{"id":"ff96ea40-6a20-4cb3-9c3f-1a11c4bc690c","arxiv_id":"2608.10118","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Negative stochastic entropy production in quantum trajectories is bounded below by a sharp function of the mean completed entropy, so apparent second-law violations cannot become rarer than a universal floor.","lead":"This paper proves a universal lower bound on how often quantum trajectories show negative entropy production, the apparent violations of the second law. The bound is controlled by a completed entropy that includes a dynamical asymmetry term, and it makes negative events frequent but mild near reversibility.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; Eq. (5) is the key imported assumption, with a minor misstatement in the median bound of Eq. (21).","rationale":"The reader's weakest assumption identifies Eq. (5) as the key external input, which is indeed the step imported from Ref. [11]. We partially agree: the reader is correct that within the formal definition of σ via the backward law, reversal-oddness and the identification with the boundary expression rely on this identity, and that failure of the identity would undermine the physical IFT and severity bounds. However, we judge this assumption as standard and likely validated in the cited construction and supplemental material; the central frequency bound is also more robust than the reader's phrasing suggests, since in the measured-record protocol σ defined via Eq. (11) is reversal-odd by construction regardless of Eq. (5). The only concrete error found is the median bound in Eq. (21), which should be ln(2/L) rather than (ln 2)/L; this is a minor non-central slip. On balance the paper's main theorem and its numerical support are sound, and the reader's ACCEPT verdict stands.","tokens_in":12041,"tokens_out":27074,"duration_ms":243609,"concrete_test":"Verify Eq. (5) directly in the random finite-coupling ensemble: explicitly construct the physically backward protocol (reversed unitary sequence, swapped ancilla labeling, initial system state r_m, ancilla states q_μ) and compute P~(γ) for every record; then check P~(γ)P~(γ′)=P(γ)P(γ′) for all 4d_E^2 records in a sample of realizations. If any violation appears, test whether the frequency bound Πσ≥L(⟨Ω⟩) still holds with σ defined via Eq. (11); if it does, the severity bounds (20)–(21) would need separate re-derivation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing flaw found in the central claim. The inequality chain Πσ≥ΠΩ≥L(⟨Ω⟩) follows from (i) reversal-oddness of σ, (ii) the sign-optimality inequality (7), and (iii) the Jensen-based floor for Ω; each step is explicitly derived. The only external input is the product identity P~(γ′)P~(γ)=P(γ)P(γ′) (Eq. 5), imported from Ref. [11] and stated to be proven in the Supplemental Material. This identity is needed to make the physically-defined σ (via the backward law) reversal-odd and to connect the boundary expression (11) with the IFT-satisfying definition; within the arbitrary-coupling construction it is a standard microreversibility condition. The numerical applications are exact enumerations and consistently satisfy the theorem. A minor issue: Eq. (21) states median(−σ|σ<0)≤(ln 2)/L, but solving e^{-m}/L≤1/2 gives m≥ln(2/L); the correct upper bound on the median is ln(2/L). This is a non-central algebraic slip and does not affect the main frequency bound.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives universal bounds on the frequency of negative physical entropy production (\"apparent second-law violations\") in monitored quantum trajectories. It works with the completed entropy Ω=ln(P/P'), the log-likelihood ratio against the reversed record; the physical entropy σ=ln(P/\\tilde P'); and their difference σ*=Ω−σ. The main theorem is the chain Π_σ ≥ Π_Ω ≥ L(⟨Ω⟩), where Π_X = P(X<0)+P(X=0)/2 and L(⟨Ω⟩)=[1−⟨Ω⟩/g(⟨Ω⟩)]/2 with g the inverse of a↦a tanh(a/2). The proof combines (i) reversal-oddness of σ under the microreversibility identity \\tilde P(γ')\\tilde P(γ)=P(γ)P(γ') (Eq. 5), (ii) sign optimality of the likelihood ratio among reversal-odd