Aberystwyth University
IMAPS
Institute of
Mathematics & Physics

Quantum Probabilistic Symmetries

3rd to 7th September 2012

2012 LMS Midlands Regional Meeting & Workshop

Workshop Poster
Meeting Programme
Public Lecture Poster (English)
Public Lecture Poster (Welsh)
Map of Venues
 
 
 
 
LMS
 
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The timetable of all events, including titles & abstracts of talks, is in this interactive google calendar.

Lectures of the LMS Midlands Regional Meeting

A poster of the progamme of the meeting is available here.

  • Matthias Christandl: The quantum marginal problem.
    • Given a set of local density matrices, are they compatible? That is, could they arise from a joint global state? This question is known as the quantum marginal problem and is of importance in many aspects of quantum theory ranging from quantum chemistry (here known as the N-representability problem) to quantum information theory. In this talk, I will give an overview over recent progress on this problem and highlight some unexpected relations to multiparticle entanglement, group representation theory (Lie and symmetric groups) and the P versus NP conjecture of computer science.
  • Masaki Izumi: Group actions on operator algebras.
    • I'll give an overview of the classification results of group actions on operator algebras. I'll start with rather classical results on injective factors, and then focus on the recent development of poly-Z group actions on classifiable C*-algebras.
  • Roland Speicher: Quantum symmetries in free probability.
    • Quantum groups describe symmetries in a non-commutative context. I will discuss a special 'easy' class of such quantum symmetries. Basic examples are quantum permutations and quantum rotations. Those strengthen the corresponding classical symmetries. I will motivate these easy quantum groups and say a few words about their representation theory. Their role as symmetries will be emphasized by a non-commutative version of a de Finetti Theorem. (Joint work with Teo Banica and with Claus Köstler.)
  • Dan-Virgil Voiculescu: Noncommutative probability aspects of trace-class commutators.
    • Abstract TBA.
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Public Evening Lecture

A poster of this evening event is available here.

  • Reinhard Werner: Einstein and entanglement.
    • The famous 1935 paper by Einstein, Podolsky and Rosen (EPR) marks the beginning of the idea of quantum entanglement. For quantum information theory it is thus as fundamental as the development of the quantum formalism in 1926. Yet at the time it was considered refuted by a reply given by Bohr, and henceforth treated as an embarrassment or as a sign of Einstein's beginning senility. For about 30 years it was hardly read at all. In this talk I will explain what the controversy was about, and what point Einstein made. I will present an entirely elementary version of the argument, or rather of a strengthened form due to Bell (1963). This strengthened form shows that local classical explanations of quantum correlations are impossible, and hence that one of Einstein's life-long projects was bound to fail. From a modern perspective Bell's version is present in nuce already in the EPR paper, so it is somewhat mysterious why Einstein did not go all the way in 1935. I will offer a tentative explanation connecting the main point he was making to a notion called 'steering', which logically lies between 'entanglement' and Bell-style 'nonlocality'.
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Lecture Series of the Workshop

To foster the communication between participants coming from different research areas the workshop includes several thematic lecture streams. The timetable of these lectures is in the interactive google calendar.

  • Lectures on the Quantum Marginal Problem.
  • Given a set of local density matrices, are they compatible? That is, could they arise from a joint global state? In four lectures we will explain the - as we believe - unexpected and beautiful pieces of mathematics that lurk behind this old question of quantum physics.

    • The Quantum Marginal Problem I: Eigenvalues and Representations.
      (Presented by Matthias Christandl.)
    • The Quantum Marginal Problem II: Eigenvalue Distributions.
      (Presented by Matthias Christandl.)
    • The Quantum Marginal Problem III: Algebraic Geometry.
      (Presented by Michael Walter.)
    • The Quantum Marginal Problem IV: Entanglement.
      (Presented by Michael Walter.)
  • Literature: http://arxiv.org/abs/quant-ph/0604183 (Part 1), http://arxiv.org/abs/1204.0741, http://arxiv.org/abs/1208.0365
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  • Lectures on Free Probability and Quantum Groups.
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  • Uwe Franz: On the quantum symmetry group of a Hadamard matrix.
  • Teodor Banica and Remus Nicoara showed that one can associate a unique quantum symmetry group to a complex Hadamard matrix. In my talk I will present a new probabilistic approach to studying this quantum group. Joint work with Teodor Banica and Adam Skalski. See also arXiv:1112.5018.

