Quantum thermodynamic uncertainty relation and macroscopic superconducting coherence

Fabio Taddei; Franco Mayo; Michele Governale; Nahual Sobrino; Rosario Fazio

arxiv: 2506.02904 · v3 · pith:RN3LQQ6Ynew · submitted 2025-06-03 · ❄️ cond-mat.mes-hall · cond-mat.supr-con

Quantum thermodynamic uncertainty relation and macroscopic superconducting coherence

Franco Mayo , Nahual Sobrino , Rosario Fazio , Fabio Taddei , Michele Governale This is my paper

Pith reviewed 2026-05-19 11:11 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.supr-con

keywords thermodynamic uncertainty relationmacroscopic coherencepair amplitudeAndreev regimehybrid normal-superconducting junctionsquantum transportnonequilibrium fluctuationssuperconducting quantum dot

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The pith

Macroscopic superconducting coherence governs departures from the normal quantum thermodynamic uncertainty relation in hybrid N-S devices.

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

The paper establishes that in subgap normal-superconducting junctions, departures from the normal quantum TUR are governed by macroscopic superconducting coherence as quantified by the pair amplitude. A dephasing probe can suppress this coherence to restore the bound. The authors derive a hybrid quantum TUR for two-terminal N-S junctions in the Andreev regime that is never violated, saturates only at vanishing current, and corresponds to the normal bound with charge doubled from e to 2e. Computations confirm that deviations track the pair amplitude in quantum dot and Cooper-pair-splitter systems. This creates a direct link between superconducting coherence and nonequilibrium fluctuations.

Core claim

In hybrid normal-superconducting devices operating in the subgap Andreev regime, departures from the normal quantum TUR are governed by macroscopic superconducting coherence quantified by the pair amplitude. A hybrid quantum TUR is derived that holds generally for two-terminal N-S junctions: it is never violated, saturated only at vanishing current, and obtained from the normal bound by the replacement e to 2e. For N-S quantum dot and Cooper-pair-splitter systems, deviations track the pair amplitude.

What carries the argument

The pair amplitude in the central region, which quantifies macroscopic superconducting coherence and controls deviations from the standard TUR bound.

Load-bearing premise

The derivation assumes the system stays in the subgap Andreev regime where transport is dominated by Andreev processes and that the pair amplitude fully captures the macroscopic coherence modifying the TUR.

What would settle it

Measure current noise and entropy production in an N-S quantum dot in the subgap regime and test whether the relative uncertainty times entropy production deviates from the normal value in proportion to the pair amplitude squared.

Figures

Figures reproduced from arXiv: 2506.02904 by Fabio Taddei, Franco Mayo, Michele Governale, Nahual Sobrino, Rosario Fazio.

**Figure 2.** Figure 2: FIG. 2. Fano factor as a function of (a) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. The value of Γ [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Violation of the quantum TUR in the CPS system [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

read the original abstract

Stability and efficiency are mutually exclusive in a thermodynamic process, e.g. in a thermal machine. Any effort to reduce the fluctuations of a certain output quantity is necessarily accompanied by an increase of entropy production, therefore lowering its efficiency. This interplay is beautifully captured by the so called Thermodynamic Uncertainty Relations (TURs) which set a lower bound on the relative uncertainty of a current for a given rate of entropy production. Their status in hybrid normal-superconducting (N-S) devices has remained unsettled. We show that, in the subgap regime, departures from the normal quantum TUR are governed by {\it macroscopic} superconducting coherence quantified by the pair amplitude, and that introducing a dephasing probe suppresses this coherence and restores the bound. We further derive a hybrid quantum TUR that is general for two-terminal N-S junctions in the Andreev regime: the inequality is never violated, is saturated only at vanishing current, and is related to the normal quantum bound under the replacement (e to 2e). For N-S quantum dot and Cooper-pair-splitter systems we compute current and noise and show that deviations from the normal bound track the pair amplitude on the central region. The results establish a direct link between superconducting macroscopic coherence and nonequilibrium fluctuations and supply a general bound for the Andreev regime.

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.

