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  • Open Access

Divisibility of Dynamical Maps: Schrödinger Versus Heisenberg Picture

Federico Settimo1,*, Andrea Smirne2,3, Kimmo Luoma1, Bassano Vacchini2,3, Jyrki Piilo1, and Dariusz Chruściński4

  • *Contact author: fesett@utu.fi
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PRX Quantum 7, 010340 – Published 26 February, 2026

DOI: https://doi.org/10.1103/6dt2-sq44

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 Schrödinger 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 Schrödinger 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.

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Popular Summary

Memory effects in quantum dynamics is a multifaceted phenomenon containing concepts such as dynamical divisibility and information flow. In general, considering theoretical predictions for quantum systems, Schrödinger and Heisenberg pictures are equivalent for computing single-time observables. In other words, whether evolving quantum mechanical states or operators should give equivalent descriptions in this framework and should not make a difference.

While characterization and quantification of quantum memory effects have been a very popular area of research during the recent years, our results show, surprisingly, that the use of Schrödinger and Heisenberg pictures is an independent concept within this framework. In other words, the dynamics can display memory effects in one picture and not in the other one. While there exist many quantifiers for memory effects in the Schrödinger picture, we develop a quantifier for memory effects in the Heisenberg picture. For this purpose, we also give a clear physical operational interpretation in terms of guessing probability between quantum mechanical effects.

Our results call for further investigations about similarities and differences between Schrödinger and Heisenberg pictures, in general. The results also open an unexplored area how to understand, interpret, and exploit memory effects present in the Heisenberg picture, e.g., within the quantum information framework.

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PRX Quantum 7, 010340– Published 26 February, 2026
  • Received 26 June 2025
  • Revised 8 October 2025
  • Accepted 5 January 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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