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:0711.3172 · Physical Review Letters

Assessing Non-Markovian Quantum Dynamics

Michael M. Wolf, Jens Eisert, Toby S. Cubitt, J. I. Cirac

Published Oct 6, 2008• 596 citations
arXivHTMLPDF

Abstract

We investigate what a snapshot of a quantum evolution--a quantum channel reflecting open system dynamics--reveals about the underlying continuous time evolution. Remarkably, from such a snapshot, and without imposing additional assumptions, it can be decided whether or not a channel is consistent with a time (in)dependent Markovian evolution, for which we provide computable necessary and sufficient criteria. Based on these, a computable measure of "Markovianity" is introduced. We discuss how the consistency with Markovian dynamics can be checked in quantum process tomography. The results also clarify the geometry of the set of quantum channels with respect to being solutions of time (in)dependent master equations.

DOI
10.1103/physrevlett.101.150402

Related papers

Deciding whether a Quantum Channel is Markovian is NP-hard

Toby S. Cubitt, Jens Eisert, Michael M. Wolf

We show that deciding whether a given quantum channel can be generated by a Markovian master equation is an NP-hard problem. As a consequence, this result suggests that extracting the underlyin g physics governing the evolution of a quantum system, as described by its dynamical equations, may be a hard task regardless of how much data is gathered via measurements of the system. On the other hand, if the system dimension is fixed, the problem of deciding whether a process is consistent with being Markovian can be efficiently decided, and we give an explicit algorithm in this case. This work also leads to a complexity-theoretic solution to a long-standing open problem in probability theory.

•Aug 17, 2009•2 citations

Entanglement and Non-Markovianity of Quantum Evolutions

́Angel Rivas, Susana F. Huelga, Martin Bodo Plenio

We address the problem of quantifying the non-markovian character of quantum time evolutions of general systems in contact with an environment. We introduce two different measures of non-markovianity that exploit the specific traits of quantum correlations and are suitable for opposite experimental contexts. When complete tomographic knowledge about the evolution is available, our measure provides a necessary and sufficient condition to quantify strictly the non-markovianity. In the opposite case, when no information whatsoever is available, we propose a sufficient condition for non-markovianity. Remarkably, no optimization procedure underlies our derivation, which greatly enhances the practical relevance of the proposed criteria.

arXiv:0911.4270•Physical Review Letters•Jul 28, 2010•1,031 citations

Fitting time-dependent Markovian dynamics to noisy quantum channels

Emilio Onorati, Tamara Kohler, Toby S. Cubitt

Understanding how to characterise and mitigate errors is a key challenge in developing reliable quantum architecture for near-term applications. Recent work (arXiv:2103.17243) provides an efficient set of algorithms for analysing unknown noise processes requiring only tomographic snapshots of the quantum operator under consideration, without the need of any a-priori information on the noise model, nor necessitating a particular experimental setup. The only assumption made is that the observed channel can be approximated by a time-independent Markovian map, which is a typically reasonable framework when considering short time scales. In this note we lift the time-independent assumption, presenting an extension of the scheme now able to analyse noisy dynamics with time-dependent generators from a sequence of snapshots. We hence provide a diagnostic tool for a wider spectrum of instances while inheriting all the favourable features from the previous protocol. On the theoretical side, the problem of characterising time-dependent Markovian channels has been an open problem for many decades. This work gives an approach to tackle this characterisation problem rigorously.

arXiv:2303.08936•Mar 15, 2023•1 citation

Non-Markovian master equations from piecewise dynamics

Bassano Vacchini

We construct a large class of completely positive and trace-preserving non-Markovian dynamical maps for an open quantum system. These maps arise from a piecewise dynamics for statistical operators characterized by a continuous time evolution interrupted by jumps, randomly distributed in time and described by a quantum channel. The state of the open system is shown to obey a closed evolution equation, given by a master equation with a memory kernel and an inhomogeneous term. The non-Markovianity of the obtained dynamics is explicitly assessed studying the behavior of the distinguishability of two different initial system's states with elapsing time.

arXiv:1301.2902•Physical Review A•Mar 8, 2013•61 citations

Characterizing non-Markovian quantum evolution

Dariusz Chruściński

We analyze classes of legitimate quantum dynamics using a local in time master equation. Markovian dynamics is characterized in terms of a divisible dynamical map. Moreover, we provide a family of criteria which can distinguish Markovian and non-Markovian quantum evolution. They are based on a simple observation that Markovian dynamics implies monotonic behavior of several well-known quantities such as distinguishability of states, fidelity and relative entropy.

