Table of contents

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 Committee

Kazuyuki 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)

Adiabatic quantum annealing is a paradigm of analog quantum computation, where a given computational job is converted to the task of finding the global minimum of some classical potential energy function and the search for the global potential minimum is performed by employing external kinetic quantum fluctuations and subsequent slow reduction (annealing) of them. In this method, the entire potential energy landscape (PEL) may be accessed simultaneously through a delocalized wave-function, in contrast to a classical search, where the searcher has to visit different points in the landscape (i.e., individual classical configurations) sequentially. Thus in such searches, the role of the potential energy might be significantly different in the two cases. Here we discuss this in the context of searching of a single isolated hole (potential minimum) in a golf-course type gradient free PEL. We show, that the quantum particle would be able to locate the hole faster if the hole is deeper, while the classical particle of course would have no scope to exploit the depth of the hole. We also discuss the effect of the underlying quantum phase transition on the adiabatic dynamics.

Simulated annealing and quantum annealing are algorithms for combinatorial optimization problems. The former brings solutions using thermal fluctuations and through classical dynamics, while the latter does using quantum fluctuations and through quantum dynamics. In this paper, dynamics of these two algorithms are compared by means of the Kibble-Zurek argument, employing a one-dimensional random Ising model as an exercise for algorithms. We reveal that quantum annealing reduces residual errors faster than simulated annealing with decreasing annealing rate. The result implies the advantage of quantum annealing over simulated annealing.

We study the performance of quantum annealing for systems with ground-state degeneracy by directly solving the Schrödinger equation for small systems and quantum Monte Carlo simulations for larger systems. The results indicate that quantum annealing may not be well suited to identify all degenerate ground-state configurations, although the value of the ground-state energy is often effciently estimated. The strengths and weaknesses of quantum annealing for problems with degenerate ground states are discussed in comparison with classical simulated annealing.

The adiabatic quantum evolution of the Lipkin-Meshkov-Glick (LMG) model across its quantum critical point is studied. The dynamics is realized by linearly switching the transverse field from an initial large value towards zero. We concentrate our attention on the residual energy after the quench in order to estimate the level of diabaticity of the evolution. We discuss a Landau-Zener approximation of the finite size LMG model, that is successful in reproducing the behavior of the residual energy as function of the transition rate in most of the regimes considered. The system proposed is a paradigm of infinite-range interaction or high-dimensional models.

Recently, explicit real time dynamics has been studied in various systems. These quantum mechanical dynamics could provide new recipes in information processing. We study quantum dynamics under time dependent external fields, and explore how to control the quantum state, and also how to bring the state into a target state. Here, we investigate a pure quantum mechanical dynamics, dynamics in quantum Monte Carlo simulation and also in quantum master equation. For the control magnetic states, operators which do not commute with magnetization are important. We study case of the transverse Ising model, in which we compare natures of thermal and quantum fluctuations. We also study the cases of the Dzyaloshinsky-Moriya interaction, where we find a peculiar energy level structure. Moreover we study the case of itinerant magnetic state, where we study the change from the Mott insulator to the Nagaoka ferromagnetic state. Effects of dissipation are also discussed.

This article reviews recent research on dissipative macroscopic quantum bosonic and fermionic systems. These studies address the following issues. (i) The existence of static and dynamic phase transitions; order of the critical lines and type of phases. (ii) The dynamics of systems that are unable to reach equilibrium with their (equilibrium) environment due to their intrinsic slow dynamics or driven by an external drive in the form of a time-dependent magnetic field, coupling to source and drain leads, etc. (iii) The development of an effective temperature that controls the large-scale dynamics and a two-time dependent decoherence phenomenon. (iv) The role played by the environment in the behaviour of the systems.

