Guest of Honor : Prof. Andrzej Jamiolkowski (Toru ́n)

Higher-order quantum processes and quantum causal structures

• Giulio Chiribella (QICI Quantum Information and Computation Initiative, Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong)

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.

The Choi-Jamio lkowski Isomorphism: when Maths meets Physics

• Saverio Pascazio (Dipartimento di Fisica, Universit`a di Bari, Italy)

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.

Generalizations of the Choi matrix

• Erling Størmer (Department of Mathematics, University of Oslo, 0316 Oslo, Norway)

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.

A few puzzles in open quantum dynamics

• Fabio Benatti (Dipartimento di Fisica, Universit`a degli Studi di Trieste,)

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.

The Heisenberg groups and the dimensions of Hilbert spaces

• Ingemar Bengtsson (Stockholms Universitet, Fysikum)

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.

The Operational Choi-Jamio lkowski Isomorphism

• Emily Adlam (The Rotman Institute of Philosophy,)

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.

Non-decomposable positive maps from tensor products

• Alexander M ̈uller-Hermes (Department of Mathematics, University of Oslo, 0316 Oslo, Norway)

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.

Positive Maps and Entanglement in Real Hilbert Spaces

• Vern Paulsen (Institute for Quantum Computing and Department of Pure Mathematics, University of Waterloo, Waterloo, Waterloo, ON, Canada N2L 3G1)

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.

Guest of Honor : Prof. Man-Duen Choi (Toronto)

Closing Remarks: Francesco Petruccione (Stellenbosch)