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Infinite Dimensional Analysis Quantum Probability and Related Topics

A Characterization of Pure Finitely Correlated States

Taku Matsui

Published Oct 1, 1998• 27 citations
Publisher

Abstract

We give a characterization of pure finitely correlated states (quantum Markov states) as zero energy states of UHF algebras.
DOI
10.1142/s0219025798000351

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Markov Semigroups on UHF C∗^\ast∗-algebras(White Noise Analysis and Quantum Probability)
The goal of our work is to characterize the landscape of the frustration-free quantum spin models over the Cayley graph of a finitely generated group G. This is achieved by establishing GGG-equivariant morphisms from the partially ordered space of frustration-free models to the partially ordered spaces 1) of hereditary C*-algebras of the underlying UHF algebra of quasi-local observables, 2) of open projections in its double dual, and 3) of subsets of pure state space. Our main result consists of an intrinsic characterization of the images of these morphisms, which captures the essence of frustration-freeness and enables us to extend the concept to generic AF-C*-algebras. Additionally, using well established facts about AF-C*-algebras, we prove density theorems and provide intrinsic and optimal characterizations of frustration-free ground states.
Any tripartite state which saturates the strong subadditivity relation for the quantum entropy is defined as the Markov state. A tripartite pure state describing an open system, its environment, and their purifying system is a pure Markov state if and only if the bipartite marginal state of the purifying system and environment is a product state. It has been shown that as long as the purification of the input system-environment state is a pure Markov state, the reduced dynamics of the open system can be described, on the support of the initial system state, by a quantum channel for every joint unitary evolution of the system-environment composite even in the presence of initial correlations. Entanglement, discord, and classical correlations of the initial system-environment states implied by the pure Markov states are analyzed and it has been shown that all these correlations are entirely specified by the entropy of environment. Some implications concerning perfect quantum error correction procedure and quantum Markovian dynamics are presented.