayes' rule. As an application, I give a generalized argument for the pooling rule proposed by Spekkens and Wiseman that is a direct analog of a result in classical supra-Bayesian pooling.