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  • Adam Skalski: On some categories of coloured partitions related to representations of quantum symmetry groups.
  • We will discuss certain categories of two-coloured partitions arising in the study of representation theories of quantum symmetry groups of duals of free groups and related to families of intertwiners of certain unitary matrices. Some (free)-probabilistic interpretations will be given and indications to related open problems presented. Based on joint work with Teodor Banica.

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  • Roland Speicher Free Probability and Quantum Groups I, II.
  • In this series of four lectures we will give an introduction to free probability and special classes of compact quantum groups, with particular emphasis on the interaction between the two subjects. This interaction goes in both directions: quantum groups describe symmetries of free random variables; and many intrinsic quantities for those quantum groups, like coefficients or characters, become usually asymptotically free.

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  • Lectures on the Combinatorics and Representation Theory of Large Groups.
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  • Alexander Gnedin: q-Exchangeability, random permutations and characters of the symmetric group I,II.
  • Various generalisations of exchangeable processes and de Finetti's theorem are related to random permutations with probability function depending on the value of some fixed permutation statistic, and consistent for various sizes n.

    We shall discuss a q-deformation of exchangeability related to the Mallows models for random permutations, with sufficient statistic being the number of inversions. A q-exchangeable process (for q<1) favours increasing patterns, and in the infinite model has stationarity features.

    Another kind of symmetry appears when the permutation statistic is the number of descents. We will show that this instance is related to a dual problem of describing the characters on the symmetric group that depend solely on the number of cycles of permutation.

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  • Piotr Sniady: Combinatorics of the asymptotic representation theory of the symmetric groups I-III.
  • The representation theory of the symmetric groups S(n) is intimately related to combinatorics: combinatorial objects such as Young tableaux and combinatorial algorithms such as Murnaghan-Nakayama rule. In the limit as n tends to infinity, the structure of these combinatorial objects and algorithms becomes complicated and it is hard to extract from them some meaningful answers to asymptotic questions. In order to overcome these difficulties, a kind of dual combinatorics of the representation theory of the symmetric groups was initiated in 1990s. In this series of lectures I will present this dual combinatorics and its relations to free probability theory.

    • Lecture I: Preliminaries on representations.
    • Lecture II & III: Dual combinatorics of representations of symmetric groups.
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  • Lectures on Operator Algebras and Subfactor Theory.
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  • David Evans: Braided Subfactors and Conformal Field Theory I
  • Abstract TBA

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  • Masaki Izumi: The classification of certain fusion categories and subfactors I,II.
  • It is known that a pair of fusion categories naturally arises from a nice subfactor as an important classification invariant. A typical example of a fusion category is the representation category of a finite group. In this series of talks, I'll report on a few classification results of certain fusion categories and subfactors.

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  • Mathew Pugh: Braided Subfactors and Conformal Field Theory II
  • I will discuss two applications of the nimrep theory arising from braided subfactors. The first is to almost Calabi-Yau algebras, which are finite dimensional graded algebras associated to SU(3) braided subfactors. In the second part I will discuss spectral measures for operators associated to the Lie groups SU(2), SU(3) and G2, their nimrep graphs and finite subgroups.

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  • V.S. Sunder: Hilbert-von Neumann modules.
  • (Report on joint work with Panchugopal Bikram, Kunal Mukherjee (both of IMSc) and R. Srinivasan (of CMI).)

    We introduce a way of regarding Hilbert von Neumann modules as spaces of operators between Hilbert space, not unlike Skeide, but in an apparently much simpler manner and involving far less machinery. We verify that our definition is equivalent to that of Skeide, by verifying the `Riesz lemma' or what he calls `self-duality'. An advantage with our approach is that we can totally side-step the need to go through C*-modules and avoid the two stages of completion - first in norm, then in the strong operator topology - involved in the former approach.

    We establish the analogue of the Stinespring dilation theorem for Hilbert von Neumann bimodules, and we develop our version of `internal tensor products' which we refer to as Connes fusion for obvious reasons.

    In our discussion of examples, we examine the bimodules arising from automorphisms of von Neumann algebras, verify that fusion of bimodules corresponds to composition of automorphisms in this case, and that the isomorphism class of such a bimodule depends only on the inner conjugacy class of the automorphism. We also relate Jones' basic construction to the Stinespring dilation associated to the conditional expectation onto a finite-index inclusion (by invoking the uniqueness assertion regarding the latter).

