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arxiv: 2605.26818 · v1 · pith:ENFMLY5Onew · submitted 2026-05-26 · 🪐 quant-ph · cond-mat.stat-mech

Nonclassical energy-change distribution as a witness of non-Markovian quantum dynamics

Marco Pezzutto , Anton Corr , Gabriele De Chiara , Salvatore Lorenzo , Stefano Gherardini This is my paper

Pith reviewed 2026-06-29 17:20 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords non-MarkovianityCP-divisibilityKirkwood-Dirac quasiprobabilityenergy-change distributionopen quantum systemsanomalous energy fluxesquantum witness
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The pith

Negative values in the energy-change distribution always witness violations of CP-divisibility.

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

The paper establishes that non-Markovian quantum dynamics of an open system can be identified using only measurements of the system's energy. It demonstrates that any violation of CP-divisibility produces non-positive values in the energy-change Kirkwood-Dirac quasiprobability distribution evaluated at consecutive times. The connection strengthens when system-environment interactions preserve energy. A sympathetic reader would care because this supplies a direct experimental signature for memory effects without requiring full quantum state tomography or access to the environment.

Core claim

Violations of CP-divisibility are always witnessed by non-positive values of the energy-change Kirkwood-Dirac quasiprobability distribution associated with the system's Hamiltonian, evaluated at consecutive times. The link between non CP-divisibility and non-positivity of the system's energy-change distribution is stronger when the system-environment interactions are energy-preserving. The witness works whenever anomalous energy fluxes, due to non-Markovianity, are realized. Anomalous fluxes are also detected by the non-Markovianity measure built over the quantum mutual information between the states of the open system and of a quantum correlated reference.

What carries the argument

the energy-change Kirkwood-Dirac quasiprobability distribution associated with the system's Hamiltonian

Load-bearing premise

The dynamics must realize anomalous energy fluxes due to non-Markovianity, which occurs more readily when interactions are energy-preserving.

What would settle it

An experiment realizing a CP-divisibility violation while the energy-change Kirkwood-Dirac distribution remains non-negative at every pair of consecutive times would refute the witness.

Figures

Figures reproduced from arXiv: 2605.26818 by Anton Corr, Gabriele De Chiara, Marco Pezzutto, Salvatore Lorenzo, Stefano Gherardini.

Figure 1
Figure 1. Figure 1: , we show the capability of the non-positive functional Nq[Pn(uS )] to detect violations of CP-divisibility, for a cool￾ing process modeled by a memory-mediated collision model. The system’s initial state is the completely mixed state I/2, while the memory and environment particles are initialized 0 20 40 60 80 0.10 0.05 0.00 0.05 0.10 0.15 0.20 n 76 77 78 79 80 0.005 0.000 0.005 38 39 40 41 n 0.005 0.000 … view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Energy-preserving interactions. Comparison of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Non energy-preserving interactions. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Comparison of [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
read the original abstract

We address the problem of identifying non-Markovian quantum time evolutions of an open quantum system by only performing measurements of the system's energy. We demonstrate that violations of CP-divisibility are always witnessed by non-positive values of the energy-change Kirkwood-Dirac quasiprobability distribution associated with the system's Hamiltonian, evaluated at consecutive times. The link between non CP-divisibility and non-positivity of the system's energy-change distribution is stronger when the system-environment interactions are energy-preserving. The witness works whenever anomalous energy fluxes, due to non-Markovianity, are realized. Anomalous fluxes are also detected by the non-Markovianity measure built over the quantum mutual information between the states of the open system and of a quantum correlated reference.

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

2 major / 1 minor

Summary. The manuscript claims that violations of CP-divisibility in the dynamics of an open quantum system are always witnessed by non-positive values of the energy-change Kirkwood-Dirac quasiprobability distribution associated with the system Hamiltonian, evaluated at consecutive times. The witness is effective whenever anomalous energy fluxes due to non-Markovianity occur and the connection is stronger when system-environment interactions are energy-preserving. The authors also construct a non-Markovianity measure from the quantum mutual information between the system and a correlated reference.

