**ONLINE WORKSHOP**

# Celebrating the Choi-Jamiołkowski Isomorphism

**March 1-2, 2023**

The Choi-Jamiołkowski Isomorphism is a remarkable result in the field of open quantum systems and quantum information theory establishing the correspondence between linear maps in operator algebras and bipartite operators in the corresponding Hilbert spaces. Nearly five decades ago it was established that positive maps correspond to block-positive operators [1] and completely positive maps correspond 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. Jamiołkowski,** “Linear transformations which preserve trace and positive semidefinite operators,” Rep. Math. Phys. **3**, 275–278 (1972). https://doi.org/10.1016/0034-4877(72)90011-0

[2] **M.-D. Choi**, “Completely positive linear maps on complex matrices,” Linear Algebra Appl. **10**, 285–290 (1975). https://doi.org/10.1016/0024-3795(75)90075-0

**Guests of Honour**

- Prof. Man-Duen Choi (Toronto)
- Prof. Andrzej Jamiołkowski (Toruń)

### Speakers

- Emily Adlam (London)
- Fabio Benatti (Trieste)
- Ingemar Bengtsson (Stockholm)
- Ángela Capel Cuevas (Tübingen)
- Giulio Chiribella (Hong Kong)
- Sabrina Maniscalco (Helsinki)
- Alexander Müller-Hermes (Oslo)
- Saverio Pascazio (Bari)
- Vern Paulsen (Waterloo)
- Erling Størmer (Oslo)

### Programme

**Download the programme**

##### Wednesday, 1 March 2023 (CET: GMT+2)

*Session Chair: Karol Życzkowski*

**15.00-15.10**

Karol Życzkowski (Kraków)

Opening Address

**15.10-15.40**

Guest of Honor

Andrzej Jamio lkowski (Torun)

**15.45-16.15**

Giulio Chiribella (Hong Kong)

Higher-order quantum processes and quantum causal structures

**16.20-16.50**

Saverio Pascazio (Bari)

The Choi-Jamio lkowski Isomorphism: when Maths meets Physics

**16.55-17.15****COFFEE BREAK**

*Session Chair: Dariusz Chruściński *

**17.15-17.45**

Erling Størmer (Oslo)

Generalizations of the Choi matrix

**17.50-18.20**

Fabio Benatti (Trieste)

A few puzzles in open quantum dynamics

**18.25-18.55**

Ingemar Bengtsson (Stockholm)

The Heisenberg groups and the dimensions of Hilbert spaces

##### Thursday, 2 March 2023 (CET: GMT+2)

*Session Chair: Francesco Petruccione*

**15.00-15.30**

Ángela Capel Cuevas (Tübingen)

A generic quantum Wielandt’s inequality

**15.35-16.05**

Emily Adlam (London)

The Operational Choi-Jamio lkowski Isomorphism

**16.10-16.40**

Alexander Müller-Hermes (Oslo)

Non-decomposable positive maps from tensor products

**16.45-17.00****COFFEE BREAK**

*Session Chair: Vinayak Jagadish*

**17.00-17.30**

Sabrina Maniscalco (Helsinki)

Emergent entanglement structures and self-similarity in quantum spin chains

**17.35-18.05**

Vern Paulsen (Waterloo)

Positive Maps and Entanglement in Real Hilbert Spaces

**18.10-18.40**

Guest of Honor

Prof. Man-Duen Choi (Toronto)

**18.45-19.00**

Francesco Petruccione (Stellenbosch)

Closing Remarks

### Abstracts

**Download the Book of Abstracts**

##### The Operational Choi-Jamio lkowski Isomorphism - Emily Adlam (2 Mar 15.35-16.05)

*The Rotman Institute of Philosophy, 1151 Richmond Street, London**eadlam90@gmail.com*

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 object. 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 specifc ontological picture. I then give an operational formulation of the Choi-Jamio lkowski isomorphism and show that, in an operational theory which exhibits 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.

##### A few puzzles in open quantum dynamics - Fabio Benatti (1 Mar 17.50-18.20)

*Dipartimento di Fisica, Università degli Studi di Trieste, Trieste, Italy**Fabio.Benatti@ts.infn.it*

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 (1 Mar 18.25-18.55)

*Stockholms Universitet, Fysikum**ibeng@fysik.su.se*

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.

##### Higher-order quantum processes and quantum causal structures - Giulio Chiribella (1 Mar 15.45-16.15)

*QICI Quantum Information and Computation Initiative, Department of Computer **Science, The University of Hong Kong, Pokfulam Road, Hong Kong**giulio@cs.hku.hk*

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.

##### A generic quantum Wielandt's inequality - Ángela Capel Cuevas (2 Mar 15.00-15.30)

*Fachbereich Mathematik, Universität Tübingen, 72076 Tübingen, Germany**angela.capel@uni-tuebingen.de*

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(n2 log 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.

##### Emergent entanglement structures and self-similarity in quantum spin chains - Sabrina Maniscalco (2 Mar 17.00-17.30)

*QTF Centre of Excellence, Department of Physics, University of Helsinki, Finland**sabrina.maniscalco@helsinki.fi*

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.

##### Non-decomposable positive maps from tensor products - Alexander Müller-Hermes (2 Mar 16.10-16.40)

*Department of Mathematics, University of Oslo, 0316 Oslo, Norway**muellerh@math.uio.no*

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.

##### The Choi-Jamio lkowski Isomorphism: when Maths meets Physics - Saverio Pascazio (1 Mar 16.20-16.50)

*Dipartimento di Fisica, Università di Bari, Italy**INFN, Sezione di Bari, Italy**saverio.pascazio@ba.infn.it*

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.

##### Positive Maps and Entanglement in Real Hilbert Spaces - Vern Paulsen (1 Mar 17.35-18.05)

*Institute for Quantum Computing and Department of Pure Mathematics, University of **Waterloo, ON, Canada N2L 3G1**vpaulsen@uwaterloo.ca*

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.

##### Generalizations of the Choi matrix - Erling Størmer (1 Mar 17.15-17.45)

*Department of Mathematics, University of Oslo, 0316 Oslo, Norway**erlings@math.uio.no*

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

**Organisers**

- Dariusz Chruściński (Nicolaus Copernicus University, Poland)
- Vinayak Jagadish (Jagiellonian University, Poland)
- Francesco Petruccione (Stellenbosch University, South Africa)
- Karol Życzkowski (Jagiellonian University, Poland)