Universal Dynamics and Renormalization in Many-Body-Localized Systems

Quantum mechanics, information theory, and relativity theory are the basic foundations of theoretical physics. The acquisition of information from a quantum system is the interface of classical and quantum physics. Essential tools for its description are Kraus matrices and positive operator valued measures (POVMs). Special relativity imposes severe restrictions on the transfer of information between distant systems. Quantum entropy is not a Lorentz covariant concept. Lorentz transformations of reduced density matrices for entangled systems may not be completely positive maps. Quantum field theory, which is necessary for a consistent description of interactions, implies a fundamental trade-off between detector reliability and localizability. General relativity produces new, counterintuitive effects, in particular when black holes (or more generally, event horizons) are involved. Most of the current concepts in quantum information theory may then require a reassessment.

There is no doubt that Bell’s theorem [1] is a fundamental result for our understanding of quantum physics and its relation with classical physics. Before Bell, the possibility that an intuitive classical model could exist with the same predictive power as quantum physics was valid and, in a sense, justified in view of the arguments by Einstein, Podosky and Rosen (EPR) on the incompleteness of quantum physics [2]. After Bell’s work, a classical model for quantum physics is still possible but, as discussed below, requires breaking some very natural assumptions that, in a way, make it as counter-intuitive as quantum physics. In the last decade, our understanding of Bell’s theorem, for instance of the assumptions required for its derivation and its implications, has significantly improved using concepts and ideas borrowed from quantum information theory. At the same time, concepts from foundations of quantum physics have opened new approaches to quantum information applications, especially in the so-called device-independent scenario. The purpose of this text is to provide an overview over this new research direction merging quantum foundations and information theory, with an emphasis on the motivations and some of the obtained results. Our text, however, should not be understood as a review paper, but more as a rather personal selection of results in the field, unavoidably biased to some of our works. The structure of the essay is as follows: we start by presenting the assumptions required in the derivation of Bell’s theorem and its implications. We move on and show how ideas from Bell’s arguments can be used for quantum information purposes: we introduce the device-independent approach to quantum information theory and argue that it can be interpreted as a form of Bell-type quantum information theory. Then, we reverse this direction and show how ideas from information theory help us to understand quantum physics.

Quantum mechanics and information theory are among the most important scientific discoveries of the last century. Although these two areas initially developed separately, it has emerged that they are in fact intimately related. In this review the author shows how quantum information theory extends traditional information theory by exploring the limits imposed by quantum, rather than classical, mechanics on information storage and transmission. The derivation of many key results differentiates this review from the usual presentation in that they are shown to follow logically from one crucial property of relative entropy. Within the review, optimal bounds on the enhanced speed that quantum computers can achieve over their classical counterparts are outlined using information-theoretic arguments. In addition, important implications of quantum information theory for thermodynamics and quantum measurement are intermittently discussed. A number of simple examples and derivations, including quantum superdense coding, quantum teleportation, and Deutsch's and Grover's algorithms, are also included.

Questioning of quantum information Guolin Wu The Center of Philosophy and Technology, School of Marxism South China University of Technology, Guangzhou, Guangdong, P. R. China. E-Mails: ssglwu@scut.edu.cn * The Center of Philosophy and Technology, School of Marxism. South China University of Technology, Guangzhou, Guangdong, P. R. China,510640; Tel.:0086-13660190516; Fax: 0086-2087114979 Accepted: 19 February 2015

Principles of quantum computation and information volume II

Data compression is a fundamental problem in quantum and classical information theory. A typical version of the problem is that the sender Alice receives a (classical or quantum) state from some known ensemble and needs to transmit them to the receiver Bob with average error below some specified bound. We consider the case in which the message can have a variable length and the goal is to minimize its expected length. For classical messages this problem has a well-known solution given by Huffman coding. In this scheme, the expected length of the message is equal to the Shannon entropy of the source (with a constant additive factor) and the scheme succeeds with zero error. This is a single-shot result which implies the asymptotic result, viz. Shannon's source coding theorem, by encoding each state sequentially. For the quantum case, the asymptotic compression rate is given by the von-Neumann entropy. However, we show that there is no one-shot scheme which is able to match this rate, even if interactive communication is allowed. This is a relatively rare case in quantum information theory when the cost of a quantum task is significantly different than the classical analogue. Our result has implications for direct sum theorems in quantum communication complexity and one-shot formulations of Quantum Reverse Shannon theorem.

In the past few years, there has been an increasingly active exchange of ideas and methods between the formerly rather disjunct fields of quantum information and many-body physics. This has been due, on the one hand, to the growing sophisti- cation of methods and the increasing complexity of problems treated in quantum information theory, and, on the other, to the recognition that a number of central issues in many-body quantum systems can fruitfully be approached from the quan- tum information point of view. Nowhere has this been more evident than in the context of the family of numerical methods that go under the rubric density-matrix renormalization group. In particular, the concept of entanglement and its definition, measurement, and manipulation lies at the heart of much of quantum information theory (1). The density-matrix renormalization group (DMRG) methods use proper- ties of the entanglement of a bipartite system to build up an accurate approximation to particular many-body wave functions. The cross-fertilization between the two fields has led to improvements in the understanding of interacting quantum systems in general and the DMRG method in particular, has led to new algorithms related to and generalizing the DMRG, and has opened up the possibility of studying many new physical problems, ones of interest both for quantum information theory and for understanding the behavior of strongly correlated quantum systems (2). In this line, we discuss some relevant concepts in quantum information theory, including the relation between the DMRG and data compression and entanglement. As an application, we will use the quantum information entropy calculated with the DMRG to study quantum phase transitions, in particular in the bilinear-biquadratic spin-one chain and in the frustrated spin-1/2 Heisenberg chain.

