Divisibility of dynamical maps: Schr\"odinger vs. Heisenberg picture

TL;DR

This paper explores the differences in divisibility properties of quantum dynamical maps between the Schrödinger and Heisenberg pictures, revealing they are generally inequivalent and introducing a measure for Heisenberg divisibility violations.

Contribution

It generalizes the concept of divisibility to the Heisenberg picture, demonstrating their inequivalence and proposing a quantifier for Heisenberg divisibility violations.

Findings

Schrödinger and Heisenberg divisibility are generally not equivalent.

Explicit examples of dynamics divisible only in one picture.

Introduces a measure for Heisenberg P-divisibility violation with operational meaning.

Abstract

Divisibility of dynamical maps is a central notion in the study of quantum non-Markovianity, providing a natural framework to characterize memory effects via time-local master equations. In this work, we generalize the notion of divisibility of quantum dynamical maps from the Schr\"odinger to the Heisenberg picture. While the two pictures are equivalent at the level of physical predictions, we show that the divisibility properties of the corresponding dual maps are, in general, not equivalent. This inequivalence originates from the distinction between left and right generators of time-local master equations, which interchange roles under duality. We demonstrate that Schr\"odinger and Heisenberg divisibility are distinct concepts by constructing explicit dynamics divisible only in one picture. Furthermore, we introduce a quantifier for the violation of Heisenberg P-divisibility, analogous to the trace-distance-based measure of non-Markovianity, and provide it with an operational interpretation in terms of the guessing probability between effects. Our results show that Heisenberg divisibility is an independent witness of memory effects and highlight the need to consider both pictures when characterizing non-Markovian quantum dynamics.