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GBMS Theme 1: Boole and Beyond in Quantum Information Theory

Date: 17 - 20 Aug 2015

Venue: Western Gateway Building, UCC

 

Organizers

  • Matthias Christandl (Copenhagen)
  • Benoit Collins (Kyoto)
  • Claus Köstler (UCC)

Speakers Include (*= Video-Streamed Talks)

  • Collin Bleak (St. Andrews)
  • Daniel Burgarth (Aberystwyth)
  • Motohisa Fukuda (TU Munich / Yamagata)
  • Robin Hillier (Lancaster)
  • Jukka Kiukas (Aberystwyth)
  • Robert Koenig (TU Munich)
  • Hans Maassen (Nijmegen)
  • Shane Mansfield (Oxford)
  • Michael Mc Gettrick (Galway)
  • Mauro Paternostro* (Queen's, Belfast)
  • Andreas Ruschhaupt (UCC)
  • Volkher Scholz (ETH Zurich)
  • David Sutter (ETH Zurich)
  • Ivan G Todorov* (Queen's, Belfast)
  • Michael M. Wolf (TU Munich)
  • John Wright (Carnegie Mellon)

The programme is here.

Title & Abstract of Talks

  • Collin Bleak, On the semi-decidability of the periodicity problem for elements of various groups

    We prove that both the Higman-Thompson group 2V and the rational group R_n of Grigorchuk, Nekrashevych, and Suschanskii have semi-decidable periodicity problems. That is, there are algorithms which can confirm, given an element of one of these groups, that the element has finite order. However, there is no algorithm which can confirm in finite time whether a general element of one of these groups has infinite order. The argument is based on studying the achievable dynamical systems under the partial action of 2V on a Cantor space. Joint with Jim Belk.

  • Daniel Burgarth, Quantum Computing in Plato's Cave

    We show that mere observation of a quantum system can turn its dynamics from a very simple one into a universal quantum computation. This effect, which occurs if the system is regularly observed at short time intervals, can be rephrased as a modern version of Plato's Cave allegory. More precisely, while in the original version of the myth, the reality perceived within the Cave is described by the projected shadows of some more fundamental dynamics which is intrinsically more complex, we found that in the quantum world the situation changes drastically as the "projected" reality perceived through sequences of measurements can be more complex than the one that originated it. After discussing examples we go on to show that this effect is generally to be expected: almost any quantum dynamics will become universal once "observed" as outlined above. Conversely, we show that any complex quantum dynamics can be "purified" into a simpler one in larger dimensions. This is joint work with Paolo Facchi, Vittorio Giovannetti, Hiromichi Nakazato, Saverio Pascazio and Kazuya Yuasa.

  • Motohisa Fukuda, Additivity violation and tensor powers of quantum channels

    Additivity violation of minimum output entropy of quantum channels implies that entangled inputs can increase the classical capacity of certain quantum channels. In this talk, first we learn about additivity violation, its consequences and open problems. Second, we briefly go over proofs of additivity violation pointing out important techniques: random construction, Dvoretzky's theorem, etc. Third, we go on to similar problems for tensor powers of quantum channels. To investigate behaviors of such things is important to know more about operational quantities, for example classical capacity of quantum channels.

  • Robin Hillier, A limit theorem for dynamical decoupling and intrinsic decoherence

    Dynamical decoupling is a key tool in quantum information theory designed to counterfeit environment-induced decoherence. We investigate it from the point of view of analysis, and obtain interesting new descriptions and estimates. Moreover, we prove that dynamical decoupling would not work for (potential) intrinsic/internal decoherence of a closed quantum system and thereby propose a method of partially identifying the latter.

