Quantum Information

Quantum Mechanics has been shown to provide a rigorous foundation for Statistical Mechanics. Concentration of measure, or typicality, is the main tool to construct a purely quantum derivation for the methods of Statistical Mechanics. From this point of view statistical ensembles are effective description for isolated quantum systems, since typically a random pure state of the system will have properties similar to those of the ensemble. Nevertheless, it is often argued that most of the states of the Hilbert space are not relevant for realistic systems. This talk will address this issue, presenting recent results on the emergence of typicality in the context of matrix product states, a set of physically and computationally relevant states associated to many-body Hamiltonians.

We find analytic models that can perfectly transfer, without state initialization or remote collaboration, arbitrary functions in two- and three-dimensional interacting bosonic and fermionic networks. This provides for the possible experimental implementation of state transfer through bosonic or fermionic atoms trapped in optical lattices. A significant finding is that the state of a spin qubit can be perfectly transferred through a fermionic system. Families of Hamiltonians are described that are related to the linear models and that enable the perfect transfer of arbitrary functions. Furthermore, we propose methods for eliminating certain types of errors

Topological order is a new kind of collective order which appears in two-dimensional quantum systems such as the fractional quantum Hall effect and brings about rather unusual particles: unlike bosons or fermions these anyons obey exotic statistics and can be exploited to perform quantum computation. Topological order also implies that quantum states at low energies exhibit a very subtle, yet intricate inner structure. Remarkably, both phenomena can be studied in relatively simple spin systems (like Kitaev's quantum double models and the ubiquitous toric code) which in fact capture the essential properties of entire topological phases of matter in many important cases. What is the relationship between these topological phases? Reviewing recent work I will explain how they arrange themselves in a landscape of dualities and hierarchies. In particular, I will focus on two aspects: first, a duality between electric and magnetic quasiparticles in generalized quantum double models and second, a hierarchy construction of quantum states which are related by the condensation of topological charges.

Quantum error correcting codes and topological quantum order (TQO) are inter-connected fields that study non-local correlations in highly entangled many-body quantum states. In this talk I will argue that each of these fields offers valuable techniques for solving problems posed in the other one. First, we will discuss the zero-temperature stability of TQO and derive simple conditions that guarantee stability of the spectral gap and the ground state degeneracy under generic local perturbations. These conditions thus can be regarded as a rigorous definition of TQO. Our results apply to any quantum spin Hamiltonian that can be written as a sum of geometrically local commuting projectors on a D-dimensional lattice. This large class of Hamiltonians includes Levin-Wen string-net models and Kitaev's quantum double models. Secondly, we derive upper bounds on the parameters of quantum codes with local check operators and discuss the implications for feasibility of a quantum self-correcting memory.

The counter-intuitive phenomena in quantum mechanics are often based on the counter-factual (or virtual) processes. The famous example is the Hardy paradox, which has been recently solved in two independent experiments. Also, the delayed choice experiment and one of quantum descriptions of the closed time like curves can be also examples of the counter-intuitive phenomena. The counter-factual processes can be characterized by the weak value initiated by Yakir Aharonov and his colleagues. In this talk, I will introduce the weak value from the probability theory and the connection to the counter-factual processes in these examples.

I'll describe a connection between uncertainty relations, information locking and low-distortion embeddings of L2 into L1. Exploiting this connection leads to the first explicit construction of entropic uncertainty relations for a number of measurements that is polylogarithmic in the dimension d while achieving an average measurement entropy of (1-e) log d for arbitrarily small e. From there, it is straightforward to obtain the first strong information locking scheme that is efficiently computable using a quantum computer. This locking scheme can be interpreted as a method for encrypting classical messages using a key of size much smaller than the message length. Other applications include efficient encodings for amortized quantum identification over classical channels and new string commitment protocols.

Quantum computers have emerged as the natural architecture to study the physics of strongly correlated many-body quantum systems, thus providing a major new impetus to the field of many-body quantum physics. While the method of choice for simulating classical many-body systems has long since been the ubiquitous Monte Carlo method, the formulation of a generalization of this method to the quantum regime has been impeded by the fundamental peculiarities of quantum mechanics, including, interference effects and the no-cloning theorem. We overcome those difficulties by constructing a quantum algorithm to sample from the Gibbs distribution of a quantum Hamiltonian at arbitrary temperatures, both for bosonic and fermionic systems. This is a further step in validating the quantum computer as a full quantum simulator, with a wealth of possible applications to quantum chemistry, condensed matter physics and high energy physics.

Even though the security of quantum key distribution has been rigorously proven, most practical schemes can be attacked and broken. These attacks make use of imperfections of the physical devices used for their implementation. Since current security proofs assume that the physical devices' exact and complete specification is known, they do not hold for this scenario. The goal of device-independent quantum key distribution is to show security without making any assumptions about the internal working of the devices. In this talk, I will first explain the assumptions 'traditional' security proofs make and why they are problematic. Then, I will discuss how the violation of Bell inequalities can be used to show security even when a large part of the physical devices is untrusted.

A system of spins with complicated interactions between them can have many possible configurations. Many configurations will be local minima of the energy, and to get from one local minimum to another requires changing the state of very many spins. A system like this is called a spin glass, and at low temperatures tends to get caught for very long times at a local minimum of energy, rather than reaching its true ground state. Indeed, in many cases, finding the ground state energy of a spin glass is a computationally hard problem, too hard to be solved on a classical computer or even a quantum computer in any reasonable amount of time. Which types of interactions give us computationally hard problems and spin glasses? I will survey what is known as we close in on finding the simplest complex spin systems.

We introduce a family of variational ansatz states for chains of anyons which optimally exploits the structure of the anyonic Hilbert space. This ansatz is the natural analog of the multi-scale entanglement renormalization ansatz for spin chains. In particular, it has the same interpretation as a coarse-graining procedure and is expected to accurately describe critical systems with algebraically decaying correlations. We numerically investigate the validity of this ansatz using the anyonic golden chain and its relatives as a testbed. This demonstrates the power of entanglement renormalization in a setting with non-abelian exchange statistics, extending previous work on qudits, bosons and fermions in two dimensions. This is joint work with Ersen Bilgin.

Over the last twenty years, quantum information and quantum computing have profoundly shaped our thinking about the basic concepts of quantum physics. But can these insights also shape the way we /teach/ quantum mechanics to undergraduate physics students? A recent adventure in textbook-writing suggests some strategies and dilemmas.

In this talk we will explore a "toy model" of quantum theory that is similar to actual quantum theory, but uses scalars drawn from a finite field. The set of possible states of a system is discrete and finite. Our theory does not have a quantitative notion of probability, but only makes the "modal" distinction between possible and impossible measurement results. Despite its very simple structure, our toy model nevertheless includes many of the key phenomena of actual quantum systems: interference, complementarity, entanglement, nonlocality, and the impossibility of cloning.