Measurements on a bipartite system AB are classified into ones that are freely implementable with only classical communication between A and B (LOCC measurements), and the others that require consumption of entanglement if they are to be implemented with classical communication. When we notice that measurements on a bipartite system can also be used to create entanglement, we have another natural classification: measurements that are capable of creating entanglement, and the others that have no ability of producing entanglement (separable measurements). Interestingly, there exists a separable measurement that is not an LOCC measurement, namely, a measurement that requires entanglement to implement, but is not capable of creating entanglement at all. Such an example was found for a pair of three-level systems, in conjunction with a discrimination task of an intricate set of eight or nine states. Here we show an example of such a measurement on a pair of two-level systems (qubits), which arises in a discrimination task of just two states that look quite simple and have little intricacy. Such an example suggests that this kind of irreversibility is not only shown by a limited set of cleverly constructed measurements, but is also exhibited by a much larger class of measurements than we had expected.

Unlocking of a correlation refers to the unexpected phenomenon that a small amount of communication increases that correlation (as a function of the state of the distributed system) by a disproportionate amount. Locking refers to the suppression of the correlation prior to the communication. The notion was subsequently extended to abrupt changes in a correlation due to the manipulation (in particular, the addition or removal) of a small subsystem. In this proceeding, we review the basic ideas and summarize the results known so far.

We demonstrate that entanglement shared by two or more parties can be asymptotically reversibly interconverted when one considers the set of operations which asymptotically cannot generate entanglement. In this scenario we find that the entanglement of every quantum state is uniquely characterized by a single quantity: the regularized relative entropy of entanglement. The main technical tool is a generalization of quantum Stein's Lemma, which gives optimal discrimination rates in quantum hypothesis testing, to the case in which the alternative hypothesis might vary over sets of correlated states. We analyze the connection of our approach to recent rigorous formulations of the second law of thermodynamics.

We define and analyse the concept of entanglement production during the evolution of a general quantum mechanical dissipative system. While it is important to minimise entropy production in order to achieve thermodynamical efficiency, maximising the rate of change of entanglement is important in quantum information processing. Quantitative relations are obtained between entropy and entanglement productions, under specific assumptions detailed in the text. We apply these to the processes of dephasing and decay of correlations between two initially entangled qubits. Both the Master equation treatment as well as the higher Hilbert space analysis are presented. Our formalism is very general and contains as special cases many reported individual instance of entanglement dynamics, such as, for example, the recently discovered notion of the sudden death of entanglement.

Complementarity is a very old concept in quantum mechanics, however the rigorous definition is not so old. Complementarity of orthogonal bases can be formulated in terms of maximal Abelian algebras and this may lead to avoid commutativity of the subalgebras. In some sense this means that quantum information is treated instead of classical (measurement) information. The subject is to extend to the quantum case some features from the classical case. This includes construction of complementary subalgebras. The Bell basis has also some relation. Several open questions are discussed.

We propose algorithms of the path-ingetral-based quantum Monte Carlo simulation, which is otherwise prohibitively slow. While the basic idea is the loop-cluster update, there are some important 'tricks' that are vital to make the simulation practical. In the present paper, we show two such techniques and their successful applications to the two-dimensional SU(N) Heisenberg model and the three-dimensional Bose Hubbard model. In the former, we obtain a new type of the valence-bond-solid state for the two-boson representation, while in the latter we equillibrate a system of which the size is comparable to a typical experiment of optical lattices.

We consider a long-range Ising antiferromagnet (LRIAF) put in a transverse field. Applying quantum Monte Carlo method, we study the variation of order parameter (spin correlation in Trotter time direction), susceptibility and average energy of the system for various values of the transverse field at different temperatures. The antiferromagnetic order is seen to get immediately broken as soon as the thermal or quantum fluctuations are added. We also discuss the phase diagram for the Sherrington-Kirkpatrick (SK) model with the same LRIAF bias, also in presence of a transverse field. We find that while the antiferromagnetic order is immediately broken as one adds an infinitesimal transverse field or thermal fluctuation to the system, an infinitesimal SK spin glass disorder is enough to induce a stable glass order in the antiferromagnet. This glass order eventually gets destroyed as the thermal or quantum fluctuations increased beyond their threshold values and the transition to para phase occurs. Indications of this novel phase transition are discussed. Because of the presence of full frustration, this surrogate property of the LRIAF for incubation of stable spin glass phase in it (induced by addition of a small disorder) should enable eventually the study of classical and quantum spin glass phases by using some perturbation theory with respect to the disorder.

