Weak Markov processes as linear systems

Luigi Accardi

Rolf Gohm

A noncommutative Fornasini-Marchesini system (a multi-variable version of a linear system) can be realized within a weak Markov process (a model for quantum evolution). For a discrete time parameter the resulting structure is worked out systematically and some quantum mechanical interpretations are given. We introduce subprocesses and quotient processes and then the notion of a $γ$-extension for processes which leads to a complete classification of all the ways in which processes can be built from subprocesses and quotient processes. We show that within a $γ$-extension we have a cascade of noncommutative Fornasini-Marchesini systems. We study observability in this setting and as an application we gain new insights into stationary Markov chains where observability for the system is closely related to asymptotic completeness in a scattering theory for the chain.

Mou-Hsiung Chang

In this chapter, we define and explore basic properties of a quantum Markov semigroup { I t , t ≥ 0} of linear maps on the C* -algebra or von Neumann algebra A that characterizes the quantum system. The quantum Markov semigroup (QMS) plays a key role in describing quantum Markov processes that are to be explored in the subsequent chapters. The concept of QMS extends the semigroup of probability transition operators { T t , t ≥ 0} for a classical Markov process to its noncommutative counterpart. Suppose the classical Markov process { X t , t ≥ 0} is defined on the complete filtered classical probability space (Ω, Ƒ, P ; {Ƒ t , t ≥ 0}) and with values in a measurable space (핏, B (X)). Recall that for each t ≥ 0, the probability transition operator T t : L ∞ (X, B (X)) → L ∞ (X, B (X)) by ( T t f )( X s ) = E x [ f ( X s + t ) | Ƒ s ] ∀ s , t ≥ 0. A semigroup of linear maps on the C* -algebra or von Neumann algebra A is said to be a quantum dynamical semigroup (QDS) if (i) I 0 = J (the identity operator on A ); (ii) I t I s = I t + s for all t , s ≥ 0; (iii) I t is completely positive for each t ≥ 0; and (iv) I t is σ weakly continuous on A , i.e., a ↦ tr ( ρI t ( a )) is continuous from A to ℂ for each ρ ∈ S ( A ) (the space of quantum states) and for each t ≥ 0. If in addition I t ( I ) = I , (respectively, I t ( I ) ≤ I ) for all t ≥ 0, then the QDS is said to be a quantum Makrov semigroup (QMS) (respectively, quantum sub-Markov semigroup).

Crispin W. Gardiner, P. Zoller

Crispin W. Gardiner

Youhei Fujitani, Hiroyuki Suzuki

It is known that one can derive the Jarzynski equality in a stochastic process of a classical system by assuming the local detailed balance. We study how the equality is modified in the linear feedback system. There, measurement is performed on the state following a linear Langevin equation, the measured variable is linear to the state variable with white sensor noise, the Kalman filter estimates the state by utilizing measured values in the past, and a linear regulator controls the state dynamics. Although a stochastic process produced by this dynamics is non-Markovian because of the feedback loop, it is known in the control theory that a Markov process for the estimation can be separated from the whole process. To the exponent in the Jarzynski equality, we find an additional term, whose average gives the mutual information between state variables and measured variables in the Markov process for the estimation. The resultant equality holds whether the gain is optimal or not.