Quantum Foundations

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.

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.

We will analyze different aspects of locality in causal operational probabilistic theories. We will first discuss the notion of local state and local objective information in operational probabilistic theories, and define an operational notion of discord that coincides with quantum discord in the case of quantum theory. Using such notion, we will show that the only theory in which all separable states have null discord is the classical one. We will then analyze locality of transformations, reviewing some general properties of no-signaling channels in causal theories. We will show that it is natural to define transformations on no-signaling channels that cannot be extended to all bipartite channels, and discuss the consequences of this fact on information processing.

A new ensemble interpretation of quantum mechanics is proposed according to which the ensemble associated to a quantum state really exists: it is the ensemble of all the systems in the same quantum state in the universe. Individual systems within the ensemble have microscopic states, described by beables. The probabilities of quantum theory turn out to be just ordinary relative frequencies probabilities in these ensembles. Laws for the evolution of the beables of individual systems are given such that their ensemble relative frequencies evolve in a way that reproduces the predictions of quantum mechanics. These laws are highly non-local and involve a new kind of interaction between the members of an ensemble that define a quantum state. These include a stochastic process by which individual systems copy the beables of other systems in the ensembles of which they are a member. The probabilities for these copy processes do not depend on where the systems are in space, but do depend on the distribution of beables in the ensemble. Macroscopic systems then are distinguished by being large and complex enough that they have no copies in the universe. They then cannot evolve by the copy law, and hence do not evolve stochastically according to quantum dynamics. This implies novel departures from quantum mechanics for systems in quantum states that can be expected to have few copies in the universe. At the same time, we are able to argue that the centre of masses of large macroscopic systems do satisfy Newton's laws.

We begin with a fundamental approach to quantum mechanics based on the unitary representations of the group of diffeomorphisms of physical space (and correspondingly, self-adjoint representations of a local current algebra). From these, various classes of quantum configuration spaces arise naturally. One obtains in addition the usual exchange statistics for spatial dimension d >2, induced by representations of the symmetric group, while for d = 2, the approach led to an early prediction of intermediate or “anyonic” statistics induced by unitary representations of the braid group. After reviewing these ideas, which are based on joint work with R. Menikoff and D. H. Sharp at Los Alamos National Laboratory, we shall discuss briefly some analogous possibilities for infinite-dimensional configuration spaces, including anyonic statistics for extended objects in 3-dimensional space.

Ideal measurements are described in quantum mechanics textbooks by two postulates: the collapse of the wave packet and Born’s rule for the probabilities of outcomes. The quantum evolution of a system then has two components: a unitary (Hamiltonian) evolution in between measurements and non-unitary one when a measurement is performed. This situation was considered to be unsatisfactory by many people, including Einstein, Bohr, de Broglie, von Neumann and Wigner, but has remained unsolved to date. The quantum measurement problem, that is, understanding why a unique outcome is obtained in each individual run of an experiment, is tackled by solving a Hamiltonian model within standard quantum statistical mechanics. The model describes the measurement of the z-component of a spin through interaction with a magnetic memory. The latter apparatus is modeled by a Curie–Weiss magnet having N ≫ 1 spins weakly coupled to a phonon bath. The Hamiltonian evolution exhibits several time scales. The reduction, a rapid decay of the off-diagonal blocks of the system–apparatus density matrix, arises from the many degrees of freedom of the pointer (the magnetization). The registration occurs due to a phase transition from the initial metastable state to one of the final stable states triggered by the tested system. It yields a stationary state in which the apparatus and the system are correlated. Under proper conditions the process satisfies all the features of ideal measurements, including collapse and Born’s rule. As usual, irreversibility is ensured by the macroscopic size of the apparatus, in particular by the large value of N. Nothing else than the usual quantum statistical mechanics and Schro ̈dinger equation is needed, and the results support a specified version of the statistical interpretation. The solution of the quantum measurement problem requires a combination of the reduction and the registration, the properties of which arise from the irreversible dynamics.

I consider systems that consist of a few hot and a few cold two-level systems and define heat engines as unitaries that extract energy. These unitaries perform logical operations whose complexity depends on both the desired efficiency and the temperature quotient. I show cases where the optimal heat engine solves a hard computational task (e.g. an NP-hard problem) [2]. Heat engines can also drive refrigerators and use the temperature difference between two systems for cooling a third one. I argue that these triples of systems define a classification of thermodynamic resources [1]. All the above assumes that unitaries are implemented by an external controller. To get a thermodynamically reversible process, the joint process on system and controller must be reversible. Then, the implementation of the joint process requires a "meta-controller", and so on. To study thermodynamic limits without such an infinite sequence of controllers, I introduce the model of "physically universal cellular automata", in which the boundary between system and controller can be shifted (in analogy to the Heisenberg-cut for the quantum measurement problem). I show that this model raises a lot of fundamental questions [3]. Literature: [1] Janzing et al: Thermodynamic cost of reliability and low temperatures: Tightening Landauer's principle and the second law, J. Stat. Phys. 2000 [2] Janzing: On the computation power of molecular heat engines, J. Stat. Phys. 2006 [3] Janzing: Is there a physically universal cellular automaton or Hamiltonian? arXiv:1009.1720