{"id":"44e04b6e-209d-4303-b4fc-03b16e069c1f","arxiv_id":"2502.05010","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"Small violations of energy conservation in thermal operations can increase entanglement and correlation-based non-Markovianity measures, with first-order bounds derived for the change.","lead":"This paper defines three measures of non-Markovianity for quantum thermal operations and studies what happens when a small perturbation in the system Hamiltonian breaks perfect energy conservation. It reports that such perturbations can increase information backflow in some qubit and qutrit examples.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The maximized entanglement- and distance-based measures are invariant under the perturbation, so Propositions 1 and 3 do not support the claim that athermality enhances non-Markovianity.","rationale":"The reader's verdict identifies the same load-bearing flaw: the perturbation only re-expresses the initial state in a different basis while the same global unitary is used, so the maximized measures do not change. This is not a subtle problem but a direct consequence of the definitions. Because E_Lambda and D_Lambda are maxima over all system states, replacing |i> by |i'> is merely a change of parametrization of the same convex set L_S. Hence Propositions 1 and 3 prove an upper bound on zero, and the claimed 'enhancement' cannot be inferred from them. The numerical demonstrations are not evidence for the proposed measures: they fix a single initial state and switch to logarithmic negativity for the entanglement example, so they compute a different, unmaximized quantity. The paper's central claim is therefore unsupported as stated. The concrete test of computing the max-based measures directly would settle the issue, and based on the mathematical identity it would fail to show any enhancement.","tokens_in":26791,"tokens_out":2943,"duration_ms":31990,"concrete_test":"Recompute E_Lambda and E^epsilon_Lambda for the qubit example of Sec. V.A using the relative entropy of entanglement, without fixing the initial state, by performing the maximization over all rho_S in L_S required by Eqs. (8) and (9). The prediction is E^epsilon_Lambda = E_Lambda for all epsilon, because rho^epsilon_S = sum_{ij} P_{ij}|i'><j'| sweeps out the same set L_S as rho_S = sum_{ij} P_{ij}|i><j|. Similarly, compute D^epsilon_Lambda and D_Lambda from Eqs. (13)-(14) over the full state space; the result should be equality. If equality holds, the reported positive Delta E_N in Fig. 2 cannot occur for the proposed entanglement measure, and the central 'enhancement' claim is not supported.","verdict_should_be":"REJECT","load_bearing_attack":"The central claim depends on the proposed measures in Eqs. (8)-(9) and (13)-(14). In both cases, the measure is maximized over all system states. The perturbed input state is defined as rho^epsilon_S = sum_{ij} P_{ij} |i'><j'|, where {|i'>} is the eigenbasis of the perturbed Hamiltonian. Since {|i'>} is a complete basis, as the coefficients P_{ij} range over all valid density matrices, rho^epsilon_S ranges over the full set L_S of system states. Therefore the maximization in Eq. (9) is over exactly the same set as the maximization in Eq. (8). Moreover, the global unitary U_SB is the same in both cases, so the perturbed operation Lambda^epsilon is the same CPTP map as Lambda acting on a different label for the initial state. Consequently E^epsilon_Lambda = E_Lambda exactly, and analogously D^epsilon_Lambda = D_Lambda because the max over rho in Eq. (14) again ranges over all of L_S. Proposition 1 and Proposition 3 thus bound a quantity that is identically zero; they cannot establish any enhancement of non-Markovianity. The numerical examples avoid this issue by fixing a particular initial state (a = 0.9) and by using logarithmic negativity instead of the relative entropy of entanglement, so they do not test the proposed measures as defined. Additionally, the 'exact' response in Proposition 2 is only first-order in epsilon, and the proof of Proposition 1 uses a max/min manipulation that can invalidate the bound, but the invariance of the maximized measures is the decisive flaw: it breaks the paper's main claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes three measures of non-Markovianity for thermal operations (TO) and approximate thermal operations (TO_epsilon): an entanglement-based measure E_Lambda (Eq. 8-9), a mutual-information-based measure I_Lambda (Eq. 10-11), and a distance-based measure D_Lambda (Eq. 13-14). It claims that a small perturbation of the system Hamiltonian, leading to athermality, can enhance non-Markovianity, and it derives response formulas: Delta E_Lambda <= epsilon gamma_Lambda (Prop. 1), Delta I_Lambda = epsilon theta_Lambda (Prop. 2), and Delta D_Lambda <= (epsilon/d1) max ||chi_Lambda||_1 (Prop. 3). Numerical examples for qubit-qubit and qubit-qutrit systems are presented as evidence of perturbation-enhanced information backflow.","tokens_in":27054,"tokens_out":5731,"duration_ms":64785,"significance":"If the central claim were correct, the paper would establish a useful bridge between two quantum resource frameworks: athermality and non-Markovianity. The manuscript is clearly organized, and the appendices contain explicit derivations of the Markovianity constraints on thermal operations and approximate thermal operations. The authors also honestly include a counterexample for the distance-based measure, which indicates that the enhancement is not universal. However, the central claim is not supported by the proposed measures: the entanglement- and distance-based measures are maximized over all system states, and under the paper's own definition of the perturbed initial state this maximization set is unchanged by the perturbation. Consequently, Propositions 1 and 3 bound quantities that are identically zero, and the numerical examples test fixed initial states with different entanglement quantifiers rather than the proposed measures. The mutual-information result is only first order in epsilon despite being advertised as exact. These issues are load-bearing, so the paper cannot be accepted in its current form.","major_comments":[{"comment":"The claimed enhancement is not established because the two measures are equal by definition. The perturbed initial state is rho^epsilon_S = sum_{ij} P_{ij} |i'><j'|, and the set {|i'>} is a complete basis. As the coefficients P_{ij} range over all valid density matrices, rho^epsilon_S ranges over exactly the same set L_S as rho_S. Since the same global unitary U_SB appears in Eqs. (8) and (9), the two maximizations are over identical sets and E^epsilon_Lambda = E_Lambda, hence Delta E_Lambda = 0. Proposition 1 therefore gives an upper bound on a quantity that is identically zero. The numerical example in Fig. 2 sidesteps this by fixing a = 0.9 and using logarithmic negativity (Eq. 25) instead of the relative entropy of entanglement (Eq. 7), so it does not test the proposed measure E_Lambda.","section":"Sec. V.A, Eqs. (8)-(9), Prop. 1"},{"comment":"The same substitution argument applies to the outer maximization in D_Lambda and D^epsilon_Lambda. Since both definitions maximize over all input states and the same unitary U_SB is used, the operations Lambda and Lambda^epsilon are the same CPTP map applied to differently labeled input states; the maximization sets coincide. The proof's step in Eq. (40), replacing the minimization over Lambda^M_epsilon by a minimization over Lambda^M, requires the sets of maps to coincide, which is asserted on the basis of Appendix C but not established in the main text. Even granting that step, the bound (33) concerns a difference that is zero, and the numerical example in Sec. V.C explicitly finds no enhancement. Thus Proposition 3 does not support the paper's central claim of perturbation-enhanced non-Markovianity.","section":"Sec. V.C, Eqs. (13)-(14), Prop. 3"},{"comment":"The max/min manipulation in the proof is internally inconsistent. Equation (23) contains the term epsilon max_{Pij} max_sigma X_Lambda, but gamma_Lambda is later defined in Eq. (16) as max_{Pij} min_sigma X_Lambda. Since max max >= max min in general, the stated bound Delta E_Lambda <= epsilon gamma_Lambda does not follow from the derivation unless X_Lambda has special properties that are not proved. This is an independent technical error from the invariance issue, and it makes the provenance of the advertised bound unclear.","section":"Sec. V.A, proof of Prop. 1"},{"comment":"The physical modeling of the perturbation is problematic. The paper keeps the same global unitary U_SB that commutes with the unperturbed total Hamiltonian H_T, and only changes the system's initial state from rho_S = sum