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Additional Workshop Lectures

The timetable of these lectures is in this interactive google calendar.

  • Octavio Arizmendi: Free convolutions for k-symmetric probability measures.
    • We show that free multiplicative convolution between a measure concentrated in the positive real line and a probability measure μ with k-symmetry is well defined. Analytic tools to calculate this convolution are explained. Finally, we concentrate on free additive powers of k-symmetric distributions and prove that μ⊞t is a well defined probability measure, for all t > 1. We derive central limit theorems and Poisson type ones. More generally, we consider freely infinitely divisible measures and prove that free infinite divisibility is maintained under the mapping μ → μk. We conclude by focusing on (k-symmetric) free stable distributions.
  • Edwin Beggs: From homotopy to diffusion.
    • A cochain homotopy on a differential graded algebra can be used to deform the differential graded algebra, on addition of an extra dimension, which we label time. We show how examples on the classical de-Rham complex of a manifold give diffusion and drift. There is no need that the original differential graded algebra should be (graded) commutative, giving rise to various noncommutative examples. We also consider higher order differential forms and covariant differentiation on bundles. As a noncommutative geometer rather than a probabalist, I should confess lack of knowledge and hope that others will show how this fits with existing theory, and whether it has a probabalistic interpretation in the noncommutative case. Joint work with Ghaliah Alhamzi and Andrew Neate (Swansea).
  • Tomasz Brzezinski: Symmetries in non-commutative geometry (a synthetic point of view).
    • In the first part of the talk we outline the basic ideas of synthetic approach to differential geometry. The main idea of this approach, which originates from considerations of Sophus Lie is very simple: All geometric constructions are performed within a suitable base category in which space forms are objects. In the second part we explain the foundations of noncommutative geometry and then indicate how a synthetic method could be employed in the context of noncommutative differential geometry. Our more specific aim is to explain synthetic approach to commutative and nocommutative geometry on two examples of geometric notions. We intend to explain all categorical ingredients that enter the synthetic definition of a principal bundle (in classical geometry) and then to show that noncommutative generalisation of this definition yields in particular principal comodule algebras which play the role of principal bundles in noncommutative geometry.
  • Daniel Burgarth: Quantum System Identification with limited resources.
    • The aim of quantum system identification is to estimate the ingredients inside a black box, in which some quantum-mechanical unitary process takes place, by just looking at its input-output behavior. Here we establish a basic and general Lie algebraic framework for quantum system identification, that allows us to classify how much knowledge about the quantum system is attainable, in principle, from a given experimental setup. Prior knowledge on some elements of the black box helps the system identification. When the topology of the system is known, the framework enables us to establish a general criterion for the estimability of the coupling constants in its Hamiltonian.
  • Benoit Collins: Free probability techniques in quantum information theory.
    • This talk will be an overview of a few free probability techniques that have been recently used in quantum information theory in order to understand better entanglement and the behavior of typical quantum channels.
  • Vitonofrio Crismale: A De Finetti-type theorem on the CAR algebra.
    • The symmetric states on a quasi local C*-algebra on the infinite set of indices J are those invariant under the action of the group of the permutations moving only a finite, but arbitrary, number of elements of J. The celebrated De Finetti Theorem describes the structure of the symmetric states (i.e. exchangeable probability measures) in classical probability. In the present talk we show an extension of De Finetti Theorem to the case of the CAR algebra, that is for physical systems describing Fermions. Namely, we show that the compact convex set of such states is a Choquet simplex, whose extremal (i.e. ergodic w.r.t. the action of the group of permutations previously described) are precisely the product states in the sense of Araki-Moriya. In order to do that, we present some ergodic properties naturally enjoyed by the symmetric states which have a self-containing interest. The talk is based on a joint work with Francesco Fdaleo (Department of Mathematics, University of Tor Vergata, Roma).
  • Gwion Evans: Identifying approximately finite-dimensional Cuntz-Krieger algebras of higher-rank graphs.
    • The Cuntz-Krieger algebras of higher-rank graphs have attracted much interest since they were introduced by Kumjian and Pask at the turn of the millennium. They can be analysed in much the same way as Cuntz-Krieger algebras and graph C*-algebras, but they are a richer source of examples than these special cases. For instance, simple (rank one) graph C*-algebras admit an elegant classification theorem: they are either purely infinite or approximately finite-dimensional (AF). However, this dichotomy no longer holds for rank greater than one (for example, rank 2 higher-rank graph C*-algebras include, up to strong Mortia equivalence, all irrational rotation algebras). As a step towards a classification theorem for higher-rank graph C*-algebras we investigate the question when is the Cuntz-Krieger algebra of a higher-rank graph AF. Our investigations indicate that the question is far more difficult when the rank is greater than one, and leads to the possibility of an alternative presentation that might be particularly useful to study actions of groups on AF-algebras. This is joint work with Aidan Sims.
  • Roland Friedrich: Free Probability Theory and Affine Group Schemes.
    • In this talk we show in detail that the basic algebraic notion related to freeness in non-commutative probability theory is that of an affine group scheme, which are anti-equivalent to the category of commutative Hopf algebras. In this framework Voiculescu's S-transform provides a natural isomorphism with the functor of Witt vectors. Once this correspondence is established, a very rich picture emerges, which we shall then discuss. This presentation is based on joint work with J. McKay.
  • Andreas Gärtner: Recurrence, Transience and Noncommutative Poission Integrals.
    • We present a coherent approach to recurrence and transience, starting from a version of the Riesz decomposition theorem for superharmonic elements. Our approach leads to a classification of idempotent Markov operators, from which we obtain an abstract Poisson integral for weak* mean ergodic maps.
  • Dardo Goyeneche: A new method to construct families of complex Hadamard matrices in even dimensions.
    • We present a new method for constructing affine families of complex Hadamard matrices in every even dimension. This method contains the Szöllösi's method and it has an intersection with the Dita's construction. We reproduce well-known results, we extend some known families and we present new families of complex Hadamard matrices in even dimensions. We also find a restriction for any set of four mutually unbiased bases existing in dimension six.
  • Robin Hudson: A Boson de Finetti theorem, interpretative problems of quantum mechanics and measures of entanglement.
    • An early quantum de Finetti theorem gives a unique representaion of a consistent hierarchy of symmetric normal (density-operator) states on tensor products of the algebra of bounded operators on a Hilbert space as an integral over product normal states. For the integral to be supported only by pure product states it is necessary and sufficient that the hierarchy satisfy a stronger Bose-Einstein symmetry condition. Some consequences in quantum mechanics and quantum information and a conjectured generalisation are considered.
    • [1] R L Hudson and G R Moody, Locally normal symmetric states and an analogue of de Finetti's theorem, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete (PTRF) 33 (1976), 343-351.
  • Burkhard Kümmerer: A Propp-Wilson algorithm for quantum equilibrium states.
    • (Joint work with Nadiem Sissouno.)

