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### Workshop on "Operator algebras and Quantum Information Theory"

 The conference will be held in the Institut Henri Poincaré, 11-15 September 2017OrganizersConfirmed SpeakersMichael Brannan (College station)Zeqian Chen (Wuhan Institute of Physics and Mathematics)Runyao Duan (University of Technology, Sydney)Edward Effros (Los Angeles)Li Gao (UIUC)Rolf Gohm (Aberystwyth)Fumio Hiai (Tohoku University)Seung-Hyeok Kye (Seoul National University)Nicholas LaRacuente (UIUC)Adam Majewski (Gdansk)Alexander Müller-Hermes (University of Copenhagen)Magdalena Musat (University of Copenhagen)Yoshiko Ogata (Tokyo)Hiroyuki Osaka (Ritsumeikan)Carlos Palazuelos (Instituto de Ciencias Matematicas, Madrid)Vern Paulsen (Waterloo)Tatyana Shulman (Institute of Mathematics of Warsaw)Thomas Vidick (Caltech)Dan Voiculescu (UC Berkeley)Ignacio Vllanueva (Madrid)Andreas Winter (Barcelona)Tentative Schedule TBAAbstractsMichael Brannan (College station) Title: Entangled subspaces from quantum groups and their associated quantum channels.Abstract:I will describe a class of highly entangled subspaces of bipartite quantum systems arising from the representation theory of a class of compact quantum groups, called the free orthogonal quantum groups.  This construction yields new examples of quantum channels with some interesting properties.  In particular, it is possible to obtain large lower bounds on the minimal output entropies of these channels, while at the same time we can precisely describe the behavior of tensor products of these channels under certain entangled inputs.  Our analysis of tensor products turns out to relate very nicely to the Temperley-Lieb recoupling theory and the quantum 6j-symbols associated to these quantum groups. (This is joint work with Benoit Collins).Zeqian Chen (Wuhan Institute of Physics and Mathematics) Title: The geometric phase associated with quantal observable space.Abstract: In this talk, we will report that the geometric phase is introduced associated with quantal observable space. The phase is determined by the Heisenberg equation, contrary to the usual one by the Schrödinger equation. Geometrical interpretation of the phase over the quantal observable space is also presented.Runyao Duan (Centre for Quantum Software and Information, University of Technology Sydney (UTS), Australia)Title: Asymptotic entanglement manipulation under PPT operations: new SDP bounds and irreversibilityAbstract: We study various aspects of asymptotic entanglement manipulation of general bipartite states under operations that completely preserve positivity of partial transpose (PPT). Our key findings include: i) an additive semi-definite programming (SDP) entanglement measure which is an improved upper bound of the distillable entanglement than the logarithmic negativity; ii) a succinct SDP characterization of the one-copy deterministic distillation rate and an additive upper bound; iii) nonadditivity of Rains’ bound for a class of two-qubit states; and iv) two additive SDP lower bounds to the Rains’ bound and relative entropy of entanglement, respectively. These findings enable us to efficiently evaluate the asymptotic distillable entanglement and entanglement cost for several classes of mixed states. As applications, we show that for any rank-two mixed state supporting on the 3-level anti-symmetric subspace, both the Rains’ bound and its regularization are strictly less than the asymptotic relative entropy of entanglement. That also implies the irreversibility of asymptotic entanglement manipulation under PPT operations, one of the major open problems in quantum information theory.Joint works with Mr Xin Wang (UTS), available at arXiv:1601.07940, 1605.00348 and 1606.09421Edward Effros (UC Los Angeles) Title: Some remarkable gems and persistent difficulties in quantized functional analysisAbstract: TBALi Gao (UIUC) Title:Operator Algebras Aspects of Quantum Teleportation and Superdense CodingAbstract: Quantum teleportation and superdense coding are fundamental protocols in quantum information theory. They together describe the resource trade-off between quantum communication and classical communication when assisted with remote entanglement. In terms of operator spaces, teleportation and superdense coding are interpreted as that $S_1^d$ and $l_1^{d^2}$ are completely embedded into the matrix level of each other. In this talk, I will discuss the lifted embedding of their C*-envelopes--Brown algebra and free group C*-algebra. It gives strong connections between these two kinds of universal C*-algebras, and also has applications in quantum correlation sets. This is a joint work with Marius Junge.Rolf Gohm (Aberystwyth) Title: TBAAbstract: TBAFumio Hiai (Tohoku University) Title: Different quantum divergences in general von Neumann algebrasAbstract: Different quantum divergences, including standard f-divergences, maximal f-divergences, measured f-divergences, sandwiched R\'enyi divergences, $\alpha$-z-R\'enyi relative entropies, etc., have extensively been developed in these years, with various applications to quantum information, in particular, to the reversibility of quantum operations. However, quantum divergences in the von Neumann algebra setting have not been well developed yet, apart from the earlier work on quasi-entropies (whose special case is standard f-divergences) due to D. Petz and some others. In my talk I give a comprehensive survey on quantum divergences in general von Neumann algebras, based on Haagerup's theory of non-commutative $L^p$-spaces and Kosaki's complex interpolation theory of non-commutative $L^p$-spaces. Recent works on sandwiched R\'enyi divergences in von Neumann algebras due to Jencov\'a and Berta-Scholz-Tomamichel are referred to as well.Seung-Hyeok Kye (Seoul National University, Seoul, Korea) Title: Positivity of multi-linear maps and applications to quantum information theoryAbstract: In this talk, we use the duality between n-partite separable states and positive multi-linear maps with n-1 variables, to give a necessary criterion for three qubit separability in terms of diagonal and anti-diagonal entries. If all the entries are zero except for diagonal and anti-diagonal entries, then our characterization is also sufficient for separability. Many important classes of three qubit states like Greenberger-Horne-Zeilinger diagonal states are in this class. We give the