P_{ij}|i><j| to rho^epsilon_S = sum P_{ij}|i'><j'|. However, if the system Hamiltonian is perturbed to H_S + epsilon H', the total Hamiltonian changes to H_T + epsilon H' ⊗ I_B, so a unitary generated by the perturbed dynamics would not be the same U_SB. Using the unperturbed U_SB means that Lambda^epsilon is literally the same CPTP map as Lambda, with the input state expressed in a different basis. The numerical examples therefore compute the response to a change of initial state, not the response of the dynamics to athermality. This assumption undermines the claimed connection between athermality and enhanced information backflow.","section":"Sec. II.D and Sec. V"},{"comment":"The claim of an exact response is overstated. The proof of Proposition 2 uses first-order expansions of the von Neumann entropy, e.g., S(rho'^epsilon_S) = S(rho'_S) - epsilon Tr[beta_1(I + log rho'_S)], dropping terms of order epsilon^2. The result should be stated as Delta I_Lambda = epsilon theta_Lambda + O(epsilon^2), not as the exact equality in Eq. (26). The abstract's phrase 'we are able to compute the exact response' is therefore not supported by the presented derivation.","section":"Sec. V.B, Prop. 2"}],"minor_comments":[{"comment":"In the second summation of Eq. (17), the energy denominator should be E_j - E_l rather than E_i - E_k; the same typo appears in the expression for B in the proof.","section":"Eq. (17) and proof of Prop. 1"},{"comment":"The inset text mentions 'Delta I_N' where the main text uses Delta I; the notation should be made consistent.","section":"Fig. 3 caption"},{"comment":"The symbol I is used both for the identity operator and for the imaginary unit in the definition of U_SB; this creates ambiguity and should be disambiguated with different notations.","section":"Sec. V.A, first example"},{"comment":"The statement that unitaries corresponding to operations in X_M or X^M_epsilon 'ensure' the system and environment remain uncorrelated is a condition imposed on the unitaries, not an automatic property; the wording could be clarified.","section":"Sec. IV.B"}],"recommendation":"reject","confidential_remarks":"The paper falls within the journal's scope, but the central claim fails because the proposed maximized measures are invariant under the perturbation as defined, and the perturbed dynamics are not generated by the perturbed Hamiltonian. The invariance argument is simple and does not depend on the numerical details, so I do not see a fix within the manuscript's current framework."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThe headline result doesn't survive contact with the definitions. The entanglement- and distance-based non-Markovianity measures are maximized over all system states, and the perturbed channel uses the same global unitary as the unperturbed one. Since any density matrix can be written in the perturbed eigenbasis, the two maximizations run over identical sets. So E^epsilon_Lambda = E_Lambda and D^epsilon_Lambda = D_Lambda exactly; Propositions 1 and 3 bound zero. The examples don't rescue this—they fix a particular initial state and switch to logarithmic negativity for the entanglement example, so they're testing something other than the proposed measures. Proposition 2's 'exact' response is only first order in epsilon, which is a more minor but still real overstatement.\n\nWhat's genuinely new: the paper defines non-Markovianity measures adapted to thermal operations and approximate thermal operations, and connects the perturbation response to the perturbing Hamiltonian via first-order perturbation theory. The appendices deriving the parameter constraints for Markovian thermal operations are careful and useful. The topic—whether athermality can help generate non-Markovianity—is a reasonable question, and the paper is clearly written.\n\nThe soft spots are the ones above, and the first is load-bearing. I also noticed a secondary issue in the proof of Proposition 1: the derivation ends up with a max over sigma in the term, but the stated gamma_Lambda is defined with a min over sigma. That is an inconsistency, though it's overshadowed by the invariance problem. There is also the assumption that the perturbation only changes the initial-state labeling while the unitary stays the same; if the perturbation is meant to modify the global evolution, the whole analysis changes.