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Reviews in Mathematical Physics

VARIATIONAL PRINCIPLE AND THE DYNAMICAL ENTROPY OF SPACE TRANSLATION

Hajime Moriya

Published Nov 1, 1999• 3 citations
Publisher

Abstract

The dynamical entropy of space translations is used in the variational calculus of the pressure. It is shown that for quantum spin systems on a lattice of an arbitrary dimension, the pressure obtained in this way coincides with the usual one.
DOI
10.1142/s0129055x99000416

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Quantum Information is a new area of research which has been growing rapidly since the last decade. This topic is very close to potential applications to the so called Quantum Computer. In our point of view it makes sense to develop a more "dynamical point of view" of this theory. We want to consider the concepts of entropy and pressure for "stationary systems" acting on density matrices which generalize the usual ones in Ergodic Theory (in the sense of the Thermodynamic Formalism of R. Bowen, Y. Sinai and D. Ruelle). We consider the operator L\mathcal{L}L acting on density matrices ρ∈MNρ\in \mathcal{M}_Nρ∈MN​ over a finite NNN-dimensional complex Hilbert space L(ρ):=∑i=1ktr(WiρWi∗)ViρVi∗,\mathcal{L}(ρ):=\sum_{i=1}^k tr(W_iρW_i^*)V_iρV_i^*,L(ρ):=∑i=1k​tr(Wi​ρWi∗​)Vi​ρVi∗​, where WiW_iWi​ and ViV_iVi​, i=1,2,...ki=1,2,... ki=1,2,...k are operators in this Hilbert space. L\mathcal{L}L is not a linear operator. In some sense this operator is a version of an Iterated Function System (IFS). Namely, the Vi(.)Vi∗=:Fi(.)V_i (.) V_i^*=:F_i(.)Vi​(.)Vi∗​=:Fi​(.), i=1,2,...,ki=1,2,...,ki=1,2,...,k, play the role of the inverse branches (acting on the configuration space of density matrices ρρρ) and the WiW_iWi​ play the role of the weights one can consider on the IFS. We also analyze the discrete Wigner function. We suppose that for all ρρρ we have that ∑i=1ktr(WiρWi∗)=1\sum_{i=1}^k tr(W_iρW_i^*)=1∑i=1k​tr(Wi​ρWi∗​)=1. A family W:={Wi}i=1,...,kW:=\{W_i\}_{i=1,..., k}W:={Wi​}i=1,...,k​ determines a Quantum Iterated Function System (QIFS) FW\mathcal{F}_{W}FW​, FW={MN,Fi,Wi}i=1,...,k.\mathcal{F}_W=\{\mathcal{M}_N,F_i,W_i\}_{i=1,..., k}.FW​={MN​,Fi​,Wi​}i=1,...,k​.
We study dynamical conservation laws and the propagation of thermodynamic structure in translation-invariant lattice quantum spin systems. Using Lieb-Robinson bounds [CMP 28 (1972), 251-257] in arbitrary dimension, and Araki's bounds [CMP 14 (1969), 120-157] in dimension one, we prove conservation of the specific entropy, specific energy, and specific relative entropy under suitable assumptions on the interactions. The transported local Hamiltonians admit surface energies that vanish per unit volume in the van Hove limit, and local observables remain a core for the transported generators at every finite time. Weak Gibbsianity is preserved for sufficiently small times in arbitrary dimension and for all times in dimension one. These results provide the analytical foundation for the structural theory of approach to equilibrium developed in a companion paper, and establish partial results toward the conjectures formulated there.
This paper is a contribution to the general theory of quantum spin systems. We deal with a general theory in the sense that no concrete interaction shows up but an arbitrary relatively short-range interaction is chosen. It is well known that the mean entropy plays an important role in the thermodynamics of quantum spin systems: it is one of the ingredients of the Lanford–Ruelle–Robinson variational principle. We show that in the background of the existence of the mean entropy, there is an operator convergence which resembles the McMillan theorem from information theory. Asymptotic equipartition property and several entropy densities are investigated, in particular, the relation of the Gibbs condition to relative entropies. Although this paper is not intended to be a review, we try to give an overview of the theory of quantum Gibbs states in the mathematical sense. The paper is organized as follows. Section 1 provides an introduction and further sections contain our new results. Each section is closed with a short discussion on sources and references.