{"id":"ef3686b2-3789-4b5d-8329-198f3b6e5a66","arxiv_id":"2607.06190","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Thermodynamic, kinetic, and thermokinetic uncertainty relations are derived for arbitrary currents in coherent transport, remaining valid far from equilibrium and in superconducting hybrid structures.","lead":"This paper derives thermodynamic and kinetic uncertainty relations that bound the precision of any transported quantity (charge, energy, quasiparticles) in coherent quantum conductors, including superconducting junctions. It matters because previous bounds only applied to particle currents and broke down in hybrid superconducting devices.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The involution existence and its compatibility with the FCS probability distribution are assumed without proof, but the TUR derivation's physical meaningfulness depends on it.","rationale":"The reader correctly identifies the noninteracting/FCS limitation as the primary scope boundary, and this is a real and acknowledged constraint. However, the reader treats the involution existence as a minor weakness ('the involution existence is assumed rather than proven in full generality'), when it is in fact load-bearing for the central claim of 'arbitrary currents.' The paper's headline contribution is extending uncertainty relations beyond particle currents to arbitrary Q. If the involution cannot be constructed for some physically relevant Q, the word 'arbitrary' is too strong and the result should be scoped to 'quantities Q for which an antisymmetrizing involution exists.' The mathematical derivations themselves (Cauchy-Schwarz applications, mapping between p+ and p) are sound given the involution exists. The examples (NN and NS setups) are well-chosen and the numerical demonstrations are convincing within their scope. The noninteracting limitation is clearly acknowledged and does not undermine the paper's claims within its stated domain. But the involution gap is between what is proven (bounds hold given an involution exists) and what is claimed (bounds hold for arbitrary currents). This warrants a CONDITIONAL verdict: accept the mathematical results as correct, but flag that the 'arbitrary' claim requires either a proof of involution existence or a restatement of scope. The reader's HIGH confidence and ACCEPT verdict should be tempered to reflect this gap, though the core mathematical contribution is solid.","tokens_in":10864,"tokens_out":796,"duration_ms":281285,"concrete_test":"Construct an explicit counterexample: a multi-terminal normal-superconducting setup with a specific observable Q (e.g., a linear combination of energy and charge currents) for which no involution on the FCS sample space simultaneously satisfies $q(omega^ddagger)=-q(omega)$ and yields $~sigma$ equal to the thermodynamic entropy production. If such a Q exists, the claim of 'arbitrary currents' is overstated. Alternatively, prove constructively that for any antisymmetric q on a discrete FCS sample space, such an involution always exists.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The TUR [Eq. (8)] and TKUR [Eq. (17)] both rely on the forward-backward entropy $~sigma(omega) = log[p(omega)/p(omega^ddagger)]$ defined via an involution $omega^ddagger$ satisfying $q(omega^ddagger) = -q(omega)$ [Eq. (3)]. The paper states 'we first consider an involution on the sample space' without proving that such an involution exists for arbitrary transported quantities Q in arbitrary multi-terminal or superconducting setups. For the NN example, the global reversal $(n_1,...,n_r)^ddagger = (-n_1,...,-n_r)$ works for particle number, but for a general observable q (e.g., energy current in a multi-terminal setup with Andreev processes), it is not obvious that an involution exists on the FCS sample space that simultaneously (i) is an involution, (ii) makes q antisymmetric, and (iii) yields a $~sigma$ that coincides with thermodynamic entropy production in the time-reversal-symmetric case. If no such involution exists for some Q, the fluctuation theorem [Eq. (4a)] used in the Cauchy-Schwarz argument [Eq. (6)] does not hold, and the bound [Eq. (8)] is not guaranteed. The random-observable sampling in Fig. 2(c,d) provides empirical evidence for specific BTK setups but does not constitute a proof of existence in general.