characterization in terms of a norm in the four dimensional complex spaces. We also exhibit examples of non-decomposable positive bi-linear maps which generate exposed rays in the cone of all positive bi-linear maps in 2x2 matrices. The exposedness enables us to detect three qubit PPT entanglement of nonzero volume.Nicholas LaRacuente (UIUC) Title: Non-commutative L_p Spaces and Asymmetry MeasuresAbstract: We relate a common class of entropic asymmetry measures to non-commutative L_p space norms. These asymmetry measures have operational meanings related to the resource theory of asymmetry and problems of reference frame misalignment in quantum systems. We further derive a correspondence between maximal asymmetry and the von Neumann algebra index introduced by Pimsner and Popa. This investigation is motivated by previous work of Marvian and Spekkens on extensions of Noether's theorem. This is joint work with Li Gao and Marius Junge.Adam Majewski (Gdansk) Title: Quantum correlations. Abstract: Applying the basic rules of non-commutative integrations and guided by principles of Quantum Mechanics, we present the rigorous descrip- tion of quantum correlations. This will be done for a general composite quantum system. In particular, centered on quantum probability we describe measures of quantum correlations. Our lecture will be based on the algebraic approach to quantum probability.Alexander Müller-Hermes (University of Copenhagen) Title: Tensoring Positive mapsAbstract: We investigate linear maps between matrix algebras that remain positive under tensor powers, i.e., under tensoring with n copies of themselves. Completely positive and completely co-positive maps are trivial examples of this kind. We show that for every natural number n there exist non-trivial maps with this property and that for two-dimensional Hilbert spaces there is no non-trivial map for which this holds for all n. For higher dimensions we reduce the existence question of such non-trivial “tensor-stable positive maps” to a one-parameter family of maps and show that an affirmative answer would imply the existence of NPPT boundentanglement.Magdalena Musat (University of Copenhagen) - TBAYoshiko Ogata (University of Tokyo) Title: A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterizationAbstract:We introduce a class of gapped Hamiltonians on quantum spin chains, which allows asymmetric edge ground states, and investigate its properties.Hiroyuki Osaka (Ritsumeikan)Title: Operator means and application to generalized entropiesAbstract: In this talk we present a relation between generalized entropies and operator means.  For example, as pointed by Furuichi \cite{SF11}, two upper bounds on the Tsallis entropies suggest the following inequality:  for positive operators X and Y and where the symbole stands for the weighted geometric mean, that is,Unfortunately, this inequality does not hold in general, but this is true when XY + YX \geq  0.  We can extend this inequality for a general operator mean and it is called the generalized reverse Cauchy inequality.  We also give a formulation of new Rényi relative entropies by Mosonyi and Ogawa using operator means. Carlos Palazuelos (Instituto de Ciencias Matematicas, Madrid)  Title: Classical vs Quantum communication in XOR gamesAbstract: In this talk we will study the value of XOR games G when the players are allowed to use a limited amount of one-way classical (resp. quantum) communication. This can be understood as an intermediate setting between quantum nonlocality and communication complexity problems. Then, we will show that some of the key quantities studied in these topics can be naturally described by means of tensor norms and that this description allows to find new connections between quantum nonlocality and communication complexity.Vern Paulsen (Waterloo) Title: C*-algebras and Synchronous Games. Abstract: In recent years a deep connection has been found between Connnes’ embedding problem and Tsirelson’s questions about various sets of probabilistic quantum correlations, called local, quantum, quantum approximate, and quantum commuting correlations, respectively. The most fruitful approach to studying these questions and separating these types of correlations has been through the theory of perfect strategies for finite input-output games. Synchronous games are a special family of these games. Affiliated with each synchronous game is a C*-algebra such that the game has a perfect strategy of each of these four types if and only if the C*-algebra has, respectively, a trace of one of four types. Using this theory and the work of Slofstra, we are able to construct two graphs such that their “graph isomorphism game” has a perfect quantum approximate strategy but no perfect quantum strategy. This, in turn, implies that the set of synchronous quantum correlations is not closed.Tatyana Shulman (Institute of Mathematics of Warsaw) Title: Completely positive maps in zero-error quantum information theoryThomas Vidick (Caltech) - Public lectureDan Voiculescu (UC Berkeley) Title: The Macaev operator norm, entropy and supramenability.Abstract: On the (p,1) Lorentz scale of normed ideals of compact operators, the Macaev ideal is the end at infinity. From a perturbation point of view the Macaev ideal is related to entropy, while finite p is related to Hausdorff dimension p . For discrete groups, connections to supramenability have appeared, via the regular representation. Also properties of commutants mod the Macaev ideal and of associated exotic coronas will be discussed.Ignacio Villanueva (Madrid) Title: Random quantum correlations are generically non classicalAbstract: Non-locality is certified by the existence of quantum bipartite correlations which are non-explainable by any classical model. Once we know the existence of these, we are faced with the question of how generic is non-locality among quantum correlations. In this talk I will explain recent results where we show that, under quite general assumptions on the considered distribution, a random correlation which lies on the border of the quantum set is with high probability outside the classical set. Moreover, we provide a Bell inequality certifying this fact. From the mathematical point of view, our results require precise estimates of tensor norms or random matrices. Andreas Winter (Barcelona) - TBA If you are interested in participating, please register on the IHP website.