Quantum Foundations

The effects of closed timelike curves (CTCs) in quantum dynamics, and its consequences for information processing have recently become the subject of a heated debate. Deutsch introduced a formalism for treating CTCs in a quantum computational framework. He postulated a consistency condition on the chronology-violating systems which led to a nonlinear evolution on the systems that come to interact with the CTC. This has been shown to allow tasks which are impossible in ordinary linear quantum evolution, such as computational speed-ups over (linear) quantum computers, and perfectly distinguishing non-orthogonal quantum states. Bennett and co-authors have argued, on the other hand, that nonlinear evolution allows no such exotic effects. They argued that all proofs of exotic effects due to nonlinear evolutions suffer from a fallacy they called the " linearity trap". Here we review the argument of Bennett and co-authors and show that there is no inconsistency in assuming linearity at the level of a classical ensemble, even at the presence of nonlinear quantum evolution. In fact, this is required for the very existence of empirically verifiable nonlinear evolution. The arguments for exotic quantum effects are thus seen to be based on the necessity for a fundamental distinction between proper and improper mixtures in the presence of nonlinear evolutions. We show how this leads to an operationally well-defined version of the measurement problem that we call the "preparation problem".

Tim Ralph We consider quantum mechanical particles that traverse general relativistic wormholes in such a way that they can interact with their own past, thus forming closed timelike curves. Using a simple geometric argument we reproduce the solutions proposed by Deutsch for such systems. Deutsch's solutions have attracted considerable interest because they do not contain paradoxes, however, as originally posed, they do contain ambiguities. We show that these ambiguities are removed by following our geometric derivation.

Can a density matrix be regarded as a description of the physically real properties of an individual system? If so, it may be possible to attribute the same objective significance to statistical mechanical properties, such as entropy or temperature, as to properties such as mass or energy. Non-linear modifications to the evolution of a density matrix can be proposed, based upon this idea, to account for thermodynamic irreversibility. Traditional approaches to interpreting quantum phenomena assume that an individual system is described by a pure state, with density matrices arising only through a statistical mixture or through tracing out entangled degrees of freedom. Treating the density matrix as fundamental can affect the viability of some of these interpretations, and introducting thermodynamically motivated non-linearities will not, in themselves, help in solving the quantum measurement problem.

Feynman showed that the path of least action is determined by quantum interference. The interference may be viewed as part of a quantum algorithm for minimising the action. In fact, Lloyd describes the Universe as a giant quantum computer whose purpose is to calculate its own state. Could the direction of time that the universe is apparently following be determined by a quantum algorithm? The answer lies in the violation of time reversal (T) invariance that is being observed in an increasing number of particle accelerator experiments. The violation signifies a fundamental asymmetry between the past and future and calls for a major shift in the way we think about time. Here we show that processes which violate T invariance induce destructive interference between different paths that the universe can take through time. The interference eliminates all paths except for two that represent continuously forwards and continuously backwards time evolution. This suggests that quantum interference from T violation processes gives rise to the phenomenological unidirectional nature of time. A path consisting exclusively of forward steps gives the shortest path to a point which is in the forwards direction. The quantum interference, therefore, underlies a quantum algorithm that determines shortest path through time.

In an ontological model of quantum theory that is Bell-local, one can assume without loss of generality that the outcomes of measurements are determined deterministically by the ontic states (i.e. the values of the local hidden variables). The question I address in this talk is whether such determinism can always be assumed in a noncontextual ontological model of quantum theory, in particular whether it can be assumed for nonprojective measurements. While it is true that one can always represent a measurement by a deterministic response function by incorporating ancillary degrees of freedom into one's description (for instance those of the apparatus), I show that in moving to such a representation, one typically loses the warrant to apply the assumption of measurement noncontextuality. The implications for experimental tests of measurement noncontextuality will be discussed.

