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In this talk we will explore a "toy model" of quantum theory that is similar to actual quantum theory, but uses scalars drawn from a finite field. The set of possible states of a system is discrete and finite. Our theory does not have a quantitative notion of probability, but only makes the "modal" distinction between possible and impossible measurement results. Despite its very simple structure, our toy model nevertheless includes many of the key phenomena of actual quantum systems: interference, complementarity, entanglement, nonlocality, and the impossibility of cloning.

We study a Hamiltonian system describing a three-spin 1/2 cluster like interaction competing with an Ising-like exchange. We show that a cluster state, the ground state of the Hamiltonian in the absence of the Ising term, is provided by a hidden order of topological nature. In the presence of the cluster and Ising couplings, a continuous quantum phase transition occurs in the system, directly connecting a local broken symmetry phase to a cluster phase with the hidden order. At the critical point the Hamiltonian is self-dual. We analyze the geometric entanglement and demonstrate that it can capture the transition, as a single parameter.

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.

A lattice gauge theory is described by a redundantly large vector space that is subject to local constraints, and it can be regarded as the low energy limit of a lattice model with a local symmetry. I will describe a coarse-graining scheme capable of exactly preserving local symmetries. The approach results in a variational ansatz for the ground state(s) and low energy excitations of a lattice gauge theory. This ansatz has built-in local symmetries, which are exploited to significantly reduce simulation costs. I will describe benchmark results in the context of Kitaev’s toric code with a magnetic field or, equivalently, Z2 lattice gauge theory, for lattices with up to 16 x 16 sites (16^2 x 2 = 512 spins) on a torus.

Closed quantum systems evolve unitarily and therefore cannot converge in a strong sense to an equilibrium state starting out from a generic pure state. Nevertheless for large system size one observes temporal typicality. Namely, for the overwhelming majority of the time instants, the statistics of observables is practically indistinguishable from an effective equilibrium one. In this talk we will discuss he Loschmidt echo (LE) to study this sort of unitary equilibration after a quench. In particular we ll address the issue of typicality with respect the initial state preparation and the influence of quantum criticality of the long-time probability distribution of LE.

In this talk (based on arXiv:1001.0354) we give a quantum statistical interpretation for the Kauffmann bracket polynomial state sum <K> for the Jones polynomial. We use this quantum mechanical interpretation to give a new quantum algorithm for computing the Jones polynomial. This algorithm is useful for its conceptual simplicity, and it applies to all values of the polynomial variable that lie on the unit circle in the complex plane. Letting C(K) denote the Hilbert space for this model, there is a natural unitary transformation U from C(K) to itself such that <K> = <F|U|F> where |F> is a sum over basis states for C(K). The quantum algorithm arises directly from this formula via the Hadamard Test. We then show that the framework for our quantum model for the bracket polynomial is a natural setting for Khovanov homology. The Hilbert space C(K) of our model has basis in one-to-one correspondence with the enhanced states of the bracket state summmation and is isomorphic with the chain complex for Khovanov homology with coefficients in the complex numbers. We show that for the Khovanov boundary operator d defined on C(K) we have the relationship dU + Ud = 0. Consequently, the unitary operator U acts on the Khovanov homology, and we therefore obtain a direct relationship between Khovanov homology and this quantum algorithm for the Jones polynomial. The formula for the Jones polynomial as a graded Euler characteristic is now expressed in terms of the eigenvalues of U and the Euler characteristics of the eigenspaces of U in the homology. The quantum algorithm given here is inefficient, and so it remains an open problem to determine better quantum algorithms that involve both the Jones polynomial and the Khovanov homology.

Fibonacci anyons are the simplest system of anyons capable of implementing universal topological quantum computation, an area which is of intense theoretical and experimental interest. Recent studies have shown that for nearest-neighbour interactions, the properties of the ground state of a 1-D chain of Fibonacci anyons may be modeled using a spin chain, and are related to specific conformal field theories. I will talk about the role played by boundary conditions in this mapping, and demonstrate that for these simple anyonic systems the correct spin chain models in fact correspond to conformal field theories with a defect. The presence of this defect drastically changes the excitations observable in the system.