Yoshiko Ogata

Propagation of Correlations in Quantum Lattice Systems

We provide a simple proof of the Lieb-Robinson bound and use it to prove the existence of the dynamics for interactions with polynomial decay. We then use our results to demonstrate that there is an upper bound on the rate at which correlations between observables with separated support can accumulate as a consequence of the dynamics.

QUANTUM HYPOTHESIS TESTING AND NON-EQUILIBRIUM STATISTICAL MECHANICS

We extend the mathematical theory of quantum hypothesis testing to the general W*-algebraic setting and explore its relation with recent developments in non-equilibrium quantum statistical mechanics. In particular, we relate the large deviation principle for the full counting statistics of entropy flow to quantum hypothesis testing of the arrow of time.

Diffusion of the magnetization profile in the XX model.

We investigate the magnetization profile in the intermediate time of diffusion by using the C*-algebraic method. We observe a transition from monotone profile to nonmonotone profile. This transition is purely thermal.

Variational principle for non-equilibrium steady states of the XX model

We show that non-equilibrium steady states of the one-dimensional exactly-solved XX model can be characterized by the variational principle of free energy of a long range interaction and that they cannot be a KMS state for any C*-dynamical system of the UHF algebra.

The Stability of the Non-Equilibrium Steady States

We show that the non-equilibrium steady state (NESS) of the free lattice Fermion model far from equilibrium is macroscopically unstable. The problem is translated to that of the spectral analysis of the Liouville Operator. We use the method of positive commutators to investigate it. We construct a positive commutator on the lattice Fermion system, whose dispersion relation is ω(k)=cosk−γ.

Local distinguishability of quantum states in infinite-dimensional systems

We investigate local distinguishability of quantum states by use of convex analysis of joint numerical range of operators on a Hilbert space. We show that any two orthogonal pure states are distinguishable by local operations and classical communications, even for infinite-dimensional systems. An estimate of the local discrimination probability is also given for some families of more than two pure states.

RUELLE-LANFORD FUNCTIONS AND LARGE DEVIATIONS FOR ASYMPTOTICALLY DECOUPLED QUANTUM SYSTEMS

We recover, expand, and unify quantum (and classical) large deviation results for lattice Gibbs states. The main new ingredient in this paper is a control on the overlap of spectral projections for non-commutative observables. Our proof of large deviations is based on Ruelle–Lanford functions [20, 34] which establishes the existence of a rate function directly by subadditivity arguments, as done in the classical case in [23, 32], instead of relying on Gartner–Ellis theorem, and cluster expansion or transfer operators as done in the quantum case in [21, 13, 27, 22, 16, 28]. We assume that the Gibbs states are asymptotically decoupled [23, 32], which controls the dependence of observables localized at different spatial locations. In the companion paper [29], we discuss the characterization of rate functions in terms of relative entropies.

NORMAL STATES OF TYPE III FACTORS

AbstractLet M be a factor of type III with separable predual and with normalstates ϕ 1 ,...,ϕ k ,ω with ω faithful. Let A be a finite dimensional C ∗ -subalgebra of M. Then it is shown that there is a unitary operator u∈ Msuch that ϕ i ◦ Adu = ω on A for i = 1,...,k. This follows from anembedding result of a finite dimensional C ∗ -algebra with a faithful stateinto M with finitely many given states. We also give similar embeddingresults of C ∗ -algebras and von Neumann algebras with faithful states intoM. Another similar result for a factor of type II 1 instead of type III holds. 1 Introduction Let Mbe a factor of type III with separable predual. Then two nonzero pro-jections eand fin Mare equivalent, i.e., there exists a partial isometry v∈ M ∗ Supported in part by the Global COE Program “The research and training center for newdevelopment in mathematics”, the Mitsubishi Foundation Research Grants and the Grants-in-Aid for Scientific Research, JSPS. † Supported in part by the Inoue Foundation and the Grants-in-Aid for Scientific Research,JSPS.

The Shannon-McMillan Theorem for AF C*-systems

We give a new proof of quantum Shannon-McMillan theorem, extending it to AF C*-systems. Our proof is based on the variational principle, instead of the classical Shannon-McMillan theorem.

Emptiness formation probability of XX-chain in diffusion process

Abstract We study the distribution of emptiness formation probability of XX-model in the diffusion process. There exits a Gaussian decay as well as an exponential decay. The Gaussian decay is caused by the existence of zero point in the Fermi distribution function. The correlation length for each point of scaling factor varies up to the initial condition, monotonically or non-monotonically.