      Equilibrium distributions for Markov chains on large state spaces can rarely be computed explicitly. Numerical algorithms have to be used instead. Among these the "Exact-Sampling-by-Coupling-from-the-Past-Algorithm" of J.G. Propp and D.B. Wilson dating from 1996 is one of the best known. After finitely many steps it produces a sample point with precisely the probability given by the theoretical (but unknown) equilibrium distribution.
      A straightforward generalization of this strategy to quantum Markov chains is not possible since a mixed state of a quantum system can no longer be interpreted as a distribution on a classical space of sample points, thus the idea of sampling loses its meaning.
      In this talk we start our discussion with a brief description of the classical Propp-Wilson-Algorithm. It is then reinterpreted as a statement on the existence of synchronizing words for a road-coloured graph which represents the Markov chain, thus linking it to a different representation of classical Markov chains which can be generalized to the non-commutative framework. Within this approach the concept of a synchronizing word can be formulated and finally leads us to a version of the Propp-Wilson algorithm for quantum Markov chains.
  • Franz Lehner: Characterization problems in free probability.
    • We present characterizations of the semicircle law by conditional expectations of linear and quadratic forms without boundedness assumptions. Joint work with G.P.Chistyakov.
  • Martin Lindsay: Lévy processes on compact quantum groups and noncommutative manifolds.
    • A general structure theory of Lévy processes on compact quantum groups will be described. This extends the bounded-generator case developed with my former student, Adam Skalski, and the original algebraic theory due to Schuermann and coworkers. By restriction to classical compact groups, all the classical Lévy processes are realised. If time permits, a characterisation will be given of the class of quantum stochastic flows on an `admissible' spectral triple of finite compact type.
      The key ingredients are: the representation theory of compact quantum groups, in particular the Peter-Weyl theory; quantum stochastic cocycles, and their generation via quantum stochastic differential equations in the sense of Hudson, (Evans) and Parthasarathy; and quantum isometry groups of noncommutative manifolds in the sense of Goswami.
      This is joint work with Biswarup Das. It was supported by the UKIERI Research Collaboration Network 'Quantum Probability, Noncommutative Geometry & Quantum Information'.
  • Hans Maassen: Entanglement of completely symmetric quantum states.
    • A state on n identical but distinguishable particles will be called completely symmetric if it is invariant both for permutation of the particles and simultaneous unitary rotation in their one-particle Hilbert spaces. We study the geometry of the set of completely symmetric separable states.
  • Michael Mc Gettrick: Quantum Evolutionary Game Theory.
    • We will give an introduction to a new area of research at the intersection of Quantum Game Theory and Evolutionary Computation. The idea is to play repeated 2-person quantum games on a network, where at each iteration, nearest neighbors play against one another. We then analyze how the quantum strategies evolve over time. We consider three models, with maximum bipartite entanglement, graph-dependent multipartite entanglement, and graph-independent multipartite entanglement. Partial results will be presented for the specific case of the quantum prisoners dilemma played on the cyclic graph.
  • James Mingo: Second Order Freeness and Orthogonal Random Matrices.
    • Second order freeness is a property exhibited by many ensembles of random matrices. In recent work with Mihai Popa and Emily Redelmeier we have shown that Haar distributed random orthogonal matrices are real asymptotically free of second order from independent and orthogonally invariant ensembles.
  • Graeme Mitchison: Quantum de Finetti theorems.
    • I shall describe some de Finetti-like results that have been obtained in quantum information theory. One is typically interested in systems consisting of a finite number, n, of subsystems. Given a symmetric state of such a system, the state obtained by tracing out all but k of the subsystems can be approximated, within trace norm distance proportional to k/n, by a convex sum of tensor power states. This is analogous to the de Finetti theorem of Diaconis and Freedman for finite exchangeable sequences. I shall try to illuminate the relationship between their theorem and the quantum versions.
  • Walter Reusswig: On entanglement of states on infinite tensor product algebras.
    • We investigate different concepts of entanglement for states on infinite tensor product algebras and discuss entanglement properties of pure finitely correlated states.
  • Michael Schürmann: Lévy processes on algebraic structures.
    • (Report on joint work with Stephanie Lachs)

