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arxiv: 2606.24511 · v1 · pith:QBXBW3VEnew · submitted 2026-06-23 · 🪐 quant-ph

How rare are Markovian quantum dynamics?

Charlotte B\"acker , Nick Maryshchak , Walter T. Strunz This is my paper

Pith reviewed 2026-06-25 23:53 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Markovian dynamicsnon-Markovianityqubit dynamicsopen quantum systemsmemory effectsrandom quantum mapsdecoherence
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The pith

Almost all two-step qubit dynamics are non-Markovian, with only a small finite fraction being Markovian.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper samples random two-step qubit dynamics and applies multiple witnesses of non-Markovianity to determine how common memoryless evolution is. It concludes that Markovian cases form only a small yet nonzero portion of the full set, while nearly all dynamics display memory effects. The work also tracks how this fraction shifts for restricted classes such as lower-rank or mixed-unitary maps and maps out the overlaps between different non-Markovianity criteria. A reader would care because realistic open-system modeling would then need to incorporate memory effects far more often than the Markovian approximation suggests.

Core claim

By investigating randomly generated two-step qubit dynamics with respect to different concepts and witnesses of non-Markovianity, we observe that almost all dynamics are non-Markovian, and only a small (yet finite) fraction is Markovian. Furthermore, we study how this proportion changes when considering certain subclasses such as lower rank or mixed-unitary dynamics. Importantly, our results shed light on the relative ratios of -- and interrelations between -- the sets of dynamics that are non-Markovian with respect to different criteria. Finally, we investigate the fraction of dynamics in which the memory effects are necessarily of quantum nature and establish a connection between two recen

What carries the argument

Random generation of two-step qubit dynamics together with witnesses and measures of non-Markovianity applied to the resulting maps.

If this is right

  • The relative sizes of the sets of dynamics that are non-Markovian according to distinct criteria are fixed by the sampling results.
  • A nonzero fraction of non-Markovian dynamics must involve quantum memory effects rather than classical ones.
  • Markovian approximations remain valid only for restricted subclasses whose measure is smaller than the full set.
  • Interrelations among non-Markovianity concepts become quantifiable through the observed overlaps in the sampled ensemble.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Modeling protocols for open quantum systems would default to non-Markovian descriptions unless additional structure is imposed.
  • Numerical studies of continuous-time or higher-dimensional dynamics could test whether the same rarity pattern appears beyond the two-step qubit setting.
  • The connection between different quantum-memory witnesses suggests a hierarchy that future classification schemes could exploit.

Load-bearing premise

The procedure used to generate random two-step qubit dynamics produces a representative sample of the space of all possible dynamics with respect to the chosen witnesses and measures of non-Markovianity.

What would settle it

A different random-generation method or a substantially larger sample that yields a markedly higher or lower fraction of Markovian maps would falsify the reported rarity.

Figures

Figures reproduced from arXiv: 2606.24511 by Charlotte B\"acker, Nick Maryshchak, Walter T. Strunz.

Figure 1
Figure 1. Figure 1: FIG. 1. Relations between different definitions and witnesses [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Physical motivation of the sampling strategy for a [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Visualization of the fraction of randomly sampled [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Visualization of the fraction of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: It is noticeable that once the maximal Kraus rank kmax is set to be strictly below four, not a single dynamics is CP-divisible. In addition, for lower Kraus ranks less dy￾namics can be proven to be P-indivisible. The third cri￾terion which changes significantly upon reducing kmax is the witness based on the increase of entanglement. This approaches 50%, which reflects the fact that for channels of Kraus ra… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Dependence on the maximal Kraus rank [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
read the original abstract