Desk Editor's Note private letter to a colleague

The paper derives a hybrid TUR for two-terminal N-S junctions in the Andreev regime that is never violated and saturates only at zero current, obtained by replacing e with 2e, while showing deviations track pair amplitude in the models.

read the letter

The punchline is that this paper derives a hybrid quantum thermodynamic uncertainty relation for two-terminal normal-superconducting junctions in the Andreev regime. The inequality holds without violation, saturates only when the current goes to zero, and comes from the usual quantum bound by replacing the electron charge e with 2e. What is new is this specific form of the TUR tailored to Andreev processes, along with the claim that deviations from the normal bound are governed by macroscopic superconducting coherence as quantified by the pair amplitude. They support this by calculating current and noise in two models: an N-S quantum dot and a Cooper-pair splitter. In both cases, the deviations track the pair amplitude extracted from the central region. They also show that a dephasing probe suppresses the coherence and brings the system back to the normal TUR bound. The paper does well here. The model calculations are explicit and allow direct comparison of the uncertainty with the pair amplitude. This gives a concrete illustration of how superconducting coherence affects nonequilibrium fluctuations, which is the kind of link that could matter for device applications. On the soft spots, the general derivation seems to start from expressions valid for perfect subgap Andreev reflection. While that fits the regime they target, the direct role of the pair amplitude in modifying the bound for arbitrary parameters or geometries is shown mainly through the examples rather than derived in full generality. If the hybrid TUR follows purely from the doubled charge without additional coherence dependence, then the strongest connection to macroscopic coherence rests on those numerical results. That is a moderate limitation rather than a fatal one, since the bound itself stands on its own. This work is aimed at researchers in quantum thermodynamics and mesoscopic superconductivity, particularly those interested in fluctuation bounds or noise in hybrid systems. A reader looking for extensions of TURs to superconducting setups or for bounds useful in low-noise device design would find value in the general inequality and the supporting calculations. It deserves a serious referee because the central claim is clearly stated, the models are worked out in detail, and there is enough substance for reviewers to assess the generality and the numerical evidence. I would recommend sending this to peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript derives a hybrid quantum thermodynamic uncertainty relation (TUR) for two-terminal normal-superconducting (N-S) junctions operating in the subgap Andreev regime. This bound is claimed to be never violated, saturated only at vanishing current, and obtained from the normal quantum TUR via the replacement e → 2e. The work further argues that departures from the normal TUR are governed by macroscopic superconducting coherence, quantified by the pair amplitude in the central region; this is illustrated through explicit current and noise calculations on an N-S quantum-dot model and a Cooper-pair splitter, with a dephasing probe shown to suppress the pair amplitude and restore the normal bound.

Significance. If the central claims hold, the paper supplies a general fluctuation bound specific to the Andreev regime together with a concrete link between macroscopic coherence and thermodynamic uncertainty. The explicit calculations on two distinct geometries that track the pair amplitude constitute a strength, as they provide falsifiable, model-specific predictions that can be checked against the general bound.

major comments (1)

[General derivation and model calculations] The abstract and introduction present the hybrid TUR as a general derivation for arbitrary two-terminal N-S junctions in the Andreev regime. However, the explicit demonstration that deviations track the pair amplitude is performed only for the quantum-dot and splitter geometries (see the sections containing the scattering-matrix and rate-equation calculations). The general argument appears to rest on charge-2e transport under perfect subgap Andreev reflection; it is not shown how the pair amplitude extracted from the central region enters the inequality itself for arbitrary junction parameters or geometries. This distinction is load-bearing for the claim that departures are 'governed by' macroscopic coherence.

minor comments (2)

Clarify the notation for the pair amplitude (e.g., whether it is the local or integrated quantity) and ensure it is consistently defined when comparing the general bound to the numerical results.
Add a brief discussion of the regime of validity (e.g., temperature, bias, and transparency ranges) under which the e → 2e replacement remains accurate.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The distinction between the general hybrid TUR and the model-specific illustration of coherence effects is important, and we address it directly below. We have revised the manuscript to improve clarity on this point without altering the core claims.

read point-by-point responses

Referee: [General derivation and model calculations] The abstract and introduction present the hybrid TUR as a general derivation for arbitrary two-terminal N-S junctions in the Andreev regime. However, the explicit demonstration that deviations track the pair amplitude is performed only for the quantum-dot and splitter geometries (see the sections containing the scattering-matrix and rate-equation calculations). The general argument appears to rest on charge-2e transport under perfect subgap Andreev reflection; it is not shown how the pair amplitude extracted from the central region enters the inequality itself for arbitrary junction parameters or geometries. This distinction is load-bearing for the claim that departures are 'governed by' macroscopic coherence.