Physica Scripta•Mar 1, 2013•0 citations

Complete framework for efficient characterisation of non-Markovian processes

Felix A. Pollock, César A. Rodríguez-Rosario, Thomas Frauenheim +2 more

A non-Markovian process is one that retains `memory' of its past. A systematic understanding of these processes is necessary to fully describe and harness a vast range of complex phenomena; however, no such general characterisation currently exists. This long-standing problem has hindered advances in understanding physical, chemical and biological processes, where often dubious theoretical assumptions are made to render a dynamical description tractable. Moreover, the methods currently available to treat non-Markovian quantum dynamics are plagued with unphysical results, like non-positive dynamics. Here we develop an operational framework to characterise arbitrary non-Markovian quantum processes. We demonstrate the universality of our framework and how the characterisation can be rendered efficient, before formulating a necessary and sufficient condition for quantum Markov processes. Finally, we stress how our framework enables the actual systematic analysis of non-Markovian processes, the understanding of their typicality, and the development of new master equations for the effective description of memory-bearing open-system evolution.

•Dec 2, 2015•23 citations
We investigate what a snapshot of a quantum evolution--a quantum channel reflecting open system dynamics--reveals about the underlying continuous time evolution. Remarkably, from such a snapshot, and without imposing additional assumptions, it can be decided whether or not a channel is consistent with a time (in)dependent Markovian evolution, for which we provide computable necessary and sufficient criteria. Based on these, a computable measure of "Markovianity" is introduced. We discuss how the consistency with Markovian dynamics can be checked in quantum process tomography. The results also clarify the geometry of the set of quantum channels with respect to being solutions of time (in)dependent master equations.
We show that deciding whether a given quantum channel can be generated by a Markovian master equation is an NP-hard problem. As a consequence, this result suggests that extracting the underlyin g physics governing the evolution of a quantum system, as described by its dynamical equations, may be a hard task regardless of how much data is gathered via measurements of the system. On the other hand, if the system dimension is fixed, the problem of deciding whether a process is consistent with being Markovian can be efficiently decided, and we give an explicit algorithm in this case. This work also leads to a complexity-theoretic solution to a long-standing open problem in probability theory.
We address the problem of quantifying the non-markovian character of quantum time evolutions of general systems in contact with an environment. We introduce two different measures of non-markovianity that exploit the specific traits of quantum correlations and are suitable for opposite experimental contexts. When complete tomographic knowledge about the evolution is available, our measure provides a necessary and sufficient condition to quantify strictly the non-markovianity. In the opposite case, when no information whatsoever is available, we propose a sufficient condition for non-markovianity. Remarkably, no optimization procedure underlies our derivation, which greatly enhances the practical relevance of the proposed criteria.
Understanding how to characterise and mitigate errors is a key challenge in developing reliable quantum architecture for near-term applications. Recent work (arXiv:2103.17243) provides an efficient set of algorithms for analysing unknown noise processes requiring only tomographic snapshots of the quantum operator under consideration, without the need of any a-priori information on the noise model, nor necessitating a particular experimental setup. The only assumption made is that the observed channel can be approximated by a time-independent Markovian map, which is a typically reasonable framework when considering short time scales. In this note we lift the time-independent assumption, presenting an extension of the scheme now able to analyse noisy dynamics with time-dependent generators from a sequence of snapshots. We hence provide a diagnostic tool for a wider spectrum of instances while inheriting all the favourable features from the previous protocol. On the theoretical side, the problem of characterising time-dependent Markovian channels has been an open problem for many decades. This work gives an approach to tackle this characterisation problem rigorously.
We construct a large class of completely positive and trace-preserving non-Markovian dynamical maps for an open quantum system. These maps arise from a piecewise dynamics for statistical operators characterized by a continuous time evolution interrupted by jumps, randomly distributed in time and described by a quantum channel. The state of the open system is shown to obey a closed evolution equation, given by a master equation with a memory kernel and an inhomogeneous term. The non-Markovianity of the obtained dynamics is explicitly assessed studying the behavior of the distinguishability of two different initial system's states with elapsing time.
A non-Markovian process is one that retains `memory' of its past. A systematic understanding of these processes is necessary to fully describe and harness a vast range of complex phenomena; however, no such general characterisation currently exists. This long-standing problem has hindered advances in understanding physical, chemical and biological processes, where often dubious theoretical assumptions are made to render a dynamical description tractable. Moreover, the methods currently available to treat non-Markovian quantum dynamics are plagued with unphysical results, like non-positive dynamics. Here we develop an operational framework to characterise arbitrary non-Markovian quantum processes. We demonstrate the universality of our framework and how the characterisation can be rendered efficient, before formulating a necessary and sufficient condition for quantum Markov processes. Finally, we stress how our framework enables the actual systematic analysis of non-Markovian processes, the understanding of their typicality, and the development of new master equations for the effective description of memory-bearing open-system evolution.