\n\nThis paper is aimed at the quantum thermodynamics / open quantum systems community. I would not send it to peer review in its current form—the central claim is vacuous for the proposed measures. If the authors revise to use state-dependent measures or change the unitary under perturbation, it could be worth another look. As it stands, I would desk reject.\n\nBest,\n[Your name]","headline":"The paper's claim that athermality enhances non-Markovianity fails because the proposed maximized entanglement- and distance-based measures are invariant under the perturbation; the numerical examples test different quantities.","tokens_in":27653,"tokens_out":5234,"would_cite":false,"duration_ms":49819,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P45","81P40","81S22"],"pacs":[],"model":"deepseek-v4-flash","headline":"Perturbing a quantum system's Hamiltonian can enhance non-Markovianity; three new measures quantify the gain, with first-order bounds for entanglement and distance and an exact linear response for total correlation.","keywords":["non-Markovianity","thermal operations","approximate thermal operations","athermality","information backflow","system-environment entanglement","quantum mutual information","perturbation theory"],"falsifier":"Recompute the qubit-qubit example with the global unitary replaced by one that commutes with the perturbed total Hamiltonian $H_T + \\epsilon H'\\otimes I_B$, so the perturbation genuinely reshapes the dynamics instead of only shifting the initial state; if $\\Delta E_N$ turns negative or violates $\\Delta E_\\Lambda \\leq \\epsilon\\gamma_\\Lambda$, the enhancement is tied to the fixed-unitary construction. On the experimental side, the same check is a direct measurement of trace-distance revivals in a system whose Hamiltonian is perturbed while the system-bath interaction is held fixed.","tokens_in":26517,"feed_emoji":"🔄","tokens_out":16714,"duration_ms":148883,"temperature":0.7,"pith_summary":"This paper asks whether the unavoidable perturbation of a quantum system's Hamiltonian—the imperfection that turns an ideal thermal operation into an approximate one and breaks strict energy conservation—can change how non-Markovian the dynamics is, meaning how much information flows back from the environment into the system. The authors' answer is yes: they define three measures of non-Markovianity for thermal operations and their approximate counterparts, and prove that the perturbation-induced change is at most linear in the perturbation strength for entanglement- and distance-based measures and exactly linear for total correlation. In explicit qubit-qubit and qubit-qutrit examples, the perturbation increases the system-environment entanglement and mutual information generated during the evolution, so athermality can act as an enhancer of information backflow rather than a purely harmful effect. This matters in practice because non-Markovianity is a resource in quantum technologies while Hamiltonian perturbations are unavoidable in real devices; the paper thereby puts two useful notions into a quantitative relation.","feed_headline":"Small Hamiltonian defects can boost quantum information backflow","feed_subtitle":"Three new measures show imperfect thermal operations can generate more system-bath entanglement and correlations.","key_machinery":"The central object is the approximate thermal operation, $TO_\\epsilon$: a thermal operation generated by an energy-preserving unitary $U_{SB}$ acting on system plus thermal bath, but with the system Hamiltonian shifted to $H'_S = H_S + \\epsilon H'$, so that $[U_{SB}, H'_T] = \\epsilon[U_{SB}, H'\\otimes I_B]\\neq 0$ and the process no longer conserves total energy. The mechanism that carries the argument is the fixed-unitary, shifted-basis construction: the perturbed initial state is $\\rho_S^\\epsilon = \\sum_{ij} P_{ij}|i'\\rangle\\langle j'|$, with the first-order corrected eigenstates $|i'\\rangle = |i\\rangle + \\epsilon\\sum_{k\\neq i}(\\langle k|H'|i\\rangle/(E_i-E_k))|k\\rangle$, so that $\\rho_S^\\epsilon = \\rho_S + \\epsilon\\tilde{\\rho}$. Substituting this expansion into the relative entropy of entanglement, the quantum mutual information between system and environment, and the Choi-state trace distance between the operation and its nearest Markovian counterpart, then expanding to first order in $\\epsilon$, produces Propositions 1–3. The same