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Workshop on "Operator algebras and Quantum Information Theory"

The conference will be held at the Institut Henri Poincaré, 11-15 September 2017   (link to the schedule)

Organizers
Confirmed Speakers
  1. Michael Brannan (College station) slides and Youtube video
  2. Runyao Duan (University of Technology, Sydney)
  3. Edward Effros (Los Angeles)
  4. Douglas Farenick (University of Toronto) slides
  5. Li Gao (UIUC) Youtube video
  6. Aurelian Gheondea (Bilkent University, Ankara) slides and Youtube video
  7. Rolf Gohm (Aberystwyth) slides and Youtube video
  8. Aram Harrow (MIT)
  9. Fumio Hiai (Tohoku University) slides
  10. Seung-Hyeok Kye (Seoul National University) slides
  11. Nicholas LaRacuente (UIUC)
  12. Adam Majewski (Gdansk) Youtube video
  13. Alexander Müller-Hermes (University of Copenhagen) Youtube video
  14. Magdalena Musat (University of Copenhagen) slides
  15. Yoshiko Ogata (Tokyo)
  16. Hiroyuki Osaka (Ritsumeikan) slides
  17. Carlos Palazuelos (Instituto de Ciencias Matematicas, Madrid)
  18. Vern Paulsen (Waterloo) slides and Youtube video
  19. David Pérez-García (Instituto de Ciencias Matematicas, Madrid)
  20. Thomas Vidick (Caltech)
  21. Dan Voiculescu (UC Berkeley)
  22. Ignacio Villanueva (Madrid) slides and Youtube video
  23. Reinhard Werner (Hannover) Youtube video
  24. Andreas Winter (Barcelona) Youtube video

Here is a link to a playlist with all the Youtube video

Abstracts

  1. Michael 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).


  2. 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.

  3. 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 irreversibility
    Abstract: 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 1606.09421

  4. Edward Effros (UC Los Angeles) 
    Title:
    Some remarkable gems and persistent difficulties in quantized functional analysis (QFA)
    Abstract:
    QFA was a direct outgrowth of the Heisenberg and von Neumann notions of quantized random variables. Thus, one replaces n-tuples of reals by collections of (generally unbounded) real functions on a locally compact space by unbounded self-adjoint operators. In turn, completely bounded mappings play the role of classical operators. Using ingenious replacements for such notions as the central limit theorem, it is possible to find non-commutative (i.e.. “quantum”) analogues of much of functional analysis. These developments are particularly striking when considers the tensor products of operator spaces.

  5. Douglas Farenick (University of Toronto)
    Title: Isometric and Contractive of Channels Relative to the Bures Metric
    Abstract:
    In a unital C*-algebra A possessing a faithful trace, the density operators in A are those positive elements of unit trace, and the set of all density elements forms a convex metric space with respect to the Bures metric. A linear map on A is a channel if it maps density operators to density operators. In this lecture, which is based on joint work with Samuel Jaques and Mizanur Rahaman, I will discuss the structure and properties of channels that are, respectively, isometric and contractive maps of the density space.

  6. Li Gao (UIUC) 
    Title:Operator Algebras Aspects of Quantum Teleportation and Superdense Coding
    Abstract: 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.

  7. Aurelian Gheondea (Bilkent University, Ankara)
    Title:
    Symmetry versus Conservation Laws in Dynamical Quantum Systems: A Unifying Approach through Propagation of Fixed Points
    Abstract:
    We unify recent Noether type theorems on the equivalence of symmetries with conservation laws for dynamical systems of Markov processes, of quantum operations, and of quantum stochastic maps, by means of some abstract results on propagation of fixed points for completely positive maps on C^*-algebras. We extend most of the existing results with characterisations in terms of dual infinitesimal generators of the corresponding strongly continuous one-parameter semigroups. By means of an ergodic theorem for dynamical systems of completely positive maps on von Neumann algebras we show the consistency of the condition on the standard deviation for dynamical systems of quantum operations, and hence of quantum stochastic maps as well, in case the underlying Hilbert space is infinite dimensional.

  8. Rolf Gohm (Aberystwyth
    Title:
    Asymptotic Completeness and Controllability of Open Quantum Systems
    Abstract:
    Repeated interactions of an open quantum system with copies of another system  can be interpreted as a quantum Markov process. The notion of asymptotic completeness from scattering theory is closely related to the preparability of states and hence to the controllability of the open system. We explain the main results and examples (micromaser) in [R. Gohm, F. Haag, B. Kümmerer, Universal Preparability and Asymptotic Completeness, CMP 352(1) (2017), 59-94], discuss the operator algebraic techniques developed for the proofs and reflect about their potential.