Authors: We agree that a clear separation exists between the general bound and the role of the pair amplitude. The hybrid quantum TUR is derived for arbitrary two-terminal N-S junctions in the perfect Andreev regime by noting that all subgap transport occurs via charge-2e processes; this directly yields the replacement e → 2e in the normal quantum TUR, with the bound saturated only at vanishing current. This step relies solely on the charge quantization and the absence of single-electron tunneling, without reference to a specific geometry or the pair amplitude. In contrast, the statement that departures from the normal TUR are governed by macroscopic superconducting coherence is supported by explicit calculations in two representative models (N-S quantum dot and Cooper-pair splitter). There we compute both the current-noise ratio and the pair amplitude in the central region, demonstrating that larger pair amplitude correlates with stronger violation of the normal bound, while a dephasing probe reduces the pair amplitude and restores the normal TUR. We acknowledge that we do not provide a model-independent expression in which the pair amplitude appears explicitly inside the general inequality for arbitrary junction parameters. Such an expression would require a universal definition of coherence that enters the fluctuation-dissipation relations beyond the 2e charge replacement, which lies outside the scope of the present work. We have revised the abstract, introduction, and concluding sections to state this distinction explicitly, emphasizing that the general bound follows from 2e transport while the quantitative link to coherence is illustrated and falsifiable in concrete geometries. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the hybrid TUR derivation

full rationale

The paper presents a derivation of a hybrid quantum TUR for two-terminal N-S junctions in the Andreev regime, stating that the inequality is never violated, saturated only at vanishing current, and obtained from the normal quantum bound via e to 2e replacement. This is framed as following from scattering or rate expressions under subgap Andreev reflection. The connection between deviations and pair amplitude is shown via explicit current/noise computations and pair-amplitude tracking in specific geometries (N-S quantum dot and Cooper-pair-splitter), but the general bound itself does not reduce by construction to a fitted parameter, self-definition, or load-bearing self-citation. The derivation remains self-contained against external benchmarks with independent mathematical content, consistent with a standard non-circular result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the Andreev-regime approximation and the identification of the pair amplitude as the quantifier of macroscopic coherence; no free parameters or new postulated entities are introduced in the abstract.

axioms (1)

domain assumption Standard assumptions of mesoscopic quantum transport in the subgap regime
Invoked to derive the hybrid TUR and to link deviations to the pair amplitude.

pith-pipeline@v0.9.0 · 5771 in / 1178 out tokens · 38530 ms · 2026-05-19T11:11:39.609891+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

We further derive a hybrid quantum TUR that is general for two-terminal N-S junctions in the Andreev regime: the inequality is never violated, is saturated only at vanishing current, and is related to the normal quantum bound under the replacement (e to 2e).
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the hybrid quantum TUR takes the form S/2e|J| sinh(eσ/kB|J|) ≥ 1

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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[1] [1]

A. C. Barato and U. Seifert, Thermodynamic uncertainty relation for biomolecular processes, Phys. Rev. Lett. 114, 158101 (2015)

work page 2015

[2] [2]

T. R. Gingrich, J. M. Horowitz, N. Perunov, and J. L. England, Dissipation bounds all steady-state current fluc- tuations, Phys. Rev. Lett. 116, 120601 (2016)

work page 2016

[3] [3]

Pietzonka, A

P. Pietzonka, A. C. Barato, and U. Seifert, Universal bounds on current fluctuations, Phys. Rev. E 93, 052145 (2016)

work page 2016

[4] [4]

A. M. Timpanaro, G. Guarnieri, J. Goold, and G. T. Landi, Thermodynamic uncertainty relations from ex- change fluctuation theorems, Phys. Rev. Lett.123, 090604 (2019)

work page 2019

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U. Seifert, From stochastic thermodynamics to thermo- dynamic inference, Annual Review of Condensed Matter Physics 10, 171 (2019)

work page 2019

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work page 2020

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B. K. Agarwalla and D. Segal, Assessing the validity of the thermodynamic uncertainty relation in quantum systems, Phys. Rev. B 98, 155438 (2018)

work page 2018

[8] [8]

