Generalized formalism for information backflow in assessing Markovianity and its equivalence to divisibility

Abstract

We present a general framework for the information backflow (IB) approach of Markovianity that not only includes a large number, if not all, of IB prescriptions proposed so far but also is equivalent to CP-divisibility for invertible evolutions. Following the common approach of IB, where monotonic decay of some physical property or some information quantifier is seen as the definition of Markovianity, we propose in our framework a general description of what should be called a proper 'physicality quantifier' to define Markovianity. We elucidate different properties of our framework, and use them to argue that an infinite family of non-Markovianity measures can be constructed, which would capture varied strengths of non-Markovianity in the dynamics. Moreover, we show that generalized trace-distance measure in 2 dimension serve as a sufficient criteria for IB-Markovianity for a number of prescriptions suggested earlier in literature.

Non-Markovianity of qubit evolution under the action of spin environment

“…Numerous prescriptions have been proposed to capture different aspects of quantum Markovianity. Broadly these prescriptions can be classified into two classes: information backflow 42–47 and completely positive divisibility (CP-divisibility) 1,48 .…”

Quasi-Entropies and Non-Markovianity

“…The fact that, unlike CP-divisibility, P-divisibility is not stable under double tensorisation represents a puzzle within another approach to non-Markovianity that identifies it with an information back-flow from the environment E into the open quantum system S [ 5 , 6 ]. In order to discuss this issue, we shall use a recent approach [ 8 ] that has been developed as a unifying framework for all measures, called Physicality Quantifiers, of information flows. Indeed, whether the latter goes into the system S from the environment E or vice versa from S into E can be witnessed by suitable bounded functionals from all possible statistical ensembles of states of S into the positive reals, that monotonically decrease under the action of Completely Positive maps.…”

“…This argument is at the basis of the definition of non-Markovianity proposed in [ 5 ], respectively [ 6 ], and based on the trace-distance ( 40 ), respectively generalised trace-distance ( 41 ). Both these approaches to non-Markovianity can be accommodated within the framework of Physicality Quantifiers in [ 8 ] where one can introduce various degrees of Markovianity.…”

“…The hierarchy of Markovianity degrees elaborated in [ 8 ] suggests an obvious way out of the informational puzzle presented above; namely, it might occur that the back-flow of information affecting also affects , though in a way that cannot be exposed by either the trace-distance, the generalised trace-distance or the relative entropy and that demands Physicality Quantifiers of higher degree than . Indeed, as from Theorem 5, no Physicality Quantifier with two density matrices can witness any back flow information affecting a qubit non-Markovian dynamics if this cannot be done by the generalised trace-distance.…”

“…Obviously in order to talk of back-flow of information one needs witnesses to expose it: these witnesses have recently been generically characterised as Physicality Quantifiers [ 8 ], namely as functionals on statistical ensembles consisting of d quantum states that monotonically decrease under the action of completely positive maps. Within this framework, Markovianity of a time-evolution is identified with the monotonic decrease in time of all possible Physicality Quantifiers .…”