**Quantum Channels and their Capacities **

**Alexander Holevo, Steklov Mathematical Institute, Moscow**

The notions of *channel *and* capacity* are central to the classical Shannon theory. “Quantum Shannon theory” denotes a subfield of quantum information science which uses operator analysis, convexity and matrix inequalities, asymptotic techniques such as large deviations and measure concentration to study mathematical models of quantum communication channels and their information-processing performance. From the mathematical point of view quantum channels are normalized completely positive maps of operator algebras, the analog of Markov maps in the noncommutative probability theory, while the capacities are related to certain norm-like quantities. In applications noisy quantum channels arise from irreversible evolutions of open quantum systems interacting with environment—a physical counterpart of a mathematical dilation theorem.

In the quantum case the notion of channel capacity splits into the whole spectrum of numerical information-processing characteristics depending on the kind of data transmitted (classical or quantum) as well as on the additional communication resources. An outstanding role here is played by quantum correlations — *entanglement* – inherent in tensor-product structure of composite quantum systems. These lectures present a survey of basic *coding theorems* providing analytical expressions for the capacities of quantum channels in terms of entropic quantities. Special attention is paid to infinite-dimensional channels, among which one distinguishes theoretically and practically important class of Bosonic Gaussian channels. We discuss the progress and problems in study of the structure and the entropic quantities of single- and multi-mode Gaussian channels, stressing the role played by gauge symmetry in obtaining explicit solutions.

**A Tutorial on Quantum Key Distribution**

**Robert König, Techniche Universität München**

We will cover basic principles underlying quantum key distribution. Starting with a discussion of authenticity and privacy, we will primarily focus on classical information-theoretic cryptography. Here two primitives play a fundamental role: these are privacy amplification (aka advantage distillation) and information reconciliation. We will discuss how to quantify the amount of key that can be generated in particular setups. Turning to quantum key distribution, we will consider the basic BB84 protocol and study different types of attacks. Moving to the quantum world necessitates reconsidering the notion of security and of secure keys, and requires a new analysis of privacy amplification/information reconciliation protocols. Putting all these building blocks together gives a basic outline of a security proof for standard QKD protocols.

**Quantum Information Processing with Continuous Variables**

**Seth Lloyd, MIT**

These lectures review quantum computation with continuous variables, and explore novel applications of continuous variable quantum information processing, such as enhanced sensing, quantum simulation, solution of differential equations, and quantum machine learning. After reviewing the basic quantum mechanics of continuous variables, the lectures will explore the computational power of linear, Gaussian, and non-Gaussian transformations. Continuous variable quantum error correcting codes will be presented. Technologies for continuous variable quantum information processing will be explored. Because of their analog nature, continuous variables are particularly well suited for quantum versions of machine learning, including deep quantum networks.

**Quantum Communication and Metrology in Many-Body Systems**

**Vittorio Giovannetti, Scuola Normale Superiore, Pisa**

The lectures focus on quantum information processings in the context of many-body quantum systems, with special attention to communication and metrology. In particular we shall discuss Spin-Networks Quantum Channel (SNQC) models: here at variance with the more conventional *flying-qubit *approach to quantum communication, the information exchange between the sender and receiver is assumed to be mediated by an extended medium formed by a network of interacting (spatially fixed) spins whose mutual interactions allow for the propagation of local perturbations. We shall review some special examples of SNQC and caracterize their communication efficiencies. We shall hence introduce the Lieb-Robinson bounds, and show how to use it to upper-bound the quantum information capacities of arbitary SNQC model by using a continuity argument that holds for such functionals. Next, after reviewing the notion of quantum Cramer-Rao bounds, we shall discuss quantum metrology applications for spin-network models. Finally an application of quantum machine learning procedures to quantum communication and estimantion theory will be presented.

**Raúl García-Patrón Sánchez, ****Université Libre de Bruxelles**

**Wednesday Lecture
Abstract: Review of CV-QKD**

In this lecture we will introduce the most basic protocols of continuous variables QKD that exploit squeezed and coherent states of light. We will present how to analyze their security and review the most recent development in the field.

**Friday Lecture
Abstract: Tutorial on Boson Sampling**

In this lecture we will present how the statistics of free-boson lead to permanents and discuss how this led to the well-known boson sampling quantum supremacy proposal. In the second part of the lecture we will present how experimental imperfections can be exploited to develop classical algorithm that can potentially simulate realistic boson sampling experiments.

**Distributed Quantum Sensing in a Continuous Variable Entangled Network**

**Johannes Borregaard, QMATH, University of Copenhagen**

Networking plays a ubiquitous role in quantum technology. It is an integral part of quantum communication and has significant potential for upscaling quantum computer technologies that are otherwise not scalable. Recently, it was realized that sensing of multiple spatially distributed parameters may also benefit from an entangled quantum network. Here we experimentally demonstrate how sensing of an averaged phase shift among four distributed nodes benefits from an entangled quantum network. Using a four-mode entangled continuous variable (CV) state, we demonstrate deterministic quantum phase sensing with a precision beyond what is attainable with separable probes. The techniques behind this result can have direct applications in a number of primitives ranging from biological imaging to quantum networks of atomic clocks.

**Minimal Energy Cost of Entanglement Extraction**

**Robert Jonsson, QMATH, University of Copenhagen**

We compute the minimal energy cost for extracting entanglement from the ground state of a bosonic or fermionic quadratic system. Specifically, we find the minimal energy increase ΔEmin in the system resulting from replacing an entangled pair of modes, sharing entanglement entropy ΔS, by a product state, and we show how to construct modes achieving this minimal energy cost. Thus, we obtain a protocol independent lower bound on the extraction of pure state entanglement from quadratic systems. Due to their generality, our results apply to a large range of physical systems, as we discuss with examples.

**Introduction to Quantum Gaussian Systems**

**Giacomo De Palma, QMATH, University of Copenhagen**

Quantum Gaussian systems provide the mathematical model for the electromagnetic radiation in the quantum regime. Quantum Gaussian channels are the physical operation on quantum Gaussian systems that model the propagation of electromagnetic signals through optical fibers, which are the main mean to distribute quantum states in quantum key distribution and in the forthcoming quantum internet. Quantum Gaussian states are the most relevant states of quantum Gaussian system, since they can be easily prepared experimentally and are the best codewords to communicate through quantum Gaussian channels.

This lecture will provide an introduction to quantum information with quantum Gaussian systems. The topics will include the canonical commutation relations and quantum Gaussian states, channels and measurements. Particular emphasis will be put on coherent, squeezed and thermal states, quantum Gaussian attenuator and amplifier channels and homodyne and heterodyne measurements.