  9. Aram Harrow (MIT)
    Title:
    Local Hamiltonians Whose Ground States are Hard to Approximate
    Abstract:
    Ground states of local Hamiltonians are typically highly entangled: any quantum circuit that generates them (even approximately) must be sufficiently deep to allow coupling (entanglement) between any pair of qubits. Until now this property was not known to be "robust" - the marginals of such states to a subset of the qubits containing all but a small constant fraction of them may be only locally entangled, and hence approximable by shallow quantum circuits. In this work we construct a family of commuting 16-local Hamiltonians for which any 1-10^{-9} fraction of qubits of any ground state must be highly entangled.
    This provides evidence that quantum entanglement is not very fragile, and perhaps our intuition about its instability is an artifact of considering local Hamiltonians which are not only local but spatially local. This result is related to conjectures in quantum coding theory, topological order and complexity theory which I will discuss.

    Our Hamiltonian is based on applying the hypergraph product by Tillich and Zemor to a classical locally testable code. A key tool in our proof is a new lower bound on the vertex expansion of the output of low-depth quantum circuits, which may be of independent interest.

    Based on [arXiv:1510.02082, FOCS 2017] which is joint work with Lior Eldar.

  10. Fumio Hiai (Tohoku University)
    Title: Different quantum divergences in general von Neumann algebras
    Abstract: 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.

  11. Seung-Hyeok Kye (Seoul National University, Seoul, Korea)
    Title: Positivity of multi-linear maps and applications to quantum information theory
    Abstract: 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.

  12. Nicholas LaRacuente (UIUC)
    Title: Non-commutative L_p Spaces and Asymmetry Measures

    Abstract: 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.

  13. 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.

  14. Alexander Müller-Hermes (University of Copenhagen) 
    Title: Tensoring Positive maps
    Abstract: 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 bound
    entanglement.

  15. Magdalena Musat (University of Copenhagen)
    Title: Quantum correlations, tensor norms, and factorizable quantum channels
    Abstract: In joint work with Haagerup, we established in 2015 a reformulation of the Connes embedding problem in terms of an asymptotic property of quantum channels possessing a certain factorizability property, introduced by Anantharaman-Delaroche. I will discuss new results concerning the class of channels which exactly factor through matrix algebras, and a number of open problems. I will also discuss ongoing work, joint with Brown and Rørdam, related to the remarkable recent breakthrough of Slofstra.

  16. Yoshiko Ogata (University of Tokyo)
    Title: A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization
    Abstract:We introduce a class of gapped Hamiltonians on quantum spin chains, which allows asymmetric edge ground states, and investigate its properties.

  17. Hiroyuki Osaka (Ritsumeikan)
    Title: Operator means and application to generalized entropies
    Abstract: 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,

    Unfortu
    nately, 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.

  18. Carlos Palazuelos (Instituto de Ciencias Matematicas, Madrid) 
    Title: Classical vs Quantum communication in XOR games
    Abstract: 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.

  19. 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.

  20. David Pérez-García (Instituto de Ciencias Matematicas, Madrid)
    Title:
    Matrix Product Operators and the algebras that they generate
    Abstract: Matrix Product Operators were introduced by Fannes, Nachtergaele and Werner in their seminal 1992 paper. In this talk, I will show how their study can give new insights in different subjects within quantum information, such as channels capacities, quantum cellular automata or topological order in quantum spin systems.

  21. Thomas Vidick (Caltech) - Public lecture

  22. Dan 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.

  23. Ignacio Villanueva (Madrid)
    Title: Random quantum correlations are generically non classical
    Abstract: 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.

  24. Reinhard F. Werner (Hannover)
    Title: Alice and Bob and von Neumann
    Abstract: Alice and Bob stand for the separated labs scenario, a standard setting for many quantum informational tasks, where two labs are not connected by quantum interactions, but are capable of arbitrary local operations. When “local” is formalized by two commuting von Neumann algebras, as is natural in algebraic quantum field theory, there are characteristic deviations from the case of two (usually even finite) type I algebras, which is almost exclusively studied in quantum information theory. In particular, there is a new kind of connection: It may be impossible to perform “arbitrary local operations” on the subsystems. The prototype of a forbidden operation is complete depolarization, i.e., destroying one subsystem, and replacing it with a new one in a fixed state. This operation usually has no normal extension to the whole system. It would result in a product state, and there are often no normal ones. Thus the notion of separable states loses its relevance. On the other hand, a von Neumann-algebraic system may be used as an (idealized) infinite resource for entanglement, for which the hyperfinite type II_1 factor is the canonical prototype, or even allow for exact embezzling, for which, similarly,  the hyperfinite type III_1 factor is the canonical prototype.

  25. Andreas Winter (Barcelona)
    Title: Monogamy and faithfulness of quantum entanglement
    Abstract:
    Everybody knows that quantum entanglement is monogamous,
    according to Charles Bennett's wonderful metaphor. However, it has proved surprisingly difficult to capture this intuition in quantitative terms using entanglement measures. We will discuss the few examples of measures where it works out, and show why another, equally intuitive, property, faithfulness of the measure, gets in the way of any attempt at formulating a universal monogamy relation.





If you are interested in participating, please register on the IHP website.