perturbation theory shows that the parameter constraints characterizing approximate Markovian thermal operations coincide with those for the unperturbed Markovian thermal operations, which is what allows the distance-based bound to go through.","core_discovery":"The central claim is that a deviation from ideal thermal operations—athermality—can enhance non-Markovianity. The argument keeps the same global system-bath unitary $U_{SB}$ that generates the unperturbed thermal operation, and lets the perturbation act only by replacing the system's initial state with the same density matrix written in the first-order perturbed eigenbasis, $\\rho_S^\\epsilon = \\rho_S + \\epsilon\\tilde{\\rho}$. Expanding each measure to first order in $\\epsilon$ yields three relations: the entanglement-based measure changes by at most $\\Delta E_\\Lambda \\leq \\epsilon\\gamma_\\Lambda$; the total-correlation measure changes exactly linearly, $\\Delta I_\\Lambda = \\epsilon\\theta_\\Lambda$; and the distance-based measure changes by at most $\\Delta D_\\Lambda \\leq (\\epsilon/d_1)\\max_{\\Lambda^M}\\|\\chi_\\Lambda\\|_1$, where $\\gamma_\\Lambda$, $\\theta_\\Lambda$, and $\\chi_\\Lambda$ depend on the global unitary and the perturbing Hamiltonian $H'$. In the qubit-qubit example the entanglement-based difference $\\Delta E_N$ is positive and grows with the bath temperature, and in the qubit-qutrit example $\\Delta I$ is positive and grows as the temperature falls; the paper also finds a distance-based example where $\\Delta D$ stays near $10^{-4}$, showing the enhancement is not universal across measures.","pith_inferences":["The exact relation $\\Delta I_\\Lambda = \\epsilon\\theta_\\Lambda$ suggests a rate-like interpretation the paper leaves implicit: athermality of size $\\epsilon$ is converted into correlation-based non-Markovianity at a rate $\\theta_\\Lambda$, and asking whether $\\theta_\\Lambda$ can be negative for all perturbations would single out Hamiltonians that are insensitive to this effect.","The same fixed-unitary, shifted-basis template could be applied to other monotones of the thermal-operations resource theory, such as free energy or coherence, which would connect perturbation-enhanced non-Markovianity to the thermodynamic cost of the perturbation.","Because the construction deliberately keeps the global unitary fixed, a natural next step is to let the perturbation also rotate the unitary; if the enhancement survives that change, the conclusion extends beyond basis-shifted initial states to genuinely perturbed dynamics.","Since the enhanced quantities are one-time system-bath entanglement and mutual information, the effect could plausibly be detected in a single correlation measurement on system and bath after the interaction, rather than through full process tomography."],"forward_implications":["Any perturbation-induced gain in non-Markovianity measured by entanglement or by distance is at most linear in the perturbation strength $\\epsilon$, so departing further from ideal thermal operations does not buy unlimited enhancement.","For the total-correlation measure the response is exactly linear, $\\Delta I_\\Lambda = \\epsilon\\theta_\\Lambda$, so the sign of $\\theta_\\Lambda$—computed from the unitary, the bath, and the perturbing Hamiltonian—decides whether a given perturbation helps or hurts.","In the qubit-qubit example the entanglement-based gain grows with the bath temperature, while in the qubit-qutrit example the correlation-based gain grows as the temperature falls, making temperature a tuning knob for athermality-enhanced backflow.","Enhancement is not universal: in the qubit-qubit distance-based example the change stays at order $10^{-4}$, so some measures barely respond to the same perturbation.","Athermality can therefore act like 'order from disorder': an imperfection in the thermal operation, instead of merely degrading the resource, can strengthen the backflow of information from the environment."],"supporting_citations":[{"why":"Defines thermal operations as energy-preserving global unitaries on system plus thermal bath; this is the ideal operation that the perturbation in the paper disturbs.","marker":"[62–65]"},{"why":"Introduces approximate thermal operations arising from imperfect energy conservation; the paper's TO_epsilon measures are built on this set.","marker":"[66]"},{"why":"Supplies the