Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2506.08103 · PRX Quantum

Divisibility of Dynamical Maps: Schrödinger Versus Heisenberg Picture

Federico Settimo, Andrea Smirne, Kimmo Luoma, Bassano Vacchini, Jyrki Piilo, Dariusz Chruściński

Published Jan 5, 2026• 1 citation
arXivHTMLPDF

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.

DOI
10.1103/6dt2-sq44

Related papers

Divisibility of Non-Invertible Quantum Dynamics in the Heisenberg Picture

Federico Settimo, Dariusz Chruściński

We study divisibility of finite-dimensional quantum dynamics in the Heisenberg picture when the dynamical maps need not be invertible. Heisenberg divisible dynamics determines unique reduced propagators on the accessible operator systems, and Heisenberg CP-divisibility is shown to be equivalent to an operator-norm monotonicity. Although each reduced propagator admits a unital completely positive extension, independently chosen extensions need not satisfy a composition law. Generalized inverses restore this law algebraically, but do not, in general, preserve complete positivity. We characterize the resulting obstruction and give a sufficient construction based on unital completely positive projections onto the accessible operator systems. It is shown that for qubit dynamics Heisenberg CP-divisibility always guarantees a completely positive generalized-inverse propagator. Finally, we analyze time-local generators at singular times and identify the associated gauge freedom on dynamically inaccessible directions.

quant-pharXiv:2610.07986•Oct 7, 2026•0 citations

Schrödinger and Heisenberg non-Markovianity in quantum information tasks

Federico Settimo, Kimmo Luoma, Jyrki Piilo +3 more

Quantum non-Markovianity has been widely studied and connected to the existence of memory effects in the dynamics of open systems. Surprisingly, working in the Schrödinger or in the Heisenberg picture can provide inequivalent description non-Markovianity: a process can appear to be memoryless in one picture, while displaying memory effects in the other. Here, we investigate which kind of memory is relevant for different quantum information tasks. Some of them, such as sending information via a noisy channel, require memory in both pictures in order to exhibit revivals in the task performance. For others, only one type of memory is sufficient. We also provide necessary conditions for non-Markovianity in both pictures by only considering the dynamics in one picture, showing for instance that the previously considered witness of Schrödinger non-Markovianity in terms of the volume of accessible states does indeed witness non-Markovianity in both pictures at the same time.

quant-pharXiv:2606.30751•Jul 1, 2026•0 citations

Quantum non-Markovianity: characterization, quantification and detection

Ángel Rivas, Susana F Huelga, Martin Bodo Plenio

We present a comprehensive and up-to-date review of the concept of quantum non-Markovianity, a central theme in the theory of open quantum systems. We introduce the concept of a quantum Markovian process as a generalization of the classical definition of Markovianity via the so-called divisibility property and relate this notion to the intuitive idea that links non-Markovianity with the persistence of memory effects. A detailed comparison with other definitions presented in the literature is provided. We then discuss several existing proposals to quantify the degree of non-Markovianity of quantum dynamics and to witness non-Markovian behavior, the latter providing sufficient conditions to detect deviations from strict Markovianity. Finally, we conclude by enumerating some timely open problems in the field and provide an outlook on possible research directions.

