Quantum Physics

Lee Smolin has argued that one of the barriers to understanding time in a quantum world is our tendency to spatialize time. The question is whether there is anything in physics that could lead us to mathematically characterize time so that it is not just another funny spatial dimension. I will explore the possibility(already considered by Smolin and others) that time may be distinguished from space by what I will call a measure of Booleanity. The Bell-Kochen-Specker Theorem shows that the statistics of quantum systems (unlike that of classical systems) do not in general permit of a Boolean substructure. I will outline reasons for thinking that time is the dimension in which the Booleanity of spacetime (considered as a quantum system) varies, while space is characterized by constant Booleanity. I will not be able to give a mathematically complete characterization of the Booleanity of a region of spacetime, since that would require nothing less than knowing how to quantize spacetime; however, I will argue that something like this is needed if one is to make any sense of an ontological distinction between past, present, and future in terms of modern physics. I will also briefly consider possible objections to this view arising from the relativity of simultaneity, which (on its usual interpretation) apparently places all events on an equal ontological footing. In order to get around this we need a generalized conception of simultaneity that treats Einstein's notion of simultaneity as a special case, and which allows for equivalence classes of spacelike separate events distinguished by covariant quantities such as action, phase, and (as I will argue) any reasonable measure of Booleanity.

Quantum operations are known to be the most general state transformations that can be applied to parts of compound systems compatibly with the probabilistic structure of quantum mechanics. What about the most general transformations of quantum operations? It turns out that any such general transformation can be realized by a quantum network with an open slot in which the input operation can be inserted, thus programming the resulting circuit. Moreover, one can recursively iterate this construction, generating an infinite hierarchy of admissible transformations and proving their realization within the circuit model of quantum mechanics. These results provide the basis of a new method to optimize quantum networks for information processing tasks, including e.g. gate estimation, discrimination, programming, and cloning. As examples of application, I will present here the optimal quantum networks for estimation of group transformations, for the alignment of reference frames with multiple communication rounds, and for universal cloning of unitary transformations.

More than 40 years ago, Bell ruled out completely local hidden variable models as an explanation for quantum correlations. However, a new type of hidden variable model has recently been brought to light by the work of Leggett. Such a model has both local and non-local parts. Roughly speaking, having a local part means that the measurement outcomes can be guessed with better than 50% success. In this talk, I will explain that there exist quantum correlations for which any hidden variable model must have a trivial local part. I will then discuss how an extension of the original theorem implies that these correlations can be used to enhance the quality of a private random string.

Despite over 40 years of research on Bell-type inequalities and the question of non-locality, new technical results that have general foundational relevance can still be obtained. In this talk will present a number of new results that deal with the question of how to discern local, quantum and no-signaling correlations. • 1) I will present a non-trivial no-signaling inequality that discerns no-signaling correlations from general correlations - the first to our knowledge. This inequality has a striking similarity with the CHSH inequality, yet it is crucially different. • 2) I will next discuss interesting relationships that can be inferred between some well-known conditions at different hidden- variable levels (such as the assumptions of outcome and parameter independence). The upshot of the analysis will be that which conditions are to be obeyed by different kinds of correlations and which are not, depends on the level of consideration. A conclusive picture therefore depends on which hidden-variable level is considered to be fundamental. • 3) I will further comment on interesting relationships that exist between inferences on the surface and subsurface level. Here the surface level deals with experimentally accessible probabilities (e.g., via relative frequencies) and the sub-surface level deals with probabilities that are conditioned on a hidden-variable (or the quantum state). The most interesting such a relationship is the following: any deterministic hidden-variable theory that obeys no- signaling and gives non-local correlations must show randomness at the surface, i.e., the surface probabilities cannot be deterministic. This is the case in Bohmian mechanics but this result shows it to be generic. Throughout the talk I will show how these three topics are related, and comment on the foundational impact of the results obtained.

A quantum channel models a physical process which adds noise to a quantum system by interacting with the environment. Protecting quantum systems from such noise can be viewed as an extension of the classical communication problem introduced by Shannon sixty years ago. A fundamental quantity of interest is the quantum capacity of a given channel. It measures the amount of quantum information that can be transmitted with vanishing error, in the limit of many independent transmissions over that channel. In this talk, I will show that certain pairs of channels, each with a capacity of zero, can have a strictly positive capacity when used together. This unveils a rich structure in the theory of quantum communication that is absent from Shannon\'s classical theory. This is joint work with Graeme Smith (IBM) which was published in the Sept. 26 issue of Science.

We present a method which can be used to convert certain single photon sources, such as quantum dots, into devices capable of emitting large strings of photonic cluster states in a controlled and pulsed “on demand” manner. Such sources greatly alleviate the resources required to achieve linear optical quantum computation. Standard spin errors, such as dephasing, are shown to affect only 1 or 2 of the emitted photons at a time. This allows for the use of standard fault tolerance techniques, and shows that the machine gun can be fired for arbitrarily long times. Using realistic parameters for current semiconductor quantum dot sources, we conclude high entangled-photon emission rates are achievable, with Pauli-error rates less than 0.2%.

The AdS/CFT correspondence has recently been extended to field theories satisfying the non-relativistic generalization of conformal symmetry, the Schroedinger symmetry. These holographic descriptions offer the potential to do calculations in the strong coupling regime of experimentally-realized condensed matter systems, such as fermions at unitarity. In this talk, we will outline the holographic formulation of such NRCFTs at zero temperature. We will then discuss the embedding of the appropriate geometry into IIB supergravity, and the finite temperature generalization that results. We will conclude with a brief discussion of the holographic description of non-relativistic conformal hydrodynamics and current research projects.

In recent years, the analysis of boolean functions has arisen as an important theme in theoretical computer science. In this talk I will discuss an extension of the concept of a boolean function to quantum computation. It turns out that many important classical results in the theory of boolean functions have natural quantum analogues. These include property testing of boolean functions; the Goldreich-Levin algorithm for approximately learning boolean functions; and a theorem of Friedgut, Kalai and Naor on the Fourier spectra of boolean functions. The quantum generalisation of this theorem uses a quantum extension of the hypercontractive inequality of Bonami, Gross and Beckner. This talk is based on joint work with Tobias Osborne.

Quantum entanglement has two remarkable properties. First, according to Bell\'s theorem, the statistical correlations between entangled quantum systems are inconsistent with any theory of local hidden variables. Second, entanglement is monogamous -- that is, to the degree that A and B are entangled with each other, they cannot be entangled with any other systems. It turns out that these properties are intimately related.