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A new microcanonical equilibrium state is introduced for quantum systems with finite-dimensional state spaces. Equilibrium is characterised by a uniform distribution on a level surface of the expectation value of the Hamiltonian. The distinguishing feature of the proposed equilibrium state is that the corresponding density of states is a continuous function of the energy, and hence thermodynamic functions are well defined for finite quantum systems. The density of states, however, is not in general an analytic function. It is demonstrated that generic quantum systems therefore exhibit second-order phase transitions at finite temperatures. The talk is based on work carried out in collaboration with D.W. Hook and L.P. Hughston.

What if the second law of thermodynamics, in the hierarchy of physical laws, were at the same level as the fundamental laws of mechanics, such as the great conservation principles? What if entropy were an intrinsic property of matter at the same level as energy is universally understood to be? What if irreversibility were an intrinsic feature of the microscopic dynamical law of all physical objects, including an individual qubit or qudit? This talk will show how positive answers to these questions need not contradict any of the known results of quantum mechanics. We construct a logically consistent, mathematically sound and definite, physically intriguing, non-relativistic and non-statistical quantum theory, in which the second law of thermodynamics is embedded as a fundamental microscopical law. The theory hinges upon a nonlinear extension of unitary Hamiltonian dynamics which for uncorrelated and noninteracting systems reduces to the usual Schroedinger equation for the zero entropy states, but in general generates a group (not a semi group) of irreversible time evolutions, where the non-Hamiltonian entropy generating term in the evolution equation attracts the state towards the direction of maximal entropy increase. Various examples and features of this highly non-conventional dynamical theory are discussed. References available at http://www.quantumthermodynamics.org/

Renner\'s global quantum de Finetti theorem establishes that if the state of a quantum system is invariant under permutations of its systems, then almost all of its subsystems are almost in the same state and independent of each other. Motivated by this result, we show that the most straightforward classical analogue of Renner\'s theorem is false. Joint work with Matthias Christandl (Cambridge).

The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maximally mixed state. One clean qubit computers are believed to be strictly weaker than standard quantum computers, but still capable of solving some classically intractable problems. I\'ll discuss my recent work in collaboration with Peter Shor which shows that evaluating a certain approximation to the Jones polynomial at a fifth root of unity for the trace closure of a braid is a complete problem for the one clean qubit complexity class.

The simplest algebraic curves of genus one are the nonsingular cubics in two-dimensional complex projective space. Interpreting CP^2 as the space of pure quantum states associated with a Hilbert space of dimension three, I will show how various properties of d=3 symmetric informationally complete positive operator valued measures can be understood in terms of the geometry of such curves. The resulting structure, although of considerable complexity, is very beautiful from a mathematical perspective.

We give a convenient representation for any map which is covariant with respect to an irreducible representation of SU(2), and use this representation to analyze the evolution of a quantum directional reference frame when it is exploited as a resource for performing quantum operations. We introduce the moments of a quantum reference frame, which serve as a complete description of its properties as a frame, and investigate how many times a quantum directional reference frame represented by a spin-j system can be used to perform a certain quantum operation with a given probability of success. We provide a considerable generalization of previous results on degradation of reference frame, from which follows a classification of the dynamics of spin-j system under the repeated action of any covariant map with respect to SU(2). Joint work with Lana Sheridan, Martin Laforest and Stephen Bartlett

The solution of many problems in quantum information is critically dependent on the geometry of the space of density matrices. For a Hilbert space of dimension 2 this geometry is very simple: it is simply a sphere. However for Hilbert spaces of dimension greater than 2 the geometry is much more interesting as the bounding hypersurface is both highly symmetric (it has a d^2 real parameter symmetry group, where d is the dimension) and highly convoluted. The problem of getting a better understanding of this hypersurface is difficult (it is hard even in the case of a single qutrit). It is also, we believe, both physically important and mathematically deep. In this talk we relate the problem to MUBs (mutually unbiased bases) and SIC-POVMs (symmetric informationally complete positive operator valued measures). These structures were originally introduced in connection with tomography. However, that by no means exhausts their importance. In particular their existence (non-existence???) in a given dimension is a source of significant insight into the state space geometry in that dimension. SIC-POVMs are especially important in this regard as they provide a a natural set of coordinates for state space. In this talk we give an overview of the problem. We then go on to describe some recent results obtained in collaboration with Chris Fuchs and Hoan Dang (also see recent work by Wootters and Sussman). In particular we describe the connection with minimum uncertainty states. These states, besides being interesting in themselves (they are a kind of discrete analogue of coherent states with important cryptographic applications), suggest a potentially fruitful line of attack on the still outstanding SIC existence problem.

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 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.