arXiv:1405.0303•Reports on Progress in Physics•Aug 22, 2014•1,035 citations

Quantum versus classical P -divisibility

Fabio Benatti, Dariusz Chruściński, Giovanni Nichele

$P$-divisibility is a central concept in both classical and quantum non-Markovian processes; in particular, it is strictly related to the notion of information backflow. When restricted to a fixed commutative algebra generated by a complete set of orthogonal projections, any quantum dynamics naturally provides a classical stochastic process. It is indeed well known that a quantum generator gives rise to a $P$-divisible quantum dynamics if and only if all its possible classical reductions give rise to divisible classical stochastic processes. However, this property does not hold if one operates a classical reduction of the quantum dynamical maps instead of their generators: As an example, for a unitary dynamics, $P$-divisibility of its classical reduction is inevitably lost and the latter thus exhibits information backflow. Instead, for some important classes of purely dissipative qubit evolutions, quantum $P$-divisibility always implies classical $P$-divisibility and therefore excludes information backflow in both the quantum and classical scenarios. On the contrary, for a wide class of orthogonally covariant qubit dynamics, we show that loss of classical $P$-divisibility originates from the classical reduction of a purely dissipative $P$-divisible quantum dynamics as in the unitary case. Moreover, such an effect can be interpreted in terms of information backflow due to the coherences developed by the quantumly evolving classical state.

arXiv:2405.05794•Physical Review A•Nov 12, 2024•5 citations

Open system dynamics and quantum jumps: Divisibility vs. dissipativity

Dariusz Chruściński, Kimmo Luoma, Jyrki Piilo +1 more

Several key properties of quantum evolutions are characterized by divisibility of the corresponding dynamical maps. In particular, a Markovian evolution respects CP-divisibility, whereas breaking of P-divisibility provides a clear sign of non-Markovian effects. We analyze a class of evolutions which interpolates between CP- and P-divisible classes and is characterized by dissipativity -- a long known but so far not widely used formal concept to classify open system dynamics. By making a connection to stochastic jump unravellings of master equations, we demonstrate that there exists inherent freedom in how to divide the terms of the underlying master equation into the deterministic and jump parts for the stochastic description. This leads to a number of different unravelings, each one with a measurement scheme interpretation and highlighting different properties of the considered open system dynamics. Starting from formal mathematical concepts, our results allow us to get fundamental insights in open system dynamics and to ease their numerical simulations.

•Sep 23, 2020•1 citation

Temporal self-similarity of quantum dynamical maps as a concept of memorylessness

Shrikant Utagi, R. Srikanth, Subhashish Banerjee

The problem of defining quantum non-Markovianity has proven elusive, with various in-equivalent criteria put forth to address it. The concept of CP-indivisibility and the hierarchy of stronger divisibility criteria going up to P-indivisibility, capture a fundamental aspect of memory in quantum non-Markovianity. In practice, however, there can be a memory-like influence associated with divisible channels in the form of weakening, if not reversing, the effects of decoherence. Arguably, such a facet of memory relates to CP-indivisibility as quantum discord relates to entanglement. We concretize this weaker notion of non-Markovianity by identifying it with deviation from "temporal self-similarity", the property of a system dynamics whereby the propagator between two intermediate states is independent of the initial time [Formula: see text]. We illustrate this idea through examples, and propose a geometric quantification of temporal self-similarity, and show how our approach complements the divisibility-based criterion of quantum non-Markovianity.