We perform a numerical study on the concurrence of the ground state Heisenberg XXX with next-nearest-neighbor interaction with open and periodic boundary condition.

We discuss quantum spin glasses with finite connectivity by presenting an extension of the cavity method used in studies of classical spin glasses.

We prove sufficient conditions for Topological Quantum Order at zero and finite temperatures. The crux of the proof hinges on the existence of low-dimensional Gauge-Like Symmetries, thus providing a unifying framework based on a symmetry principle. All known examples of Topological Quantum Order display Gauge-Like Symmetries. Other systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin exchange and Jahn-Teller effects in transition metal orbital compounds, short-range frustrated Klein spin models, and p+ip superconducting arrays. We analyze the physical consequences of Gauge-Like Symmetries (including topological terms and charges) and, most importantly, show the insufficiency of the energy spectrum, (recently defined) entanglement entropy, maximal string correlators, and fractionalization in establishing Topological Quantum Order. Duality mappings illustrate that not withstanding the existence of spectral gaps, thermal fluctuations may impose restrictions on suggested topological quantum computing schemes. Our results allow us to go beyond standard topological field theories and engineer new systems with Topological Quantum Order.

We have constructed universal codes for quantum lossless source coding and classical-quantum channel coding. In this construction, we essentially employ group representation theory. In order to treat quantum lossless source coding, universal approximation of multi-copy states is discussed in terms of the quantum relative entropy.

We explore the relation between the techniques of statistical mechanics and information theory for assessing the performance of channel coding. We base our study on a framework developed by Gallager in IEEE Trans. Inform. TheoryIT-11, 3 (1965), where the minimum decoding error probability is upper-bounded by an average of a generalized Chernoff's bound over a code ensemble. We show that the resulting bound in the framework can be directly assessed by the replica method, which has been developed in statistical mechanics of disordered systems, whereas in Gallager's original methodology further replacement by another bound utilizing Jensen's inequality is necessary. Our approach associates a seemingly ad hoc restriction with respect to an adjustable parameter for optimizing the bound with a phase transition between two replica symmetric solutions, and can improve the accuracy of performance assessments of general code ensembles including low density parity check codes, although its mathematical justification is still open.

We numerically examine a quantum version of TAP (Thouless-Anderson-Palmer)-like mean-field algorithm for the problem of error-correcting codes. For a class of the so-called Sourlas error-correcting codes, we check the usefulness to retrieve the original bit-sequence (message) with a finite length. The decoding dynamics is derived explicitly and we evaluate the average-case performance through the bit-error rate (BER).

We discuss a classical reinterpretation of quantum-mechanics-based analysis of classical Markov chains with detailed balance, that is based on the quantum-classical correspondence. The classical reinterpretation is then used to demonstrate that it successfully reproduces a sufficient condition for cooling schedule in classical simulated annealing, which has the inverse-logarithmic scaling.

Density matrices are a central tool in quantum physics, but it is also used in machine learning. A positive definite matrix called kernel matrix is used to represent the similarities between examples. Positive definiteness assures that the examples are embedded in an Euclidean space. When a positive definite matrix is learned from data, one has to design an update rule that maintains the positive definiteness. Our update rule, called matrix exponentiated gradient update, is motivated by the quantum relative entropy. Notably, the relative entropy is an instance of Bregman divergences, which are asymmetric distance measures specifying theoretical properties of machine learning algorithms. Using the calculus commonly used in quantum physics, we prove an upperbound of the generalization error of online learning.

In this paper the variational Bayesian approximation for partially observed continuous time stochastic processes is studied. We derive an EM-like algorithm and describe its implementation. The variational Expectation step is explicitly solved using the method of conditional moment generating functions and stochastic partial differential equations. The numerical experiments demonstrate that the variational Bayesian estimate is more robust than the EM algorithm.

The mathematical structures of loopy belief propagation are reviewed for graphical models in probabilistic information processing in the stand point of cluster variation method. An extension of adaptive TAP approaches is given by introducing a generalized scheme of the cluster variation method. Moreover the practical message update rules in loopy belief propagation are summarized also for quantum systems. It is suggested that the loopy belief propagation can be reformulated for quantum electron systems by using density matrices of ideal quantum lattice gas system.