  • Jukka Kiukas, Information geometry and local asymptotic normality for the estimation of open quantum dynamics

    Input-output formalism is a well-known framework for describing continual monitoring of a Markovian open quantum system via measurements made on its environment (typically a quantised radiation field); mathematically, the environmental noise is described in terms of quantum stochastic Wiener processes on the field Fock space. We consider the problem of identifying and estimating unknown dynamical parameters (Hamiltonian and the quantum jump operators) from the output field state. For this purpose, we first use quantum Ito calculus to derive an information geometric structure on the set of parameters, arising from the quantum Fisher information of the output state. The geometry comes with an associated CCR-algebra, and we then show that local estimation reduces asymptotically (with long observation times) to a Gaussian estimation problem on that CCR-algebra.

  • Robert Koenig, Protected gates for topological quantum field theories

    We give restrictions on locality-preserving unitary automorphisms U, which are protected gates, for 2-dimensional topologically ordered systems. For generic anyon models, we show that such unitaries only generate a finite group, and hence do not provide universality. For non-abelian models, we find that such automorphisms are very limited: for example, there is no non-trivial gate for Fibonacci anyons. More generally, systems with computationally universal braiding have no such gates. For Ising anyons, protected gates are elements of the Pauli group.These results are derived by relating such automorphisms to symmetries of the underlying anyon model: protected gates realize automorphisms of the Verlinde algebra. We additionally use the compatibility with basis changes to characterize the logical action This is joint work with M. Beverland, O. Buerschaper, F. Pastawski, J. Preskill and S. Sijher.

  • Hans Maassen, The ergodic decomposition of measurement records

    We consider a finite but otherwise general measurement on a finite quantum system, repeated infinitely often. We prove that observation of the asymptotic or `macroscopic' behavior of the measurement record amounts to a von Neumann measurement on the system. In the course of time the type of asymptotic behavior can be viewed as establishing itself, or as revealing itself. This phenomenon was known in the `non-demolition' case and has been named by Fröhlich et al. `the emergence of facts in quantum mechanics'.

  • Shane Mansfield, The reality of the quantum state: a stronger psi-ontology theorem

    The Pusey-Barrett-Rudolph no-go theorem provides an argument for the reality of the quantum state based on certain assumptions, most of which are common to the familiar no-go theorems of Bell, Kochen & Specker, etc. The exception is their assumption of preparation independence, which has been subject to a number of criticisms. We propose a much weaker, physically motivated notion of independence, which merely requires well-defined marginal statistics for joint preparation procedures. This is a minimum requirement for maintaining a reasonable notion of subsystem, and prohibits the possibility of super-luminal causal influences in the preparation process. Under the weaker condition, it is shown that the argument of PBR becomes invalid. We propose an experiment involving randomly sampled preparations that recovers an approximation of the result, which becomes exact in the limit as the sample space of preparations becomes infinite, thereby proving a stronger theorem asserting the reality of the quantum state. The analysis employs a finite version of the de Finetti theorem for conditional probabilities due to Christandl and Toner. Unlike that of PBR, the result holds even in the presence of non-local correlations in the global ontic state.

  • Michael Mc Gettrick, Quantum walks

    An overview will be provided of our recent results in the analysis of certain models of quantum random walks in 1 and 2 dimensions. In particular, we are interested in the asymptotic behaviour of the probability distribution and how this differs from that for classical random walks. We will present results for alternating walks, walks with history dependence ("memory"), and lazy walks.

  • Mauro Paternostro*, Distributing entanglement without any entanglement

    Entanglement is, allegedly, 'the' resource that makes quantum computing more advantageous than its classical counterpart. Some interesting, experimentally motivated architectures for quantum computation are based on distributed networks of computational nodes connected by entangled communication channels. In this context, distributing entanglement between two such nodes is paramount. A way of doing it is by sending a mediator that shuttles between two nodes. However, one would expect that, by doing this, the shuttle will eventually become entangled with the nodes themselves. This was proven not to be necessarily the case. However, the reasons for this remained obscure until recently. In this talk I will demonstrate that quantum discord, a weaker form of quantum correlations, is the responsible for such 'distribution of entanglement without entanglement' to occur and illustrate a recent experiment proving such intriguing relation.