arXiv:1911.04162•Scientific Reports•Sep 14, 2020•50 citations
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.
We study divisibility of finite-dimensional quantum dynamics in the Heisenberg picture when the dynamical maps need not be invertible. Heisenberg divisible dynamics determines unique reduced propagators on the accessible operator systems, and Heisenberg CP-divisibility is shown to be equivalent to an operator-norm monotonicity. Although each reduced propagator admits a unital completely positive extension, independently chosen extensions need not satisfy a composition law. Generalized inverses restore this law algebraically, but do not, in general, preserve complete positivity. We characterize the resulting obstruction and give a sufficient construction based on unital completely positive projections onto the accessible operator systems. It is shown that for qubit dynamics Heisenberg CP-divisibility always guarantees a completely positive generalized-inverse propagator. Finally, we analyze time-local generators at singular times and identify the associated gauge freedom on dynamically inaccessible directions.
Quantum non-Markovianity has been widely studied and connected to the existence of memory effects in the dynamics of open systems. Surprisingly, working in the Schrödinger or in the Heisenberg picture can provide inequivalent description non-Markovianity: a process can appear to be memoryless in one picture, while displaying memory effects in the other. Here, we investigate which kind of memory is relevant for different quantum information tasks. Some of them, such as sending information via a noisy channel, require memory in both pictures in order to exhibit revivals in the task performance. For others, only one type of memory is sufficient. We also provide necessary conditions for non-Markovianity in both pictures by only considering the dynamics in one picture, showing for instance that the previously considered witness of Schrödinger non-Markovianity in terms of the volume of accessible states does indeed witness non-Markovianity in both pictures at the same time.
We present a comprehensive and up-to-date review of the concept of quantum non-Markovianity, a central theme in the theory of open quantum systems. We introduce the concept of a quantum Markovian process as a generalization of the classical definition of Markovianity via the so-called divisibility property and relate this notion to the intuitive idea that links non-Markovianity with the persistence of memory effects. A detailed comparison with other definitions presented in the literature is provided. We then discuss several existing proposals to quantify the degree of non-Markovianity of quantum dynamics and to witness non-Markovian behavior, the latter providing sufficient conditions to detect deviations from strict Markovianity. Finally, we conclude by enumerating some timely open problems in the field and provide an outlook on possible research directions.
PPP-divisibility is a central concept in both classical and quantum non-Markovian processes; in particular, it is strictly related to the notion of information backflow. When restricted to a fixed commutative algebra generated by a complete set of orthogonal projections, any quantum dynamics naturally provides a classical stochastic process. It is indeed well known that a quantum generator gives rise to a PPP-divisible quantum dynamics if and only if all its possible classical reductions give rise to divisible classical stochastic processes. However, this property does not hold if one operates a classical reduction of the quantum dynamical maps instead of their generators: As an example, for a unitary dynamics, PPP-divisibility of its classical reduction is inevitably lost and the latter thus exhibits information backflow. Instead, for some important classes of purely dissipative qubit evolutions, quantum PPP-divisibility always implies classical PPP-divisibility and therefore excludes information backflow in both the quantum and classical scenarios. On the contrary, for a wide class of orthogonally covariant qubit dynamics, we show that loss of classical PPP-divisibility originates from the classical reduction of a purely dissipative PPP-divisible quantum dynamics as in the unitary case. Moreover, such an effect can be interpreted in terms of information backflow due to the coherences developed by the quantumly evolving classical state.
Several key properties of quantum evolutions are characterized by divisibility of the corresponding dynamical maps. In particular, a Markovian evolution respects CP-divisibility, whereas breaking of P-divisibility provides a clear sign of non-Markovian effects. We analyze a class of evolutions which interpolates between CP- and P-divisible classes and is characterized by dissipativity -- a long known but so far not widely used formal concept to classify open system dynamics. By making a connection to stochastic jump unravellings of master equations, we demonstrate that there exists inherent freedom in how to divide the terms of the underlying master equation into the deterministic and jump parts for the stochastic description. This leads to a number of different unravelings, each one with a measurement scheme interpretation and highlighting different properties of the considered open system dynamics. Starting from formal mathematical concepts, our results allow us to get fundamental insights in open system dynamics and to ease their numerical simulations.
The problem of defining quantum non-Markovianity has proven elusive, with various in-equivalent criteria put forth to address it. The concept of CP-indivisibility and the hierarchy of stronger divisibility criteria going up to P-indivisibility, capture a fundamental aspect of memory in quantum non-Markovianity. In practice, however, there can be a memory-like influence associated with divisible channels in the form of weakening, if not reversing, the effects of decoherence. Arguably, such a facet of memory relates to CP-indivisibility as quantum discord relates to entanglement. We concretize this weaker notion of non-Markovianity by identifying it with deviation from "temporal self-similarity", the property of a system dynamics whereby the propagator between two intermediate states is independent of the initial time [Formula: see text]. We illustrate this idea through examples, and propose a geometric quantification of temporal self-similarity, and show how our approach complements the divisibility-based criterion of quantum non-Markovianity.