observables (Eq. 7), and (iii) the DFT floor for Ω (Eq. 6 plus Jensen). This yields Eq. (3) and the two-sided form Eq. (9). The paper also derives a frequency–severity law (Eqs. 20–21) and an inference witness for the hidden dynamical-asymmetry term Σ* (Eq. 22). Applications are exact enumerations over random finite-coupling collision models and a coherently driven qubit with thermal ancillas, all consistent with the bounds. The central derivation is clean; the only external input is Eq. (5), with proof deferred to the Supplemental Material.","tokens_in":12235,"tokens_out":10216,"duration_ms":90371,"significance":"If correct, the result is significant: it extends sharp fluctuation-theorem control to the sign frequency of physical entropy production in regimes where σ itself has no forward detailed fluctuation theorem, and it identifies the completed entropy ⟨Ω⟩ as the controlling cost. The bound is parameter-free and sharp, and the frequency–severity law converts the physical integral fluctuation theorem into a quantitative \"frequent but mild\" statement. The sign-imbalance witness provides an operational lower bound on Σ* that is testable from paired record frequencies. The paper is unusually transparent: Eq. (7) is a short self-contained proof, Eq. (6) plus Jensen is explicit, and the numerics are exact enumerations with no fitted parameters or post-hoc exclusions. I found no load-bearing flaw; the main proof chain is sound.","major_comments":[],"minor_comments":[{"comment":"The displayed conditional tail bound P(−σ≥a|σ<0) ≤ e^{-a}/L yields, upon setting e^{-a}/L = 1/2, the upper bound median(−σ|σ<0) ≤ ln(2/L). The stated bound with (ln 2)/L is weaker, though still true for L≤1/2 because ln(2/L) ≤ (ln 2)/L in that range; please state the tighter form or at least correct the derivation to avoid the appearance of an algebraic slip.","section":"Eq. (21)"},{"comment":"The function h(a)=a tanh(a/2) is invoked in ψ(E[h(A)]) but is never explicitly defined in the main text; please define h before Eq. (6) so that the Jensen step is unambiguous.","section":"Eq. (6) and surrounding text"},{"comment":"The product identity in Eq. (5) is the only nontrivial imported assumption, and its pathwise proof is deferred to the Supplemental Material. Because the central theorem rests on this identity, please clarify in the main text whether the proof is taken verbatim from Ref. [11] or derived in the Supplement, and ensure the Supplement is self-contained.","section":"Formalism, Eq. (5)"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is within the journal's scope and the central claim is sound. The only issues I found are local: an algebraic/derivation point in Eq. (21), an undefined h in the Jensen step, and some reliance on Supplemental Material for the proof of Eq. (5) and for equality/sharpness. These do not undermine the main theorem. I recommend minor revision. If the Supplemental Material is not available to all referees, the author should either include the proof of Eq. (5) in the main text or give a precise pointer to where it appears in Ref. [11]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know: this paper proves Πσ ≥ ΠΩ ≥ L(⟨Ω⟩), i.e. the tie-corrected frequency of negative physical entropy production is bounded below by a universal function of the mean completed entropy. The transfer from Ω to σ via sign-optimality of the likelihood ratio is genuinely new, and the proof is transparent. I think the main theorem is correct.\n\nWhat's new: Eq. (7) shows that among all reversal-odd observables, the sign of Ω minimizes the probability of negative sign; this lets the established DFT floor for Ω (Ref. [29]) carry over to σ even when σ itself has no forward DFT. The frequency–severity law (Eq. (20)) and the Σ* witness (Eq. (22)) are useful consequences. The paper is honest about what it does not do: there is no Σ-only floor, and physical sharpness of the upper branch is left open.