Stochastic processes and commutation relationships

Hendra I. Nurdin, John Edward Gough

A general theory of quantum stochastic processes was formulated by Accardi, Frigerio and Lewis in 1982 within the operator-algebraic framework of quantum probability theory, as a non-commutative extension of the Kolmogorovian classical stochastic processes. More recently, studies on non-Markovian quantum processes have led to the discrete-time process tensor formalism in the Schrödinger picture to describe the outcomes of sequential interventions on open quantum systems. However, there has been no treatment of the relationship of the process tensor formalism to the quantum probabilistic theory of quantum stochastic processes. This paper gives an exposition of quantum stochastic processes and the process tensor and the relationship between them. In particular, it is shown how the latter emerges from the former via extended correlation kernels incorporating ancillas.

Luigi Accardi, I. Ya. Aref’eva, Igor Vasil'evich Volovich

A. O. Bolivar

Fattah Sakuldee, Philip Taranto, Simon Milz

Understanding the demarcation line between classical and quantum is an important issue in modern physics. The development of such an understanding requires a clear picture of the various concurrent notions of ``classicality'' in quantum theory presently in use. Here we focus on the relationship between Kolmogorov consistency of measurement statistics---the foundational footing of classical stochastic processes in standard probability theory---and the commutativity (or absence thereof) of measurement operators---a concept at the core of quantum theory. Kolmogorov consistency implies that the statistics of sequential measurements on a (possibly quantum) system could be explained entirely by means of a classical stochastic process, thereby providing an operational notion of classicality. On the other hand, commutativity of measurement operators is a structural property that holds in classical physics and its breakdown is the origin of the uncertainty principle, a fundamentally quantum phenomenon. We formalize the connection between these two a priori independent notions of classicality, demonstrate that they are distinct in general and detail their implications for memoryless multitime quantum processes.

G. O. S. Ekhaguere

Introduces a general notion of non-commutative stochastic processes over complete locally convex *-algebras and proves a theorem concerning the existence and uniqueness of such processes. The author shows that quantum fields are stochastic processes in the sense of this paper. Moreover, the author applies the theory to infer the existence and uniqueness of quantum fields.

Aubrey Truman, Ian M Davies