Ehrlich and G

T. Ehrlich and G. Schaller, Broadband frequency filters with quantum dot chains, Phys. Rev. B 104, 045424 (2021)

work page 2021

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Gerry and D

M. Gerry and D. Segal, Absence and recovery of cost- precision tradeoff relations in quantum transport, Phys. Rev. B 105, 155401 (2022)

work page 2022

[10] [10]

Guarnieri, G

G. Guarnieri, G. T. Landi, S. R. Clark, and J. Goold, Thermodynamics of precision in quantum nonequilibrium steady states, Phys. Rev. Res. 1, 033021 (2019)

work page 2019

[11] [11]

Kheradsoud, N

S. Kheradsoud, N. Dashti, M. Misiorny, P. P. Potts, 6 J. Splettstoesser, and P. Samuelsson, Power, effi- ciency and fluctuations in a quantum point contact as steady-state thermoelectric heat engine, Entropy 21, 10.3390/e21080777 (2019)

work page doi:10.3390/e21080777 2019

[12] [12]

Proesmans and J

K. Proesmans and J. M. Horowitz, Hysteretic thermody- namic uncertainty relation for systems with broken time- reversal symmetry, Journal of Statistical Mechanics: The- ory and Experiment 2019, 054005 (2019)

work page 2019

[13] [13]

Liu and D

J. Liu and D. Segal, Thermodynamic uncertainty relation in quantum thermoelectric junctions, Phys. Rev. E 99, 062141 (2019)

work page 2019

[14] [14]

Pietzonka and U

P. Pietzonka and U. Seifert, Universal trade-off between power, efficiency, and constancy in steady-state heat en- gines, Phys. Rev. Lett. 120, 190602 (2018)

work page 2018

[15] [15]

Prech, P

K. Prech, P. Johansson, E. Nyholm, G. T. Landi, C. Ver- dozzi, P. Samuelsson, and P. P. Potts, Entanglement and thermokinetic uncertainty relations in coherent meso- scopic transport, Phys. Rev. Res. 5, 023155 (2023)

work page 2023

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Ptaszy´ nski, Coherence-enhanced constancy of a quan- tum thermoelectric generator, Phys

K. Ptaszy´ nski, Coherence-enhanced constancy of a quan- tum thermoelectric generator, Phys. Rev. B 98, 085425 (2018)

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[17] [17]

Saryal, H

S. Saryal, H. M. Friedman, D. Segal, and B. K. Agar- walla, Thermodynamic uncertainty relation in thermal transport, Phys. Rev. E 100, 042101 (2019)

work page 2019

[18] [18]

Saryal, M

S. Saryal, M. Gerry, I. Khait, D. Segal, and B. K. Agarwalla, Universal bounds on fluctuations in continuous thermal machines, Phys. Rev. Lett. 127, 190603 (2021)

work page 2021

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A. M. Timpanaro, G. Guarnieri, and G. T. Landi, Quan- tum thermoelectric transmission functions with minimal current fluctuations, Phys. Rev. B 111, 014301 (2025)

work page 2025

[20] [20]

Brandner, T

K. Brandner, T. Hanazato, and K. Saito, Thermody- namic bounds on precision in ballistic multiterminal trans- port, Phys. Rev. Lett. 120, 090601 (2018)

work page 2018

[21] [21]

Macieszczak, K

K. Macieszczak, K. Brandner, and J. P. Garrahan, Uni- fied thermodynamic uncertainty relations in linear re- sponse, Phys. Rev Lett. 121, 130601 (2018)

work page 2018

[22] [22]

Saryal, S

S. Saryal, S. Mohanta, and B. K. Agarwalla, Bounds on fluctuations for machines with broken time-reversal sym- metry: A linear response study, Phys. Rev. E 105, 024129 (2022)

work page 2022

[23] [23]

Braggio, M

A. Braggio, M. Governale, M. G. Pala, and J. K¨ onig, Superconducting proximity effect in interacting quantum dots revealed by shot noise, Solid State Communications 151, 155 (2011)

work page 2011

[24] [24]

Taddei and R

F. Taddei and R. Fazio, Thermodynamic uncertainty re- lations for systems with broken time reversal symmetry: The case of superconducting hybrid systems, Phys. Rev. B 108, 115422 (2023)

work page 2023

[25] [25]

Kamijima, S

T. Kamijima, S. Otsubo, Y. Ashida, and T. Sagawa, Higher-order efficiency bound and its application to non- linear nanothermoelectrics, Phys. Rev. E 104, 044115 (2021)

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