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This Letter derives thermodynamic, kinetic, and thermokinetic uncertainty relations (TUR, KUR, TKUR) for arbitrary transported quantities in coherent, strongly-coupled, linear systems out of equilibrium. The central results are Eqs. (8), (13), and (17). The approach starts from the Levitov-Lesovik full counting statistics framework [Eq. (2)], defines a forward-backward stochastic entropy via an involution on the sample space [Eq. (4a)], and applies Cauchy-Schwarz to obtain bounds on the precision of any current. The near-equilibrium [Eqs. (9, 14)] and weak-coupling limits recover known classical results. The framework is illustrated in normal-metal (NN) and normal-superconducting (NS) hybrid setups, demonstrating that the bounds remain valid far from equilibrium and in superconducting structures where previous particle-current-specific bounds are violated.","tokens_in":11475,"tokens_out":941,"duration_ms":277207,"significance":"The main contribution is the generalization of uncertainty relations from particle currents to arbitrary transported quantities (charge, energy, quasiparticle currents) in coherent transport. The derivation is parameter-free: the bounds follow directly from Cauchy-Schwarz applied to the fluctuation theorem [Eq. (4a)] and the activity definition [Eq. (10)], with no fitted parameters. The NS setup example [Fig. 2] provides a concrete and physically relevant demonstration where existing bounds fail and the new ones hold. The random-observable sampling in Fig. 2(c,d) provides falsifiable numerical evidence for the bounds across a range of parameters. The framework correctly identifies that higher-order fluctuations of entropy production and activity enter the cost far from equilibrium, explaining apparent violations of classical bounds.","major_comments":[{"comment":"The involution existence and its compatibility with the FCS probability distribution are assumed without proof. The TUR [Eq. (8)] and TKUR [Eq. (17)] both rely on the forward-backward entropy defined via an involution satisfying q(ω†) = -q(ω) [Eq. (3)]. The paper states 'we first consider an involution on the sample space' without proving that such an involution exists for arbitrary transported quantities Q in arbitrary multi-terminal or superconducting setups. For the NN example, the global reversal (n1,...,nr)† = (-n1,...,-nr) works for particle number, but for a general observable q (e.g., energy current in a multi-terminal setup with Andreev processes), it is not obvious that an involution exists on the FCS sample space that simultaneously (i) is an involution, (ii) makes q antisymmetric, and (iii) yields a σ̃ that coincides with thermodynamic entropy production in the time-reversal-","section":null}],"minor_comments":[{"comment":"Eq. (2): the notation h is introduced without explicit definition; it appears to be a measure factor (possibly related to the density of states or the quantum of conductance), but this should be stated.","section":null},{"comment":"The transition from Eq. (6a) to Eq. (6b) involves mapping expectations from p+ back to p; the step E+[q tanh(σ̃/2)]² = E[q]² should be made more explicit for the reader.","section":null},{"comment":"Fig. 1(b,c,e,f): the y-axis labels show multiple quantities (P_N, P_E, costs); a legend or clearer labeling would improve readability.","section":null},{"comment":"The comparison with Eq. (19) is dropped after Fig. 1(f) because it does not bound P_E; it would be helpful to note that Eq. (19) was derived for particle currents (tunneling events only).","section":null},{"comment":"Conclusions: the statement about time-reversal symmetry breaking could benefit from a brief clarification that the bound [Eq. (7)] remains mathematically valid but the physical interpretation of σ̃ changes.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The stress-test concern about involution existence is valid and should be addressed, but it does not appear to be a load-bearing error that invalidates the central claim for the cases studied. The examples (NN and NS setups) use standard involutions that are well-defined, and the random-observable sampling in Fig. 2(c,d) provides empirical evidence for the NS case. The concern is about the generality claim for 'arbitrary' Q in 'arbitrary' setups, which could be addressed by clarifying the scope or providing a construction argument."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful reading and a constructive report. The referee correctly identifies the central contributions of our Letter: the generalization of uncertainty relations to arbitrary transported quantities in coherent transport, the parameter-free derivation from Cauchy-Schwarz applied to the fluctuation theorem, and the demonstration in normal and superconducting hybrid structures. The one substantive concern raised—regarding the existence and compatibility of the involution for general observables in multi-terminal and superconducting setups—is well-taken. We agree that the manuscript should state more explicitly the conditions under which the involution exists and how it is constructed in the examples considered, and we will revise accordingly.","responses":[{"response":"The referee raises a valid