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This book collects independent contributions on current developments in quantum information theory, a very interdisciplinary field at the intersection of physics, computer science and mathematics. Making intense use of the most advanced concepts from each discipline, the authors give in each contribution pedagogical introductions to the main concepts underlying their present research and present a personal perspective on some of the most exciting open problems.

Keeping this diverse audience in mind, special efforts have been made to ensure that the basic concepts underlying quantum information are covered in an understandable way for mathematical readers, who can find there new open challenges for their research. At the same time, the volume can also be of use to physicists wishing to learn advanced mathematical tools, especially of differential and algebraic geometric nature.

### Chapter 1. Introduction

Abstract
The development of quantum mechanics has been one of the greatest scientific achievements of the early twentieth century. In spite of its remarkable success in explaining and predicting an amazing number of properties of our physical world, its interpretation has raised strong controversies among a wide community of scientists and philosophers. One of the hottest points of discussion is the meaning of the so-called quantum entanglement that, for systems of two or many particles, allows in particular the possibility for each particle of the system to be simultaneously located at different spatial positions. Entangled states display a special kind of correlations. Generally speaking, differently from the statistical correlations that are usually found in classical probability theory, quantum entanglement cannot be understood in terms of statistically distributed hidden variables and must involve the possibility for quantum systems of particles to be simultaneously in different single particle pure quantum states. Entangled states therefore present facets of the quantum worlds which are even more complicated than the famous example of a superposition of states in the so-called Schrödinger’s cat which is simultaneously classically dead and alive. The peculiar phenomenology of quantum mechanics goes far beyond this paradoxical case: in contrast to the usual chain rules of classical conditional probability, the probability for a physical event to occur in a quantum framework is computed by the interference of the complex-valued amplitudes corresponding to the different classical states. In dynamical processes, these classical positional states are described by paths that the system can follow during its evolution. This description of the physical world is commonly known as Feynman integral and implicitly requires that the system be simultaneously in different classical states at all intermediate times [1]. The mathematical counterpart of this picture is that quantum states of a composite system are described by a tensor product structure where each product entry represents a component of the system. In this picture, entanglement is encoded in quantum superpositions, that is linear combinations of completely decomposed tensors. In this sense, if the tensor product involves different states of a given component which are localized in far and causally separated spatial regions, a single component of the system may be simultaneously located in different places.
Edoardo Ballico, Alessandra Bernardi, Iacopo Carusotto, Sonia Mazzucchi, Valter Moretti

### Chapter 2. A Very Brief Introduction to Quantum Computing and Quantum Information Theory for Mathematicians

Abstract
This is a very brief introduction to quantum computing and quantum information theory, primarily aimed at geometers. Beyond basic definitions and examples, I emphasize aspects of interest to geometers, especially connections with asymptotic representation theory. Proofs can be found in standard references such as Kitaev et al. (Classical and quantum computation, vol. 47. American Mathematical Society, Providence, 2002) and Nielson and Chuang (Quantum computation and quantum information. Cambridge University Press, Cambridge, 2000) as well as Landsberg (Quantum computation and information: Notes for fall 2017 TAMU class, 2017).
Joseph M. Landsberg

### Chapter 3. Entanglement, CP-Maps and Quantum Communications

Abstract
In this chapter we review the employment of quantum entanglement as a resource for information processing and transmission. In particular we introduce and discuss the notion of completely positive maps operating on observable algebras of physical systems in order to have a model to construct communication channels based on quantum processes. Then we discuss advantages and limitations of entanglement-assisted quantum communication schemes like quantum teleportation and dense coding.
Davide Pastorello

### Chapter 4. Frontiers of Open Quantum System Dynamics

Abstract
We briefly examine recent developments in the field of open quantum system theory, devoted to the introduction of a satisfactory notion of memory for a quantum dynamics. In particular, we will consider a possible formalization of the notion of non-Markovian dynamics, as well as the construction of quantum evolution equations featuring a memory kernel. Connections will be draw to the corresponding notions in the framework of classical stochastic processes, thus pointing to the key differences between a quantum and classical formalization of the notion of memory effects.
Bassano Vacchini

### Chapter 5. Geometric Constructions over and for Quantum Information

Abstract
In this review paper I present two geometric constructions of distinguished nature, one is over the field of complex numbers $${\mathbb {C}}$$ and the other one is over the two elements field $$\mathbb {F}_2$$. Both constructions have been employed in the past 15 years to describe two quantum paradoxes or two resources of quantum information: entanglement of pure multipartite systems on one side and contextuality on the other. Both geometric constructions are linked to representation of semi-simple Lie groups/algebras. To emphasize this aspect one explains on one hand how well-known results in representation theory allows one to see all the classification of entanglement classes of various tripartite quantum systems (three qubits, three fermions, three bosonic qubits…) in a unified picture. On the other hand, one also shows how some weight diagrams of simple Lie groups are encapsulated in the geometry which deals with the commutation relations of the generalized N-Pauli group.
Frédéric Holweck

### Chapter 6. Hilbert Functions and Tensor Analysis

Abstract
We show how well known tools of algebraic geometry for the study of finite sets can be fruitfully applied to the study of Waring decompositions of symmetric tensors (forms). We mainly focus on the uniqueness of a given decomposition (the identifiability problem), and show how, in some cases, one can effectively determine the uniqueness even in some range in which the Kruskal’s criterion does not apply.
Luca Chiantini

### Chapter 7. Differential Geometry of Quantum States, Observables and Evolution

Abstract
The geometrical description of Quantum Mechanics is reviewed and proposed as an alternative picture to the standard ones. The basic notions of observables, states, evolution and composition of systems are analysed from this perspective, the relevant geometrical structures and their associated algebraic properties are highlighted, and the Qubit example is thoroughly discussed.
F. M. Ciaglia, A. Ibort, G. Marmo