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Characterizing (non-)Markovianity through Fisher Information

by Paolo Abiuso, Matteo Scandi, Dario De Santis, Jacopo Surace

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Submission summary

Authors (as registered SciPost users): Paolo Abiuso
Submission information
Preprint Link: scipost_202207_00020v1  (pdf)
Date submitted: 2022-07-14 23:23
Submitted by: Abiuso, Paolo
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Quantum Physics
Approach: Theoretical


A non-isolated physical system typically loses information to its environment, and when such loss is irreversible the evolution is said to be Markovian. Non-Markovian effects are studied by monitoring how information quantifiers, such as the distance between physical states, evolve in time. Here we show that the Fisher information metric emerges as the natural object to study in this context; we fully characterize the relation between its contractivity properties and Markovianity, both from the mathematical and operational point of view. We prove, for classical dynamics, that Markovianity is equivalent to the monotonous contraction of the Fisher metric at all points of the set of states. At the same time, operational witnesses of non-Markovianity based on the dilation of the Fisher distance cannot, in general, detect all non-Markovian evolutions, unless specific physical postprocessing is applied to the dynamics. Finally, we show for the first time that non-Markovian dilations of Fisher distance between states at any time correspond to backflow of information about the initial state of the dynamics at time 0, via Bayesian retrodiction. All the presented results can be lifted to the case of quantum dynamics by considering the standard CP-divisibility framework.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2022-8-26 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202207_00020v1, delivered 2022-08-26, doi: 10.21468/SciPost.Report.5586


1) This paper proposes to study Markovian and non-Markovian dynamics using the Fisher distance instead of more conventional quantifiers of non-Markovianity. This idea is interesting since the Fisher distance is an unavoidable tool in the analysis of random processes.
2) Bayesian retrodiction is used to facilitate the analysis of experimental systems.


1) Some mathematical subtleties are ignored.
2) Interpretation of the results are questionable.


The idea of the paper is interesting. Linking the Fisher information to the study of Markovianity may be very useful to connect several ideas in quantum information processing, where the Fisher information has been widely used in recent studies of open quantum systems.

However, several points must be clarified (or corrected if necessary).

1) Efforts have been made to present state of the art results of Markovian processes (this is welcome) but the presentation of the Fisher distance is reduced to almost nothing. Since it is a key notion of the paper, I think that it must be presented with more details. I think that it will be useful for most of the readers, which may be more knowledgeable on Markovian processes than Fisher's distance.

2) The authors use extensively the approximation of $D_{Fish}$ (defined in Eq. 2), which gives us a local information on the space of probability measure, in the vicinity of a measure $p$. This is not a problem for most of the results, but this is very dangerous when it is extrapolated to the global space. Very strong constrains are necessary to keep local properties true at the scale of the full space. No particular problems may come for a Markov-process with a unique invariant probability measure, but if not, the local analysis is not sufficient. The conditions where such a 'change of scale' can be made must be clearly identified, specifically in Thm. 1.

2) In thm 1 (or before) it may be interesting to specify that this is not a "true" non-Markovian process which is studied, but the linearization of a process whose domain of definition is extended to the full space, and not a small set around the point used for the linearization. This may simplify the comment on the points that "cannot be achieved physically" (page 6).

3) I am not convinced by the comparison between Fisher distance and trace-distance. If we take the following equation as the definition of $D_{Fish}$ (see footnote page 2):
D_{Fish} (p,q)= \sqrt{2} \arccos (\langle \sqrt{p}|\sqrt{q}\rangle),
with $\langle a |b \rangle = \sum_i a_i b_i$. If we make a change of variable $p \rightarrow \tilde p =\sqrt{p}$, it is clear that $\sum_i \tilde p_i = 1$ which gives us a point on a sphere. Since all the $p_i$ are positive, this is fine and we have a one-to-one mapping between the simplex and a spherical cap. Then, it is clear that $D_{Fish}$ measure the angle between two points on the sphere. So, the trace distance is a distance adapted to measure points in the simplex, and the Fisher distance is adapted to measure distances in the sphere. Since there is a diffeomorphism between the two spaces, I expect that $D_{Tr}$ and $D_{Fish}$ have the same properties. If a dynamical map contracts $D_{Tr}$ it must contract $D_{Fish}$ and reciprocally, and that the translation invariance of $D_{Tr}$ is replaced by a rotation invariance.

6) Following the point 5), I'm not convinced by the fact that "the Fisher information metric [...] natural object to study". But I agree that it gives us an interesting point of view that may simplify some calculations.

4) The derivation of Eq. 21 must be clarified. It is not clear to me if the level of approximation is consistent with the one of Eq. 2, and hence, if a hidden mistake may be hidden in Thm. 4.

Requested changes

1) Clarify the working hypothesis of Thm.1, according to the previous comments.

2) Don't the use of the approximation of $D_{Fish}$ when it can be easily avoided. The approximation is not always helpful, and it can lead to false conclusions.

3) Typo in Eq. 14 ?

4) (optional) Remove or specify the sections in the last sentence of Appendix B.

  • validity: ok
  • significance: good
  • originality: ok
  • clarity: ok
  • formatting: good
  • grammar: good

Anonymous Report 1 on 2022-8-25 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202207_00020v1, delivered 2022-08-25, doi: 10.21468/SciPost.Report.5582


Dear Editor,
please find here enclosed my review of the manuscript entitled
"Characterizing (non-)Markovianity through Fisher Information", submitted for publication in SciPost Physics.
The authors investigate how the Fisher information metric can be used as a natural object to characterize the non-Markovianity of a dynamical open system. In particular, they prove that Markovianity is equivalent to a monotonous contraction of the Fisher metric. All the results are established in a classical sense but the authors claim that they can be lifted to the case of quantum dynamics.

The description and the characterization of the dynamics of open quantum systems is a subject of fundamental interest and a basic prerequisite for applications in quantum computing and more generally in quantum technologies. The goal of this paper is a key objective of this field. In the past few years, many papers have proposed different measures and ways to characterize the non-Markovian character of an open quantum system. The results of this paper are interesting in this direction. However, I think that this paper cannot be published in its current version in SciPost Physics and should be submitted to a more mathematical journal.

The derivation of the different results is quite technical and difficult to follow for a non-expert. Moreover, much of the paper focuses on classical dynamics whereas (as mentioned above) the role and characterization of non-Markovianity has been mainly studied in quantum physics. This aspect is only briefly described in the conclusion of the paper. In addition, the different theoretical results are not applied to standard physical examples in order to show their importance and their relevance.

  • validity: good
  • significance: good
  • originality: high
  • clarity: low
  • formatting: good
  • grammar: good

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