point: the manuscript does not explicitly state the conditions under which the involution exists or how it is constructed in the examples. We will revise the manuscript to clarify this. To address the substance of the comment, we note the following. The involution we use in both the NN and NS examples is the time-reversal involution on the FCS sample space, which exchanges the initial and final states of each scattering event. This involution is well-defined for the Levitov-Lesovik FCS framework: each scattering event ω is characterized by the set of transferred particle numbers (n_1, ..., n_r) across terminals, and the involution maps (n_1, ..., n_r) → (-n_1, ..., -n_r). This map is always an involution regardless of the observable q. The additional requirement is that q be antisymmetric under this map [Eq. (3)]. For particle number, charge, and energy currents in the NN setup, this antisymmetry holds because the transferred quantity changes sign when all particle numbers are reversed. In the NS setup, the BTK scattering amplitudes define the FCS distribution, and the same global reversal involution applies; the quasiparticle number, charge, and energy transferred all change sign under this map, so antisymmetry is satisfied. The key point is that the involution is defined on the sample space Ω (independent of q), and the antisymmetry condition [Eq. (3)] is a property of the observable q with respect to that involution—not a requirement that the involution be constructed separately for each q. For observables that are odd under time reversal (which includes all standard currents: particle, charge, energy, and quasiparticle currents), the global reversal involution satisfies all three conditions (i)-(iii) simultaneously. We agree that the paper","revision_made":"partial","referee_comment":"The involution existence and its compatibility with the FCS probability distribution are assumed without proof. The TUR [Eq. (8)] and TKUR [Eq. (17)] both rely on the forward-backward entropy defined via an involution satisfying q(ω†) = -q(ω) [Eq. (3)]. The paper states 'we first consider an involution on the sample space' without proving that such an involution exists for arbitrary transported quantities Q in arbitrary multi-terminal or superconducting setups. For the NN example, the global reversal (n1,...,nr)† = (-n1,...,-nr) works for particle number, but for a general observable q (e.g., energy current in a multi-terminal setup with Andreev processes), it is not obvious that an involution exists on the FCS sample space that simultaneously (i) is an involution, (ii) makes q antisymmetric, and (iii) yields a σ̃ that coincides with thermodynamic entropy production in the time-reversal-"}],"tokens_in":10544,"tokens_out":772,"duration_ms":173980,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The main thing to know: this paper extends thermodynamic and kinetic uncertainty relations from particle currents only to arbitrary transported quantities in coherent transport. That is a real gap being filled — prior TUR and KUR results (Refs. [17, 20, 21]) were proven only for particle currents, and the extension matters especially for superconducting hybrids where charge, quasiparticle, and energy currents all differ. The paper shows the bounds hold where prior particle-current-specific bounds were shown to be violated (Refs. [22–24]). That is the core contribution, and it is earned. The Cauchy-Schwarz strategy on the symmetrized distribution is clean and parameter-free. The near-equilibrium limit [Eq. (9)] recovers the classical TUR, the weak-coupling limit [Eq. (14)] recovers the classical KUR, and the thermokinetic unification [Eq. (17)] interpolates between them. The examples are well chosen: the NN setup with double-boxcar transmission shows the KUR bound [Eq. (13)] constrains energy-current precision where the prior particle-current KUR [Eq. (19)] simply does not apply, and the NS setup with BTK theory shows the TUR bound [Eq. (8)] correctly bounds charge-current precision where Andreev reflections break the old bound [Eq. (18)]. The random-observable sampling in Fig. 2(c,d) is a nice sanity check. The soft spot is the involution. The paper assumes an involution ω† exists satisfying q(ω†) = −q(ω) and then defines σ̃ = log[p(ω)/p(ω†)], which automatically satisfies a fluctuation theorem by construction. The mathematical bound follows whenever such an involution exists. The question is whether it always does for arbitrary observables in arbitrary multi-terminal or superconducting setups, and whether σ̃ has physical meaning as entropy production. On the first point: the paper does not prove general existence, but the framework is constructive — you bring your own involution, and the bound follows. For the examples shown (global or local reversal on FCS