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Civil War Book Review

MARK OVIAN SYSTEMS OF TRANSITION EXPECTATIONS

Volkmar Liebscher, Michael Skeide

Published Jan 1, 2009• 0 citations
Publisher

Abstract

Abstract. We propose a definition of markovian systems of transition expec-tation as a generalization of Liebscher’s continuous time version of Accardi’s quantum Markov chains and we show in a reconstruction theorem that the transition expectations may be recovered with the help product systems of Hilbert modules and units for them in the sense of Bhat and Skeide. 1.

DOI
10.31390/cosa.3.2.01

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Abstract. We propose a definition of markovian systems of transition expec-tation as a generalization of Liebscher’s continuous time version of Accardi’s quantum Markov chains and we show in a reconstruction theorem that the transition expectations may be recovered with the help product systems of Hilbert modules and units for them in the sense of Bhat and Skeide. 1.
Starting from the notion of algebraic hidden processes introduced in Ref. 9 , we propose a definition of quantum hidden Markov process which extends the notion of quantum Markov chain (QMC) in the same sense in which classical hidden Markov processes (HMPs) extend classical Markov chains. We give several structure theorems for this new family of quantum states which shares with the QMCs the unique property that the finite dimensional joint expectations are explicitly given without these states being linearly equivalent to product states (as the Gaussian ones). The special feature of this new class of quantum states is best exemplified by looking at their restrictions to diagonal sub-algebras: we prove that they lead to a new family of classical stochastic processes that generalize in a non-trivial way the classical HMPs.
A framework for finite-dimensional quantum Markov chains on Hilbert spaces is introduced. Quantum Markov chains generalize both classical Markov chains with possibly hidden states and existing models of quantum walks on finite graphs. Quantum Markov chains are based on Markov operations that may be applied to quantum systems and include quantum measurements, for example. It is proved that quantum Markov chains are asymptotically stationary and hence possess ergodic and entropic properties. With a quantum Markov chain one may associate a quantum Markov process, which is a stochastic process in the classical sense. Generalized Markov chains allow a representation with respect to a generalized Markov source model with definite (but possibly hidden) states relative to which observables give rise to classical stochastic processes. It is demonstrated that this model allows for observables to violate Bell’s inequality.
Entangled Markov chains, which can be recognized as quantum Markov chains in the sense of (1∣(1|(1∣ , were inti∨oducedinti^{\vee}oducedinti∨oduced by Accardi and Fidaleo in order to extend the notion of classical random walk to quaritum systems. In (3), using our entanglement criterion for pure states (16) which is based on the notion of the quantum mutual information, we proved that the vector states defining the EMC's on infinite tensor products of matrix algebra ∣∣genericallyI†||generically^{I\dagger}∣∣genericallyI† are entangled.
Abstract. Quantum dynamics, both reversible (i.e., closed quantum sys-tems) and irreversible (i.e., open quantum systems), gives rise to product systems of Hilbert spaces or, more generally, of Hilbert modules. When we consider reversible dynamics that dilates an irreversible dynamics, then the product system of the latter is equal to the product system of the former (or is contained in a unique way). Whenever the dynamics is on a proper subalgebra of the algebra of all bounded operators on a Hilbert space, in par-ticular, when the open system is classical (commutative) it is indispensable that we use Hilbert modules. The product system of a reversible dynamics is intimately related to a filtration of subalgebras that are independent in a state or conditionally in-dependent in a conditional expectation of the reversible system. This has been illustrated in many concrete dilations that have been obtained with the help of quantum stochastic calculus. Here the underlying Fock space or mod-ule determines the sort of quantum independence underlying the reversible system. The mini-workshop brought together experts from quantum dynamics, product systems and quantum independence who have contributed to the general theory or who have studied intriguing examples. As the implications of the tight relationship between product systems and independence had so far been largely neglected, we expect from our mini-workshop a strong innovative impulse to this field.