Traditionally, the focus on determining characteristic properties of quantum mechanics has been on properties such as entanglement. However, entanglement is a property of multiple systems. Another interesting question is to ask what properties are characteristic of single quantum systems. Two answers to this question are: 1.There is a continuous path of pure quantum states connecting any two quantum states [1], and, 2.Quantum mechanics is preparation noncontextual [2]. In this talk, I will discuss a link between these two answers to this question. In particular, I will establish some strict upper bounds on the maximum size of the set of quantum states that can be modelled in a preparation noncontextual, nonnegative theory and show that this set contains pure states that cannot be connected to any other pure state in the set. I will also discuss a common example of a preparation noncontextual model that allows negative values, namely, a discrete Wigner function, and establish necessary and sufficient conditions for bases of an arbitrary dimensional Hilbert space to have nonnegative Wigner functions, i.e., to admit a classical model. I will conclude with a discussion of some open problems. [1] L. Hardy, quant-ph/0101012v4 (2001). [2] R. W. Spekkens, Phys. Rev. A, 71, 052108 (2005)

Matt Palmer An explicit description of a physical system is necessarily written with respect to a particular reference frame. It is important to know how to adapt the description when a different, equally valid, reference frame is chosen. In the case of classical frames there is a well-defined covariance of the description. The question we want to address is: How can we extend this description of change of reference frame to the case where the frames are quantum objects? We study this problem within specific toy models, and approach it operationally. We define a procedure that will change the quantum reference frame with which a quantum system is described. We find this procedure induces decoherence in the system and is described by a non-unitary CP map, which is in interesting distinction to the reversible nature of the classical change of frame procedures.

Information theory provides a novel approach to study of the consequences of symmetry of dynamics which goes far beyond the traditional conservation laws and Noether's theorem. The conservation laws are not applicable to the dissipative and open systems. In fact, as we will show, even in the case of closed system dynamics if the state of system is not pure the conservation laws do not capture all the consequences of symmetry. Using information theoretic approach to this problem we introduce new quantities called asymmetry monotones, that if the system is closed they are constant of motion and otherwise, if the system is open, they are always non-increasing. We also explain how different results in quantum information theory can have non-trivial consequences about the symmetric dynamics of quantum systems.

Ken Wharton An analysis of the path-integral approach to quantum theory motivates the hypothesis that two experiments with the same classical action should have dual ontological descriptions. If correct, this hypothesis would not only constrain realistic interpretations of quantum theory, but would also act as a constructive principle, allowing any realistic model of one experiment to generate a corresponding model for its action-dual. Two pairs of action-dual experiments will be presented, including one experiment that violates the Bell inequality and yet is action-dual to a single particle. Demanding a consistent, realistic ontology leads to a highly restricted parameter space of possible interpretations.

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.

Quantum correlations cannot be given any classical explanation that would satisfy Bell's local causality assumption. This quite intriguing feature of quantum theory, known as quantum non-locality, has fascinated physicists for years, and has more recently been proven to have interesting applications in quantum information processing. To properly understand the power of quantum non-locality, it is important to be able to quantify it. One way for that is to compare it to other "non-local resources", such as classical communication or "non-local Popescu-Rohrlich (PR) boxes", and try to use these alternative resources to reproduce the quantum correlations. I will review known results on this subject, and present new simulations of multipartite non-local correlations.

To model statistical correlations that violate Bell inequalities (such as singlet state correlations), one must relax at least one of three physically plausible postulates: measurement independence (experimenters can freely choose measurement settings independently of any underlying variables describing the system); no-signalling (underlying marginal distributions for one observer cannot depend on the measurement setting of a distant observer), and determinism (all outcomes can be fully determined by the values of underlying variables). It will be shown that, for any given model, one may quantify the degrees of measurement dependence, signalling and indeterminism, by three numbers M, S and I. It will further be shown how the Bell-CHSH inequality may be generalised to a "relaxed" Bell inequality, of the form ++-<=B(I,S,M), where the upper bound is tight and ranges between 2 and 4. The usual Bell-CHSH inequality corresponds to I=S=M=0. More generally, the bound B(I,S,M) quantifies the necessary mutual tradeoff between I, S and M that is required to model a given violation of the Bell-CHSH inequality. Some information-theoretic implications will be briefly described, as well as a no-signalling deterministic model of the singlet state that allows up to 86% experimental free will.