sample spaces, BTK theory), the involution is well-defined. On the second point: the paper is explicit that σ̃ coincides with thermodynamic entropy production only in time-reversal symmetric systems, and flags the broken-symmetry case as open. This is honest and does not undermine the results as stated. The restriction to noninteracting systems [Eq. (2)] is also acknowledged. These are scope limitations, not flaws. This paper is for people working on quantum thermodynamic bounds in mesoscopic transport. It deserves a serious referee. The involution existence question is worth raising in review, but it does not sink the paper — the results are correct for the setups shown, and the framework is clearly stated.","headline":"Solid extension of TUR/KUR to arbitrary currents in coherent transport; the involution existence is assumed but the framework is constructive and the examples land.","tokens_in":11556,"tokens_out":1602,"would_cite":true,"duration_ms":132407,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Precision bounds for any current in coherent nanoscale transport","keywords":[],"falsifier":"A concrete counterexample would be a coherent, strongly coupled, linear system out of equilibrium where the precision of some antisymmetric transported quantity exceeds the bound C (or A, or the thermokinetic bound) while Eq. (2) remains valid. More realistically, the bounds could be tested numerically or experimentally in normal-superconducting junctions with energy-dependent transmissions: if any observable's precision-to-noise ratio exceeds the integral bound while the factorization holds, the Cauchy-Schwarz step would be contradicted. The authors' own numerical sampling of random observals","tokens_in":11039,"feed_emoji":"🔬","tokens_out":1319,"duration_ms":210334,"temperature":0.7,"pith_summary":"This paper proves that thermodynamic and kinetic uncertainty relations — which bound how precise a transported current can be relative to its fluctuations — hold for arbitrary transported quantities in coherent, strongly coupled, linear systems out of equilibrium, not just for particle currents as previously shown. The authors derive three bounds (thermodynamic, kinetic, and a unified thermokinetic relation) starting from the probability distribution of individual scattering events. The key mechanism is a two-step application of the Cauchy-Schwarz inequality: first at the level of single-scattering-event statistics over a symmetrized probability distribution built from fluctuation theorems, then again at the level of energy-resolved integrals over those events. The thermodynamic bound uses the forward-backward stochastic entropy (a quantity that satisfies a fluctuation theorem by construction and coincides with thermodynamic entropy production in time-reversal symmetric systems), while the kinetic bound uses a stochastic activity variable counting events where the transported quantity changes. Crucially, both bounds naturally incorporate higher-order fluctuations of entropy and activity, which become significant far from equilibrium and at strong coupling. This allows the bounds to remain valid where classical (near-equilibrium or weak-coupling) limits are violated — including in superconducting hybrid structures where Andreev reflections transfer charge without entropy change, causing previous particle-current-specific bounds to fail for charge-current precision. The authors demonstrate the bounds in normal-metal and normal-superconducting junctions, showing they are tighter and more general than existing formulations.","feed_headline":"Precision bounds for any current in coherent nanoscale transport","feed_subtitle":"New uncertainty relations hold for arbitrary transported quantities — charge, energy, or quasiparticles — even in superconducting hybrids.","key_machinery":"1) Forward-backward stochastic entropy sigma = log[p(omega)/p(omega^double_dagger)], which satisfies a fluctuation theorem by construction and coincides with thermodynamic entropy production in time-reversal symmetric systems. 2) A symmetrized probability distribution p_+ = (p(omega) + p(omega^double_dagger))/2 on which Cauchy-Schwarz is applied. 3) A stochastic activity variable a(omega) = 1 - delta_{q(omega),0} counting events where the transported quantity changes. 