We present quantum mechanics (QM) as theory of special classical random signals. On one hand, this approach provides a possibility to go beyond conventional QM: to create a finer description of micro processes than given by the QM-formalism. In fact, we present a model with hidden variables of the wave-type. On the other hand, our approach establishes coupling between quantum and classical information theories. We recall that quantum information theory has already been used for description of the entropy of Gaussian input signals for noisy channels. The entropy of a classical random input was invented as the entropy of the quantum density operator corresponding to the covariance operator of the input process.1 In this paper, we proceed the other way around: we apply classical signal theory to create a measurement model which reproduces quantum probabilities.

Emergent phenomena of interacting quantum many-body systems arguably pose some of the greatest challenges to modern theoretical physics. Many of these systems are characterized by drastic changes in their properties as they are driven into a regime where quantum effects and interactions become relevant. Examples are the fractional quantum Hall effect and high-temperature superconductors, to name just two. New theoretical tools are necessary to study such systems, as even simple model Hamiltonians are not fully understood after decades of intense research. Numerical simulations are emerging as an indispensable tool alongside more traditional analytic approaches and have seen immense progress in recent years. In this thesis, we study tensor network states, a new class of numerical methods that have been developed in recent years drawing on insights from quantum information theory. Combining renormalization group methods with knowledge about the entanglement structure of ground states of quantum systems, tensor network states aim to efficiently describe ground states of strongly correlated quantum systems. Center stage is taken by the projected entangled-pair states. This ansatz for two-dimensional systems is discussed in detail and its capabilities are assessed by comparison to established methods and by exploring a frustrated system. We then develop a general formalism to exploit Abelian symmetries in a tensor network algorithm and demonstrate its application to projected entangled-pair states. A large part of this thesis is devoted to the discussion of a class of supersymmetric models for interacting lattice fermions. After introducing the concept of supersymmetry in quantum mechanics and an extensive overview of conformal field theory and its relation to numerical simulations, we study the critical theory of the supersymmetric model on the square ladder. We use two tensor network state algorithms, the density-matrix renormalization group and multi-scale entanglement renormalization along with exact diagonalization to explore this challenging system. In the final two chapters, we discuss two applications of tensor network states, namely simulations of the SU(3) Heisenberg model in two dimensions using two different tensor network state algorithms with the goal of illuminating the nature of its ground state, and the indistinguishability, a measure to detect phase transitions and classify wave functions which is particularly suitable for calculations based on tensor network states.

This work introduces a modification to the Heisenberg Uncertainty Principle (HUP) by incorporating quantum complexity, including potential nonlinear effects. Our theoretical framework extends the traditional HUP to consider the complexity of quantum states, offering a more nuanced understanding of measurement precision. By adding a complexity term to the uncertainty relation, we explore nonlinear modifications such as polynomial, exponential, and logarithmic functions. Rigorous mathematical derivations demonstrate the consistency of the modified principle with classical quantum mechanics and quantum information theory. We investigate the implications of this modified HUP for various aspects of quantum mechanics, including quantum metrology, quantum algorithms, quantum error correction, and quantum chaos. Additionally, we propose experimental protocols to test the validity of the modified HUP, evaluating their feasibility with current and near-term quantum technologies. This work highlights the importance of quantum complexity in quantum mechanics and provides a refined perspective on the interplay between complexity, entanglement, and uncertainty in quantum systems. The modified HUP has the potential to stimulate interdisciplinary research at the intersection of quantum physics, information theory, and complexity theory, with significant implications for the development of quantum technologies and the understanding of the quantum-to-classical transition.

The Bures-Wasserstein distance is a Riemannian distance on the space of positive definite Hermitian matrices and is given by: $d(\Sigma,T) = \left[\text{tr}(\Sigma) + \text{tr}(T) - 2 \text{tr} \left(\Sigma^{1/2}T\Sigma^{1/2}\right)^{1/2}\right]^{1/2}$. This distance function appears in the fields of optimal transport, quantum information, and optimisation theory. In this paper, the geometrical properties of this distance are studied using Riemannian submersions and quotient manifolds. The Riemannian metric and geodesics are derived on both the whole space and the subspace of trace-one matrices. In the first part of the paper a general framework is provided, including different representations of the tangent bundle for the SLD Fisher metric. The last part of the paper unifies up till now independent arguments and results from quantum information theory and optimal transport. The Bures-Wasserstein geometry is related to the Fubini-Study metric and the Wigner-Yanase information.