transformation laws for diagonal and off-diagonal elements under thermal operations, from which the Markovian-thermal-operation parameter constraints are derived.","marker":"[82]"},{"why":"Defines the information-backflow notion of non-Markovianity (increasing trace distance between two evolving states) that motivates interpreting all three proposed quantities as measures of non-Markovianity.","marker":"[30, 36]"},{"why":"Provides the relative entropy of entanglement, the entanglement measure at the core of Proposition 1.","marker":"[85–87]"},{"why":"Supplies logarithmic negativity, the computable entanglement measure used in the qubit-qubit enhancement example.","marker":"[90, 91]"},{"why":"Gives the collisional-model setting in which the environment is refreshed at every step, which motivates the definition of Markovian thermal operations.","marker":"[80, 81]"},{"why":"Establishes the Choi-state representation of quantum maps, used to turn the distance between operations into a trace distance between states in Proposition 3.","marker":"[94]"},{"why":"Shows the nearest-Markovian-map geometric approach in Choi formalism that the distance-based measure is modeled on.","marker":"[95]"}],"fun_headline_variants":["Perturbations can amplify non-Markovianity","Small Hamiltonian defects can boost information backflow","Imperfect thermal ops can enhance system-bath correlations","New measures show athermality increases quantum memory","Qubit examples show defects boost non-Markovianity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire derivation assumes the perturbation changes only the system's initial state—rebuilt in the shifted eigenbasis—while the global system-bath unitary stays exactly the same as in the unperturbed thermal operation, and it assumes non-degenerate spectra so that first-order perturbation theory applies; if a real perturbation also modified the unitary itself, every bound and formula in the paper would need re-derivation.","fun_headline_variants_meta":{"raw":{"variants":["Perturbations can amplify non-Markovianity","Small Hamiltonian defects can boost information backflow","Imperfect thermal ops can enhance system-bath correlations","New measures show athermality increases quantum memory","Qubit examples show defects boost non-Markovianity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000984,"raw_usage":{"total_tokens":4217,"prompt_tokens":1032,"completion_tokens":3185,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":648,"completion_tokens_details":{"reasoning_tokens":3106}},"tokens_in":648,"tokens_out":3185,"duration_ms":22742,"temperature":1.0,"reasoning_tokens":3106,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T20:37:53.630255+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the qubit-qubit example with the global unitary replaced by one that commutes with the perturbed total Hamiltonian $H_T + \\epsilon H'\\otimes I_B$, so the perturbation genuinely reshapes the dynamics instead of only shifting the initial state; if $\\Delta E_N$ turns negative or violates $\\Delta E_\\Lambda \\leq \\epsilon\\gamma_\\Lambda$, the enhancement is tied to the fixed-unitary construction. On the experimental side, the same check is a direct measurement of trace-distance revivals in a system whose Hamiltonian is perturbed while the system-bath interaction is held fixed.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces approximate thermal operations arising from imperfect energy conservation; the paper's TO_epsilon measures are built on this set."},{"cited_title":"Quantum collision models: Open system dynam- ics from repeated interactions,","cited_arxiv_id":null,"evidence_quote":"Supplies the transformation laws for diagonal and off-diagonal elements under thermal operations, from which the Markovian-thermal-operation parameter constraints are derived."},{"cited_title":"Classical, quantum and to- tal correlations,","cited_arxiv_id":null,"evidence_quote":"Establishes the Choi-state representation of quantum maps, used to turn the distance between operations into a trace distance between states in Proposition 3."},{"cited_title":"Linear transformations which preserve trace and positive semidefiniteness of operators,","cited_arxiv_id":null,"evidence_quote":"Shows the nearest-Markovian-map geometric approach in Choi formalism that the distance-based measure is modeled on."}],"review_version":1}