4) The factorization of average current and zero-frequency noise into energy-resolved integrals over single-scattering-event statistics (Eq. 2), which connects the single-event Cauchy-Schwarz bounds to macroscopic transport.","core_discovery":"The central result is that the Cauchy-Schwarz inequality, applied first to the symmetrized probability distribution of individual scattering events and then to the energy-resolved integral over those events, yields uncertainty relations for any antisymmetric transported quantity. The thermodynamic bound has cost C = integral of E[tanh(sigma/2)] / (1 - E[tanh(sigma/2)]), the kinetic bound has cost A = integral of E[a] / (1 - E[a]), and a unified thermokinetic bound combines both. These bounds reduce to the classical TUR and KUR near equilibrium and at weak coupling respectively, but remain valid far from equilibrium and in superconducting structures where all prior bounds — which were proven只","pith_inferences":["The factorization requirement (Eq. 2) means these bounds are fundamentally limited to noninteracting or mean-field systems. Extending to genuinely interacting systems would require replacing the single-scattering-event probability distribution with a many-body counting statistics framework, where the Cauchy-Schwarz strategy may not directly apply.","The connection between the kinetic activity defined here and quantum Fisher information (noted in related work by the authors) suggests that the thermokinetic bound might be interpretable as a quantum Cramér-Rao bound for a specific measurement channel, which would connect precision limits to parameter estimation theory.","The dependence of the thermokinetic bound's tightness on the choice of involution (global vs. local reversal) suggests an optimization principle: the tightest bound for a given observable is obtained by choosing the involution that maximizes the number of null-activity events, which could be formulated as a variational problem."],"forward_implications":["The bounds apply to any antisymmetric transported quantity — charge, energy, quasiparticle number, or arbitrary observables — removing the restriction to particle currents that limited all prior coherent-transport uncertainty relations.","In superconducting hybrid structures, the bounds correctly constrain charge-current precision even though Andreev reflections transfer charge without entropy production, a scenario where previous bounds were violated.","Far from equilibrium, higher-order moments of entropy production and activity provide positive corrections to the cost, meaning coherent conductors can exceed classical precision limits without violating any bound.","The unified thermokinetic bound interpolates between the thermodynamic and kinetic costs, and its tightness depends on how restrictively one defines the class of events contributing to the activity.","The framework extends to time-reversal symmetry breaking (e.g., magnetic fields), where the forward-backward entropy differs from thermodynamic entropy production but the mathematical bound remains valid."],"fun_headline_variants":["Uncertainty bounds for any current in coherent transport","Thermodynamic and kinetic bounds for arbitrary coherent currents","Precision-cost bounds valid far from equilibrium in nanoscale transport","Cauchy-Schwarz yields current uncertainty bounds in superconducting hybrids","Unified thermokinetic uncertainty relations for arbitrary currents"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The entire derivation requires that the average current and zero-frequency noise factorize into energy-resolved integrals over single-scattering-event statistics. This factorization is valid only for noninteracting (or mean-field) systems where the Levitov-Lesovik full counting statistics applies. If particle-particle interactions cannot be neglected, the probability distribution over scattering events is not well-defined in this form and the Cauchy-Schwarz argument collapses","fun_headline_variants_meta":{"raw":{"variants":["Uncertainty bounds for any current in coherent transport","Thermodynamic and kinetic bounds for arbitrary coherent currents","Precision-cost bounds valid far from equilibrium in nanoscale transport","Cauchy-Schwarz yields current uncertainty bounds in superconducting hybrids","Unified thermokinetic uncertainty relations for arbitrary currents"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":507,"prompt_tokens":443,"completion_tokens":64,"prompt_tokens_details":null},"tokens_in":443,"tokens_out":64,"duration_ms":43597,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T13:43:03.050226+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"A concrete counterexample would be a coherent, strongly coupled, linear system out of equilibrium where the precision of some antisymmetric transported quantity exceeds the bound C (or A, or the thermokinetic bound) while Eq. (2) remains valid. More realistically, the bounds could be tested numerically or experimentally in normal-superconducting junctions with energy-dependent transmissions: if any observable's precision-to-noise ratio exceeds the integral bound while the factorization holds, the Cauchy-Schwarz step would be contradicted. The authors' own numerical sampling of random observals","supporting_citations":[],"review_version":1}