Significance. If the central link holds under the stated conditions, the result supplies a witness for non-Markovianity that requires only energy measurements on the system, avoiding full tomography. This is potentially useful for experimental platforms in which energy is the accessible observable. The use of Kirkwood-Dirac quasiprobabilities to connect nonclassical energy statistics to divisibility violations adds a new diagnostic tool in the study of open-system memory effects.

major comments (2)
  1. [Abstract] Abstract: the opening claim that CP-divisibility violations 'are always witnessed' by negativity of the energy-change KD distribution is immediately qualified by the statements that the witness 'works whenever anomalous energy fluxes... are realized' and is stronger for energy-preserving interactions. This tension must be resolved by an explicit statement, early in the introduction, of the precise conditions under which the witness is guaranteed to function.
  2. [Main derivation] Derivation of the witness (likely §3 or §4): the link between KD negativity and CP-divisibility violation appears to invoke either the existence of anomalous fluxes or the commutation [H_int, H_S] = 0. The manuscript should supply the full derivation steps, state the assumptions explicitly, and either prove that the witness is tight under those assumptions or exhibit a counter-example in which CP-divisibility is violated without producing KD negativity.
minor comments (1)
  1. [Abstract] Abstract and introduction: the phrase 'energy-change Kirkwood-Dirac quasiprobability distribution' should be accompanied by a brief definition or a forward reference to its precise mathematical definition on first appearance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify the scope and presentation of our results. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the opening claim that CP-divisibility violations 'are always witnessed' by negativity of the energy-change KD distribution is immediately qualified by the statements that the witness 'works whenever anomalous energy fluxes... are realized' and is stronger for energy-preserving interactions. This tension must be resolved by an explicit statement, early in the introduction, of the precise conditions under which the witness is guaranteed to function.

    Authors: We agree that the abstract and opening paragraphs create an apparent tension between the unqualified claim of 'always witnessed' and the subsequent qualifications. In the revised manuscript we will insert, immediately after the first paragraph of the introduction, an explicit statement of the precise conditions: the witness is guaranteed whenever anomalous energy fluxes due to non-Markovianity occur, and the link is strengthened (but not required) when the interaction Hamiltonian commutes with the system Hamiltonian. This removes the ambiguity while preserving the original claims under the stated conditions. revision: yes

  2. Referee: [Main derivation] Derivation of the witness (likely §3 or §4): the link between KD negativity and CP-divisibility violation appears to invoke either the existence of anomalous fluxes or the commutation [H_int, H_S] = 0. The manuscript should supply the full derivation steps, state the assumptions explicitly, and either prove that the witness is tight under those assumptions or exhibit a counter-example in which CP-divisibility is violated without producing KD negativity.

    Authors: We will expand the derivation (currently in §§3–4) to include every intermediate step, explicitly list all assumptions (including the occurrence of anomalous fluxes and the optional commutation condition), and prove that the witness is tight under these assumptions: any CP-divisibility violation that produces an anomalous energy flux necessarily yields a negative value of the energy-change KD quasiprobability. The proof follows directly from the relation between the sign of the quasiprobability and the sign of the energy flux together with the definition of CP-divisibility; the expanded section will make this chain fully transparent. revision: yes

Circularity Check

0 steps flagged

No circularity: witness derived from standard channel properties and KD quasiprobability definition

full rationale

The paper's central claim links CP-divisibility violations to negativity in the energy-change Kirkwood-Dirac distribution via the definition of the quasiprobability and properties of quantum channels. No step reduces by construction to a fitted parameter renamed as prediction, a self-citation chain, or an ansatz smuggled from prior work by the same authors. The explicit qualification that the witness applies when anomalous fluxes occur is stated as a regime condition rather than a hidden definitional equivalence. The derivation remains self-contained against external benchmarks of CP-divisibility and quasiprobability theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, ad-hoc axioms, or new entities are introduced in the provided text. The claim rests on standard notions of CP-divisibility and quasiprobability distributions.

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Forward citations

Cited by 1 Pith paper

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    Then, we consider the following projectors on some of these eigenstates: P0 = |0⟩ ⟨0| = 1 0 0 0 ! ,P 1 = |1⟩ ⟨1| = 0 0 0 1 ! , P+ = |+⟩ ⟨+| = 1 2 1 1 1 1 ! ,P R = |R⟩ ⟨R| = 1 2 1−i i1 ! , (A.27) from which one ends up to the following identities: I=P 0 +P 1, σ x =2P + −(P 0 +P 1), σy =2P R −(P 0 +P 1), σ z =P 0 −P 1.(A.28) Thanks to the linearity of the m...

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    tl” stands for “time-local

    Construct the vectors{ ⃗cn}, Eq. (A.25) and matrices {Mn}, Eq. (A.26) for alln. In this way, we can compute the evolved state via Eqs. (A.21)- (A.23), initializing the system in an arbitrary input state. Inverted map AssumingΛhas been successfully reconstructed, we show how to deriveΛ −1. As in (A.23), we can express the action of Λ−1 n on a generic (evol...

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