In his classical argument, Rao derives the Riemannian distance corresponding to the Fisher metric using a mapping between the space of positive measures and Euclidean space. He obtains the Hellinger distance on the full space of measures and the Fisher distance on the subset of probability measures. In order to highlight the interplay between Fisher theory and quantum information theory, we extend this construction to the space of positive-definite Hermitian matrices using Riemannian submersions and quotient manifolds. The analog of the Hellinger distance turns out to be the Bures–Wasserstein (BW) distance, a distance measure appearing in optimal transport, quantum information, and optimisation theory. First we present an existing derivation of the Riemannian metric and geodesics associated with this distance. Subsequently, we present a novel derivation of the Riemannian distance and geodesics for this metric on the subset of trace-one matrices, analogous to the Fisher distance for probability measures.

Statistical mechanical informatics (SMI) is an approach that applies physics to information science, in which many-body problems in information processing are tackled using statistical mechanics methods. In the last decade, the use of SMI has resulted in great advances in research into classical information processing, in particular, theories of information and communications, probabilistic inference and combinatorial optimization problems. It is expected that the success of SMI can be extended to quantum systems. The importance of many-body problems is also being recognized in quantum information theory (QIT), for which quantification of entanglement of bipartite systems has recently been almost completely established after considerable effort. SMI and QIT are sufficiently well developed that it is now appropriate to consider applying SMI to quantum systems and developing many-body theory in QIT. This combination of SMI and QIT is highly likely to contribute significantly to the development of both research fields. The International Workshop on Statistical-Mechanical Informatics has been organized in response to this situation. This workshop, held at Sendai International Conference Center, Sendai, Japan, 14–17 September 2008, and sponsored by the Grant-in-Aid for Scientific Research on Priority Areas `Deepening and Expansion of Statistical Mechanical Informatics (DEX-SMI)' (Head investigator: Yoshiyuki Kabashima, Tokyo Institute of Technology) (Project http://dex-smi.sp.dis.titech.ac.jp/DEX-SMI), was intended to provide leading researchers with strong interdisciplinary interests in QIT and SMI with the opportunity to engage in intensive discussions. The aim of the workshop was to expand SMI to quantum systems and QIT research on quantum (entangled) many-body systems, to discuss possible future directions, and to offer researchers the opportunity to exchange ideas that may lead to joint research initiatives. We would like to thank the contributors of the workshop as well as all the participants, who have enjoyed the workshop as well as their stay in Sendai, one of the most beautiful cities in Japan. This successful workshop will stimulate further development of the interdisciplinary research field of QIT and SMI. Masahito Hayashi, Jun-ichi Inoue, Yoshiyuki Kabashima and Kazuyuki Tanaka Editors The IW-SMI 2008 Organizing CommitteeKazuyuki Tanaka, General Chair (Tohoku University) Yoshiyuki Kabashima, Vice-General Chair (Tokyo Institute of Technology) Jun-ichi Inoue, Program Chair (Hokkaido University) Masahito Hayashi, Pulications Chair (Tohoku University) Hidetoshi Nishimori (Tokyo Institute of Technology) Toshiyuki Tanaka (Kyoto University)

The development of quantum information theory over the last 20 years has produced a plethora of interesting new results and along with them a host of claims have been made by physicists and philosophers about how quantum information theory helps us understand the quantum world. When one examines such claims with any attention to detail, it is quite obvious that radically different and incompatible claims are being made about the quantum world. Christopher Timpson’s Quantum Information Theory and the Foundations of Quantum Mechanics provide a sober and thorough critical guide to these claims. Claims made about how quantum information theory can help us understand the quantum world are a motley collection, and so too are the chapters of Timpson’s book. That said, they can almost be sorted into two broad categories: those that aim to define quantum information and to understand the quantum world in terms of it, and those that aim to reconstruct quantum mechanics in terms of informationtheoretic axioms which will render the quantum world understandable to us. Timpson’s Chapter 6, on quantum computation is the exception. Oftentimes, different concepts of information are deployed in the literature associated with quantum information theory that are as much related as Michael Jackson the pop music icon and Michael Jackson the beer and whisky expert. It is strikingly ironic how much confusion has been generated by appeal to concepts of information. Timpson’s second chapter, ‘‘What is information?’’, is the antidote to such confusion. Timpson claims that there are really only two proper ways to talk about information in this context: the everyday way, which has a strong connection to knowledge, and the technical way, whose roots lie in Shannon (1948). With respect to both, Timpson’s most pressing point is to guard against a view that

Coding technology is used in several information processing tasks. In particular, when noise during transmission disturbs communications, coding technology is employed to protect the information. However, there are two types of coding technology: coding in classical information theory and coding in quantum information theory. Although the physical media used to transmit information ultimately obey quantum mechanics, we need to choose the type of coding depending on the kind of information device, classical or quantum, that is being used. In both branches of information theory, there are many elegant theoretical results under the ideal assumption that an infinitely large system is available. In a realistic situation, we need to account for finite size effects. The present paper reviews finite size effects in classical and quantum information theory with respect to various topics, including applied aspects.