  • Andreas Ruschhaupt, Quantum control with shortcuts to adiabaticity

    Shortcuts to adiabaticity are mostly-analytical derived schemes for the fast and robust control of quantum systems. In this talk, we review these "shortcut" schemes. We especially concentrate on the optimization of the stability of shortcut schemes versus different sources of errors. We apply these tools to design a trap trajectory for shuttling a single ion in a harmonic trap. We develop trap trajectories with low sensitivity on spring-constant noise and position noise fluctuations for several noise spectra. In addition, we derive shortcuts to adiabatic population inversion of the internal state of an atom. Again, we optimize the stability of the shortcut schemes versus different sources of errors like noise or systematic errors.

  • Volkher Scholz, Quantum Bilinear Optimization

    We study optimization programs given by a bilinear form over noncommutative variables subject to linear inequalities. Problems of this form include the entangled value of two-prover games, entanglement assisted coding for classical channels and quantum-proof randomness extractors. We introduce an asymptotically converging hierarchy of efficiently computable semidefinite programming (SDP) relaxations for this quantum optimization. This allows us to give upper bounds on the quantum advantage for all of these problems. Compared to previous work of Pironio, Navascu\'es and Ac\'in, our hierarchy has additional constraints. By means of examples, we illustrate the importance of these new constraints both in practice and for analytical properties. We also discuss implications of Connes’ embedding conjecture to our framework. Joint work with Mario Berta and Omar Fawzi, arXiv:1506.08810

  • David Sutter, Relative entropy, recovery maps, and approximate quantum Markov chains

    The Shannon and von Neumann entropies quantify the uncertainty in a system. They are operationally motivated by natural information processing tasks such as compression, channel coding or randomness extraction. A mathematical consequence of the postulates of quantum physics are several entropy inequalities such as the strong subadditivity of quantum entropy and the monotonicity of quantum relative entropy under physical processes. A series of recent works showed that these inequalities can be strengthened in the context of recoverability, i.e., by considering the question of how well a physical process can be reversed. This also provides an operational definition of approximate quantum Markov chains. In this talk, I will give an overview about the recent works on recoverability, present some new results, and discuss open problems. Based on joint work with Omar Fawzi, Renato Renner (arXiv:1504.07251) and Marco Tomamichel, Aram Harrow (arXiv:1507.00303).

  • Ivan G Todorov*, Quantum chromatic numbers of graphs

    The chromatic number of a graph is a well-known and widely used parameter, with far reaching applications both within and outside Graph Theory. It is defined as the smallest number of colours that are needed in order to colour the vertices of the graph in such a way that adjacent vertices receive different colours. A quantum version of the chromatic number was defined in 2007 by P. Cameron, A. Montanaro, M. Newman, S. Severini and A. Winter, utilising an entangled state, shared between two players, and it was demonstrated that this new chromatic number can be strictly smaller than the classical one. In this talk, based on a joint work with S. Severini, D. Stahlke, V. Paulsen and A. Winter, I will describe how ideas from operator algebra theory can be used to define other useful quantum versions of the classical chromatic number. At the heart of this approach lies the connection with quantum correlations between systems sharing entangled states, and their link to tensor theory of operator algebras. Bounds for the quantum chromatic numbers will be displayed, and their connections to Tsirelson's Problem and to Connes' Embedding Problem will be discussed.

  • Michael M. Wolf, (Un-)decidable problems in quantum theory

    In the talk I will review recent results on the (un-)decidability of problems in quantum many-body physics and quantum information theory. In both fields there is a natural integer limit that opens the door to undecidability of some of the central properties: the thermodynamic limit in quantum many-body theory and the large block-size limit in information theory. I will try to illuminate the thin line between computable and uncomputable and to illustrate the physical consequences of unprovable properties.

  • John Wright, Efficient quantum tomography

    In the quantum state tomography problem, one wishes to estimate an unknown d-dimensional quantum state given as few copies as possible. In this talk, we describe new work showing that d^2 copies are sufficient, essentially matching known lower bounds. In addition, we show that the top k eigenvalues of an unknown mixed state can be learned using only k^2 copies, independent of the dimension d.