      A full classification of noncommutative independences satisfying only the first three of Muraki's four axioms will be given. Corresponding to the five independences of Muraki's classification, we obtain five families of independences which in the tensor, free and monotone/anti-monotone cases are parametrized by one complex number, in the Boolean case by a pair of complex numbers. The independences, in general, are no longer positive. We define Lévy processes for each independence and, in the tensor case, show how these processes can be realized on a Bose Fock space over a dual pair. Using a generalization of Franz' theory of reduction of independences, the two-parameter Boolean independences can be reduced to the tensor case.
  • Kay Schwieger: Asymptotics of diagonal quantum couplings.
    • In this talk we provide a Quantum Coupling Inequality, similar to the well-known inequality of the classical coupling method. Moreover, we present the construction of the diagonal coupling associated to a tensor dilation and, finally, state conditions for this coupling to converge.
  • Otgonbayar Uuye: Weak semiprojectivity of cones.
    • Weak semiprojectivity is the C*-algebraic analogue of the notion of absolute neighbourhood retract.

      The cone of a finite-dimensional C*-algebra is known to be weakly semiprojective. We show that the cone of an AF algebra is weakly semiprojective if and only if the algebra itself is weakly semiprojective.
  • Reinhard Werner: The universal embezzling algebra.
    • Joint work with Uffe Haagerup and Volkher Scholz (in preparation).

      'Embezzling' has been introduced as a term for a quantum operation in which entanglement is extracted from a large system (the 'bank') without noticeably changing the state of the large system, and without exchanging classical information between the entangled partners. The large system can be idealized as an infinite bipartite system given by a von Neumann algebra and its commutant. We give a criterion for a normal state to have the embezzling property and show that all normal states have this property if and only if the algebra is type III1. This is contrasted with other infinite entanglement resources like the hyperfinite II1 factor, which according to the Araki-Woods construction corresponds to a supply of infinitely many qubit singlets.

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