\n\nWhat it does well: the derivation is self-contained apart from the product identity (Eq. (5)) imported from Ref. [11]; that identity is the load-bearing external assumption. The numerical applications are exact enumerations over all records, with no fitted parameters, and they include cases where Πσ exceeds 1/2, which is a nice demonstration that the DFT ceiling does not transfer. The citation pattern looks appropriate; the self-citations are to prior work that this paper genuinely builds on.\n\nSoft spots: the median bound in Eq. (21) is wrong as stated. The paper asserts median(−σ|σ<0) ≤ ln 2 / L, but solving e^{−m}/L ≤ 1/2 gives m ≥ ln(2/L), so the correct upper bound is ln(2/L). This is a minor algebraic slip and does not affect the main frequency bound. A referee should also check the equality and sharpness claims in the Supplemental Material, since the main text defers them, but nothing in the text suggests they are in trouble.\n\nBottom line: this is a solid within-subfield theorem. It is not a paradigm shift, but it answers a well-defined question and the answer is likely right. I would send it to peer review; after the median bound is fixed, I'd accept. If you work on quantum trajectory thermodynamics or fluctuation theorems, worth citing.","headline":"A clean proof of a new sign-frequency bound for quantum trajectories; the transfer step is the real contribution, and the paper deserves refereeing despite a minor slip in one median bound.","tokens_in":12803,"tokens_out":2607,"would_cite":true,"duration_ms":22870,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Negative entropy events in quantum trajectories obey a universal probability floor","keywords":["quantum trajectories","stochastic entropy production","detailed fluctuation theorem","second-law violations","completed entropy","collision models","sign statistics","integral fluctuation theorem"],"falsifier":"Enumerate every record in a finite-collision model that satisfies the product identity of Eq. (5) and the physical integral fluctuation theorem, and search for a model with $P(\\sigma\\le 0)<[1-\\langle\\Omega\\rangle/g(\\langle\\Omega\\rangle)]/2$; finding one would refute the universal floor. Equivalently, run the paper's random-collision ensemble but with a deterministic backward protocol that is not the time-reversal partner (for instance, reversing only the time order of ancilla interactions but not their state preparations) and check whether $\\Pi_\\sigma$ drops below $L(\\langle\\Omega\\rangle)$.","tokens_in":11827,"feed_emoji":"⚛️","tokens_out":6174,"duration_ms":54227,"temperature":0.7,"pith_summary":"This paper proves a quantitative lower bound on the probability that a quantum trajectory produces negative physical entropy, the event usually called an apparent violation of the second law. Even when physical entropy production $\\sigma$ does not obey a detailed fluctuation theorem, the paper shows that its sign statistics are controlled by a completed entropy $\\Omega$ that does obey one. The main inequality is $\\Pi_\\sigma \\ge \\Pi_\\Omega \\ge L(\\langle\\Omega\\rangle)$, with $L(\\langle\\Omega\\rangle)=[1-\\langle\\Omega\\rangle/g(\\langle\\Omega\\rangle)]/2$, so in particular the probability of $\\sigma\\le 0$ is at least that floor. The result makes apparent violations a predictable, frequency-bounded feature rather than an unexplained exception, and it turns measured sign statistics into a witness for hidden dynamical asymmetry.","feed_headline":"Apparent second-law violations get a universal probability floor","feed_subtitle":"Even without a detailed fluctuation theorem, negative entropy events are bounded below by a function of the completed entropy.","key_machinery":"The load-bearing object is the completed entropy production $\\Omega(\\gamma)=\\ln[P(\\gamma)/P(\\gamma')]$, the log-likelihood ratio of a record against its time-reversed record under the same forward law, which decomposes as $\\Omega=\\sigma+\\sigma^*$ with $\\sigma^*$ the forward-backward dynamical-asymmetry term. The argument relies on the product identity $\\tilde P(\\gamma')\\tilde P(\\gamma)=P(\\gamma)P(\\gamma')$ linking forward and physically backward laws, which makes $\\sigma$ reversal-odd and gives $\\Omega$ its detailed fluctuation theorem. The transfer step is an inequality