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Celebrating the Choi-Jamiołkowski Isomorphism

Africa/Johannesburg
Description

The Choi-Jamio lkowski Isomorphism is a remarkable result in the field of open quantum systems and quantum information theory establishing the correspon-
dence between linear maps in operator algebras and bipartite operators in the cor-responding Hilbert spaces. Nearly five decades ago it was established that positive maps correspond to block-positive operators [1] and completely positive maps cor-respond to positive operators [2]. The workshop aims to celebrate this fundamental result and is devoted to new frontiers in the research in open quantum systems,entanglement and quantum information theory.
[1] A. Jamio lkowski, “Linear transformations which preserve trace and positive
semidefiniteness of operators”, Rep. Math. Phys. 3, 275-278 (1972).
[2] M.-D. Choi, “Completely positive linear maps on complex matrices”, Linear
Algebra Appl. 10, 285-290 (1975).

  • Wednesday, 1 March
    • 1
      Opening Address : Karol Zyczkowski (Krakow)
    • Session Chair: Karol Zyczkowski
    • 2
      Guest of Honor : Prof. Andrzej Jamiolkowski (Toru ́n)
    • 3
      Higher-order quantum processes and quantum causal structures

      One of the most profound insights of the Choi-Jamio lkowski isomorphism is that
      quantum processes can be treated as quantum states. Following this idea, it is
      natural to consider a kind of super-processes that transform quantum processes into
      quantum processes, in a similar way as ordinary processes transform quantum states
      into quantum states. This construction can be iterated recursively, generating an
      infinite hierarchy of processes of increasingly higher orders. Physically, this hierarchy
      corresponds to an extension of the framework of quantum circuits, including the
      ordinary acyclic circuits considered in quantum computing, as well as a new type
      of quantum circuits with cycles. In this talk I will present the main notions in the
      study of higher order quantum processes, discussing their application to quantum
      information and their connection with the study of causal structure in quantum
      mechanics.

      Speaker: Giulio Chiribella (QICI Quantum Information and Computation Initiative, Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong)
    • 4
      The Choi-Jamio lkowski Isomorphism: when Maths meets Physics

      I discuss the interplay between Physics and Mathematics, from a personal perspective, in the light of the celebrated Choi-Jamio lkowski Isomorphism. I argue that it is desirable that a physical law, when expressed in terms of a differential equation, should admit any initial conditions. Choi-Jamio lkowski isomorphism, complete positivity, correlations and entanglement form a facet pattern that defines the correct formulation of the dynamics of open quantum systems. A formulation that requires coordination and teamwork if all its ingredients are to be kept in perfect alignment.

      Speaker: Saverio Pascazio (Dipartimento di Fisica, Universit`a di Bari, Italy)
    • 16:55
      COFFE BREAK
    • 5
      Generalizations of the Choi matrix

      We first study generalizations of Choi matrices for linear maps of the n×n matrices
      into themselves. Then we generalize this to certain maps of Von Neumann algebras
      into themselves.

      Speaker: Erling Størmer (Department of Mathematics, University of Oslo, 0316 Oslo, Norway)
    • 6
      A few puzzles in open quantum dynamics

      Despite its long history, research in open quantum dynamics still provides unexpected facets. We shall discuss some of them that are connected with entanglement generation and super-activation of back-flow of information by means of tensor products of dynamical maps.

      Speaker: Fabio Benatti (Dipartimento di Fisica, Universit`a degli Studi di Trieste,)
    • 7
      The Heisenberg groups and the dimensions of Hilbert spaces

      Finite Heisenberg groups have a certain universal status. In every finite dimensional
      Hilbert space there is at least one Heisenberg group that acts irreducibly in this
      dimension, and in no other. I will describe some dimension dependent structures
      that arise in this way, and some connections between seemingly different dimensions
      that arise from them.

