Talks

We present a general hydrodynamic theory of transport in the vicinity of superfluid-insulator transitions in two spatial dimensions described by ``Lorentz\'\'-invariant quantum critical points. We allow for a weak impurity scattering rate, a magnetic field $B$, and a deviation in the density, $rho$, from that of the insulator. We show that the frequency-dependent thermal and electric linear response functions, including the Nernst coefficient, are fully determined by a single transport coefficient (a universal electrical conductivity), the impurity scattering rate, and a few thermodynamic state variables. With reasonable estimates for the parameters, our results predict a magnetic field and temperature dependence of the Nernst signal which resembles measurements in the cuprates, including the overall magnitude. Our theory predicts a ``hydrodynamic cyclotron mode\'\' which could be observable in ultrapure samples. We also present exact results for the zero frequency transport co-efficients of a supersymmetric conformal field theory (CFT), which is solvable by the AdS/CFT correspondence. This correspondence maps the $rho$ and $B$ perturbations of the 2+1 dimensional CFT to electric and magnetic charges of a black hole in the 3+1 dimensional anti-de Sitter space. These exact results are found to be in full agreement with the general predictions of our hydrodynamic analysis in the appropriate limiting regime. The mapping of the hydrodynamic and AdS/CFT results under particle-vortex duality is also described.

Resent research seems to indicate that charged extremal black holes in D=4 supersymmetric theories should be most naturally described in terms of more primitive atomic constituents. I will briefly describe what I mean by these atomic constituents and how they appear to play a role in both BPS and non-BPS extremal black holes.

Every restriction on quantum operations defines a resource theory, determining how quantum states that cannot be prepared under the restriction may be manipulated and used to circumvent the restriction. A superselection rule is a restriction that arises through the lack of a classical reference frame. The states that circumvent it (the resource) are quantum reference frames. We consider the resource theories that arise from three types of superselection rule, associated respectively with lacking: (i) a phase reference, (ii) a frame for chirality, and (iii) a frame for spatial orientation. Focussing on pure unipartite quantum states, we identify the necessary and sufficient conditions for a deterministic transformation between two resource states to be possible and, when these conditions are not met, the maximum probability with which the transformation can be achieved. We also determine when a particular transformation can be achieved reversibly in the limit of arbitrarily many copies and find the maximum rate of conversion. (joint work with Gilad Gour)

Entanglement plays a fundamental role in quantum information processing and is regarded as a valuable, fungible resource, The practical ability to transform (or manipulate) entanglement from one form to another is useful for many applications. Usually one considers entanglement manipulation of states which are multiple copies of a given bipartite entangled state and requires that the fidelity of the transformation to (or from) multiple copies of a maximally entangled state approaches unity asymptotically in the number of copies of the original state. The optimal rates of these protocols yield two asymptotic measures of entanglement, namely, entanglement cost and distillable entanglement. It is not always justified, however, to assume that the entanglement resource available, consists of states which are multiple copies, i.e.,tensor products, of a given entangled state. More generally, an entanglement resource is characterized by an arbitrary sequence of bipartite states which are not necessarily of the tensor product form. In this seminar, we address the issue of entanglement manipulation for such general resources and obtain expressions for the entanglement cost and distillable entanglement.

The manifold of pure quantum states can be regarded as a complex projective space endowed with the unitary-invariant Fubini-Study metric. The physical characteristics of a given quantum system can then be represented by a variety of geometrical structures that can be identified in this manifold. This talk will review a number of examples of such structures as they arise in the state spaces of spin-1/2, spin-1, spin-3/2, and spin-2 systems, and various types of entangled systems, all of which have fascinating and beautiful geometries associated with them. The geometric approach offers interesting insights into the nature of quantum systems, and is also useful in the consideration of foundational issues such as those related to the measurement problem.

It has been known for a long time that instanton effects control the large order behavior of the perturbation series in quantum mechanics and gauge theories. I present a study of this connection in the context of matrix models in 1/N-expansion and topological strings. I will show how to compute the one-instanton corrections for a generic matrix model. Due to a recent matrix model inspired formalism for the topological string amplitudes on local Calabi-Yau manifolds, this can be used to compute nonperturbative effects in topological string theory and make predictions about the asymptotics of the string perturbation series. I discuss various cases where our predictions can be tested, yielding spectacular agreement with the asymptotics extracted by standard numerical methods.

It is known that finite fields with d elements exist only when d is a prime or a prime power. When the dimension d of a finite dimensional Hilbert space is a prime power, we can associate to each basis state of the Hilbert space an element of a finite or Galois field, and construct a finite group of unitary transformations, the generalised Pauli group or discrete Heisenberg-Weyl group. Its elements can be expressed, in terms of the elements of a Galois field. This group presents numerous applications in Quantum Information Science e.g. tomography, dense coding, teleportation, error correction and so on. The aim of our talk is to give a general survey of these properties and to present recently obtained results in connection with three problems: -the so-called ''Mean King's problem'' in prime power dimension, -discrete Wigner distributions, -and quantum tomography . Finally we shall discuss a limitation of the possible dimensions in which the so-called epistemic interpretation can be consistently formulated, in relation with the existence of finite affine planes, Euler's conjecture and the 36 officers problem.