on signs: for any reversal-odd observable $X$, $\\langle\\operatorname{sgn} X\\rangle\\le\\langle\\operatorname{sgn}\\Omega\\rangle$, with the difference equal to a distinguishability-weighted penalty for sign disagreement; this proves $\\Pi_\\sigma\\ge\\Pi_\\Omega$. The frequency floor $\\Pi_\\Omega\\ge L(\\langle\\Omega\\rangle)$ comes from applying Jensen's inequality to the concave function $\\psi(y)=\\tanh[g(y)/2]$ in the DFT relation, with equality only for a binary flipped-coin law.","core_discovery":"The discovery is a transfer theorem: the tie-corrected sign statistic $\\Pi_\\sigma=P(\\sigma<0)+P(\\sigma=0)/2$ for physical entropy production is bounded below by the same sharp floor that the completed entropy $\\Omega=\\ln[P(\\gamma)/P(\\gamma')]$ satisfies via its detailed fluctuation theorem. The proof shows that the sign of $\\Omega$ is optimal among all reversal-odd trajectory observables, so $\\Pi_\\sigma\\ge\\Pi_\\Omega$, and the DFT for $\\Omega$ gives $\\Pi_\\Omega\\ge L(\\langle\\Omega\\rangle)$ where $L(\\langle\\Omega\\rangle)=[1-\\langle\\Omega\\rangle/g(\\langle\\Omega\\rangle)]/2$ and $g$ inverts $a\\mapsto a\\tanh(a/2)$. Consequently $P(\\sigma\\le 0)\\ge L(\\langle\\Omega\\rangle)$, and the two-sided form $1-L(\\langle\\Omega\\rangle)\\ge\\Pi_\\sigma\\ge L(\\langle\\Omega\\rangle)$ holds under the same assumptions. Combining this floor with the integral fluctuation theorem $\\langle e^{-\\sigma}\\rangle=1$ yields a \"frequent but mild\" law: negative $\\sigma$ events are bounded below in frequency but their conditional magnitude is exponentially suppressed, and the observed sign imbalance certifies a minimum hidden mean $\\Sigma^*$.","pith_inferences":["The sign-optimality result means the probability of apparent second-law violation is at least the error of the optimal equal-prior classifier deciding whether a record came from the forward or reversed law, suggesting fluctuation-theorem sign bounds can be read as thermodynamic performance guarantees for an arrow-of-time decision rule.","In experiments that already measure endpoint populations and ancilla energy changes, the inequality turns a simple count of negative-$\\sigma$ records into an estimate of the unmeasured dynamical asymmetry $\\Sigma^*$, offering a probe of hidden driving or non-Markovianity without full trajectory reconstruction.","Near reversibility the two-sided bound forces both signs of $\\sigma$ close to equal frequency, so a measured imbalance away from $1/2$ implies a completed-entropy cost at least proportional to the square of that imbalance; this could serve as a model-free consistency check for entropy estimators.","The paper leaves finite-sample confidence bounds on $q_\\sigma$ open; deriving them from binomial tail inequalities would make the sign-witness inequality directly applicable to experimental runs of finite length."],"forward_implications":["Negative physical entropy production cannot become rare faster than the square root of the completed dissipation near reversibility: $L(\\langle\\Omega\\rangle)=1/2-\\sqrt{\\langle\\Omega\\rangle/8}+O(\\langle\\Omega\\rangle^{3/2})$.","When $\\sigma$ has no forward detailed fluctuation theorem, the completed entropy still supplies a sharp sign floor, so the bound applies in arbitrary-coupling and driven settings including the coherently driven qubit model.","The integral fluctuation theorem converts the floor into a quantitative \"frequent but mild\" law: $P(-a<\\sigma<0)+\\tfrac12 P(\\sigma=0)\\ge[L(\\langle\\Omega\\rangle)-e^{-a}]_+$, and the conditional severity satisfies $P(-\\sigma\\ge a\\mid\\sigma<0)\\le e^{-a}/L(\\langle\\Omega\\rangle)$.","A measured sign imbalance $q_\\sigma$ certifies a minimum hidden asymmetry: $\\langle\\Omega\\rangle\\ge I(q_\\sigma)$ and $\\Sigma^*\\ge[I(q_\\sigma)-\\Sigma]_+$, using only sign statistics plus the mean physical entropy.","The bound extends to continuous monitoring whenever the forward path measure and its composition with record reversal are mutually absolutely continuous."],"supporting_citations":[{"why":"Supplies the arbitrary-coupling quantum-trajectory