      Speaker: Ingemar Bengtsson (Stockholms Universitet, Fysikum)
    • 18:55
      END OF DAY 1
  • Thursday, 2 March
    • 8
      A generic quantum Wielandt’s inequality

      In this talk, I will provide a generic version of quantum Wielandt’s inequality, which
      gives an optimal upper bound on the minimal length such that products of that
      length of n-dimensional matrices in a generating system span the whole matrix
      algebra with probability one. I will show that this length generically is of order
      Θ(log n), as opposed to the general case, in which the best bound to the date is
      O(n2log n). We will discuss the implications of this result as a new bound on the
      primitivity index of a random quantum channel, as well as to show that almost any
      translation-invariant (with periodic boundary conditions) matrix product state with
      length of order Ω(log n) is the unique ground state of a local Hamiltonian. Finally,
      we will comment on the possibility of extending these results to Lie algebras. This
      is based on joint work with Yifan Jia.

      Speaker: Angela Capel Cuevas (Fachbereich Mathematik, Universit ̈at T ̈ubingen, 72076 T ̈ubingen, Germany)
    • Session Chair: Francesco Petruccione
    • 9
      The Operational Choi-Jamio lkowski Isomorphism

      I use an operational formulation of the Choi-Jamio lkowski isomorphism to explore
      an approach to quantum mechanics in which the state is not the fundamental ob-
      ject. I first situate this project in the context of generalized probabilistic theories
      and argue that this framework may be understood as a means of drawing conclusions
      about the intratheoretic causal structure of quantum mechanics which are independent of any specific ontological picture. I then give an operational formulation of the Choi-Jamio lkowski isomorphism and show that, in an operational theory whichexhibits this isomorphism, several features of the theory which are usually regarded as properties of the quantum state can be derived from constraints on non-local correlations. This demonstrates that there is no need to postulate states to be the bearers of these properties, since they can be understood as consequences of a fundamental equivalence between multipartite and temporal correlations.

      Speaker: Emily Adlam (The Rotman Institute of Philosophy,)
    • 10
      Non-decomposable positive maps from tensor products

      Understanding how positivity of maps behaves under tensor products is linked to
      several open problems in quantum information theory. In my talk, I will present
      some recent results on how non-decomposable positive maps can arise from tensor
      products and tensor powers of decomposable maps. The Choi-Jamio lkowski isomorphism is an indispensable tool in this line of research.

      Speaker: Alexander M ̈uller-Hermes (Department of Mathematics, University of Oslo, 0316 Oslo, Norway)
    • 16:45
      COFFE BREAK
    • 11
      Emergent entanglement structures and self-similarity in quantum spin chains

      We introduce an experimentally accessible network representation for many-body
      quantum states based on entanglement between all pairs of its constituents. We illustrate the power of this representation by applying it to a paradigmatic spin chain
      model, the XX model, and showing that it brings to light new phenomena. The
      analysis of these entanglement networks reveals that the gradual establishment of
      quasi-long range order is accompanied by a symmetry regarding single-spin concurrence distributions, as well as by instabilities in the network topology. Moreover,
      we identify the existence of emergent entanglement structures, spatially localised communities enforced by the global symmetry of the system that can be revealed
      by model-agnostic community detection algorithms. The network representation
      further unveils the existence of structural classes and a cyclic self-similarity in the
      state, which we conjecture to be intimately linked to the community structure. Our
      results demonstrate that the use of tools and concepts from complex network theory
      enables the discovery, understanding, and description of new physical phenomena
      even in models studied for decades.

      Speaker: Sabrina Maniscalco (QTF Centre of Excellence, Department of Physics, University of Helsinki, Finland)
    • Session Chair: Vinayak Jagadish
    • 12
      Positive Maps and Entanglement in Real Hilbert Spaces

      Partially motivated by recent research in quantum physics, we take a closer look
      at the similarities and differences between the study of positive maps, separability,
      and entanglement in the real and complex case. It is possible for real matrices to
      be entangled as operators on a real Hilbert space and yet separable when regarded
      as acting on a complex space. These two distinct theories of entanglement in the
      real case correspond to two different theories of entanglement breaking maps in the
      real case. Finally, we see what these differences have to say about real versions
      of the PPT-squared conjecture. Based on joint research with G. Chiribella, K.R.
      Davidson, and M. Rahaman.

      Speaker: Vern Paulsen (Institute for Quantum Computing and Department of Pure Mathematics, University of Waterloo, Waterloo, Waterloo, ON, Canada N2L 3G1)
    • 13
      Guest of Honor : Prof. Man-Duen Choi (Toronto)
    • 14
      Closing Remarks: Francesco Petruccione (Stellenbosch)