A profound understanding of decoherence and dissipation in quantum dynamics is crucial for the realistic modeling of the evolution of quantum systems. In open quantum dynamics one distinguishes between a memoryless, so-called Markovian evolution and dynamics incorporating memory effects, termed non-Markovian. In this work we study how prevalent memory effects are in the set of all such dynamics. We thus investigate how often a Markovian description is applicable. This question is approached by investigating randomly generated two-step qubit dynamics with respect to different concepts and witnesses of non-Markovianity. We observe that almost all dynamics are non-Markovian, and only a small (yet finite) fraction is Markovian. Furthermore, we study how this proportion changes when considering certain subclasses such as lower rank or mixed-unitary dynamics. Importantly, our results shed light on the relative ratios of -- and interrelations between -- the sets of dynamics that are non-Markovian with respect to different criteria. Finally, we investigate the fraction of dynamics in which the memory effects are necessarily of quantum nature and establish a connection between two recently developed concepts that characterize the quantumness of memory in non-Markovian dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper claims that Markovian dynamics are rare among all possible open quantum dynamics. By Monte Carlo sampling of randomly generated two-step qubit CPTP maps and testing them against multiple non-Markovianity witnesses, the authors report that only a small yet finite fraction of the sampled maps are Markovian while almost all exhibit memory effects. The work further examines how this fraction changes for lower-rank and mixed-unitary subclasses, quantifies the relative sizes and overlaps of sets defined by different non-Markovianity criteria, and studies the subset in which memory effects are necessarily quantum, establishing a link between two recent characterizations of quantum memory.

Significance. If the sampling procedure induces a representative measure on the space of two-step dynamics, the numerical result would indicate that Markovian descriptions are atypical rather than generic, with direct consequences for the modeling of decoherence and the applicability of Markovian master equations. The comparative analysis of different witnesses and the connection drawn between quantum-memory concepts add value beyond the headline fraction. The significance is nevertheless conditional on the ensemble used to generate the random maps.

major comments (1)
  1. [Numerical sampling / Methods] The procedure used to generate the ensemble of random two-step qubit dynamics is not described with sufficient detail (distribution over Kraus operators or Choi matrices, handling of CPTP constraints, sampling algorithm, and statistical error control) to determine whether the reported Markovian fraction is an artifact of the chosen measure rather than a property of the space itself. This directly affects the central claim that 'almost all dynamics are non-Markovian'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and constructive criticism. We address the sole major comment below and will revise the manuscript to incorporate the requested methodological details.

read point-by-point responses

  1. Referee: [Numerical sampling / Methods] The procedure used to generate the ensemble of random two-step qubit dynamics is not described with sufficient detail (distribution over Kraus operators or Choi matrices, handling of CPTP constraints, sampling algorithm, and statistical error control) to determine whether the reported Markovian fraction is an artifact of the chosen measure rather than a property of the space itself. This directly affects the central claim that 'almost all dynamics are non-Markovian'.

    Authors: We agree that the current description of the sampling procedure lacks sufficient detail for reproducibility and for readers to evaluate the underlying measure. In the revised manuscript we will expand the relevant section to specify: the precise distribution (uniform sampling with respect to the Hilbert-Schmidt volume on the convex set of two-step CPTP maps, realized via random Choi matrices projected onto the CPTP cone), the algorithm (generation of random Kraus operators followed by normalization and enforcement of trace preservation and complete positivity), and statistical controls (sample size, convergence checks, and bootstrap-derived error bars on the reported fractions). We will also explicitly state that all claims are with respect to this ensemble and discuss its relation to other natural measures on the space of quantum channels. These additions will directly address the concern without changing the numerical findings or the overall conclusion. revision: yes

Circularity Check

0 steps flagged

No circularity: prevalence estimates arise from direct Monte Carlo sampling.

full rationale

The paper computes fractions of Markovian vs. non-Markovian dynamics by generating random two-step qubit maps and applying witnesses directly to the samples. No derivation reduces a claimed result to a fitted parameter, self-definition, or self-citation chain; the output is the empirical count under the chosen ensemble. The representativeness of the sampling measure is an external modeling assumption rather than an internal logical reduction. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are described in the abstract; the work is framed as a numerical survey.

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discussion (0)

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    Examples where both concepts coincide We have seen that any dynamics with quantum mem- ory according to Eq. (14) necessarily also shows genuine backflow with respect to Eq. (13). Hence, there is a class of dynamics having both properties. One example for such a dynamics is given by the dynamicsD= (E AD,1) where1is the identity channel andE AD is the full ...

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    Differences between the concepts According to Ref. [8], any revival that occurs in the presence of an initially pure environment E can never show only noncausal information revival, genuine back- flow is necessary. Let us thus consider such a dynamics, which is given in terms of the two-qubit unitary USE(t) = exp −it 2 σz ⊗σ x .(B18) We consider the evolu...