construction and the product identity linking forward and physically backward laws, which yields the decomposition $\\Omega=\\sigma+\\sigma^*$.","marker":"[11]"},{"why":"Establishes the sharp frequency floor $\\Pi_\\Omega\\ge L(\\langle\\Omega\\rangle)$ for a variable satisfying the detailed fluctuation theorem, the floor that is transferred to $\\Pi_\\sigma$.","marker":"[29]"},{"why":"Provides the involution-based thermodynamic uncertainty relation whose reversal-odd current framework the sign-optimality theorem extends.","marker":"[26]"},{"why":"Defines stochastic entropy production along a trajectory and supplies the integral fluctuation theorem $\\langle e^{-\\sigma}\\rangle=1$ used for the frequency-severity law.","marker":"[41]"},{"why":"Gives the finite-time probabilistic-violation context and the standard Markov/IET ceiling $P(\\sigma\\le -a)\\le e^{-a}$ combined with the new floor.","marker":"[63]"}],"fun_headline_variants":["Quantum entropy dips get a hard probability floor","Apparent second-law breaks now have a proven bound","Negative entropy events: bounded in frequency, mild in size","Quantum trajectories: negative entropy is frequent but mild","Universal floor on apparent entropy violations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The bound collapses unless the forward trajectory law and the physically backward trajectory law are linked by the product identity $\\tilde P(\\gamma')\\tilde P(\\gamma)=P(\\gamma)P(\\gamma')$, meaning the backward protocol must be the exact time-reversal of the forward one in the sense that the completed entropy $\\Omega$ is a proper log-likelihood ratio; if that identity fails, $\\sigma$ need not be reversal-odd and the transfer $\\Pi_\\sigma\\ge\\Pi_\\Omega$ has no basis.","fun_headline_variants_meta":{"raw":{"variants":["Quantum entropy dips get a hard probability floor","Apparent second-law breaks now have a proven bound","Negative entropy events: bounded in frequency, mild in size","Quantum trajectories: negative entropy is frequent but mild","Universal floor on apparent entropy violations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000642,"raw_usage":{"total_tokens":3010,"prompt_tokens":1057,"completion_tokens":1953,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":673,"completion_tokens_details":{"reasoning_tokens":1883}},"tokens_in":673,"tokens_out":1953,"duration_ms":14795,"temperature":1.0,"reasoning_tokens":1883,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:13:26.919646+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate every record in a finite-collision model that satisfies the product identity of Eq. (5) and the physical integral fluctuation theorem, and search for a model with $P(\\sigma\\le 0)<[1-\\langle\\Omega\\rangle/g(\\langle\\Omega\\rangle)]/2$; finding one would refute the universal floor. Equivalently, run the paper's random-collision ensemble but with a deterministic backward protocol that is not the time-reversal partner (for instance, reversing only the time order of ancilla interactions but not their state preparations) and check whether $\\Pi_\\sigma$ drops below $L(\\langle\\Omega\\rangle)$.","supporting_citations":[{"cited_title":"Van Vu, V","cited_arxiv_id":null,"evidence_quote":"Establishes the sharp frequency floor $\\Pi_\\Omega\\ge L(\\langle\\Omega\\rangle)$ for a variable satisfying the detailed fluctuation theorem, the floor that is transferred to $\\Pi_\\sigma$."},{"cited_title":"Manzano and R","cited_arxiv_id":null,"evidence_quote":"Provides the involution-based thermodynamic uncertainty relation whose reversal-odd current framework the sign-optimality theorem extends."},{"cited_title":"Campisi, P","cited_arxiv_id":null,"evidence_quote":"Defines stochastic entropy production along a trajectory and supplies the integral fluctuation theorem $\\langle e^{-\\sigma}\\rangle=1$ used for the frequency-severity law."},{"cited_title":"Maillet, P","cited_arxiv_id":null,"evidence_quote":"Gives the finite-time probabilistic-violation context and the standard Markov/IET ceiling $P(\\sigma\\le -a)\\le e^{-a}$ combined with the new floor."}],"review_version":1}