A Generalized Landauer's Principle for Unitarily Transformed Thermal Reservoirs

Hao Xu

arxiv: 2602.04552 · v2 · submitted 2026-02-04 · 🪐 quant-ph · gr-qc· hep-th

A Generalized Landauer's Principle for Unitarily Transformed Thermal Reservoirs

Hao Xu This is my paper

Pith reviewed 2026-05-16 07:37 UTC · model grok-4.3

classification 🪐 quant-ph gr-qchep-th

keywords Landauer's principlesqueezed thermal stateentropy productionUnruh-DeWitt detectorquantum fieldunitary transformationeffective Hamiltonianquantum thermodynamics

0 comments

The pith

Defining an effective Hamiltonian extends Landauer's principle to unitarily transformed thermal states

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

The paper establishes a generalized version of Landauer's principle that applies when a thermal reservoir undergoes a unitary transformation, such as becoming a squeezed thermal state. By introducing an effective Hamiltonian for these transformed states, the authors prove a generalized inequality that reduces to the standard Landauer bound for ordinary thermal reservoirs. This framework also allows a consistent definition of entropy production, which they show is always non-negative. The approach is applied to an Unruh-DeWitt detector moving in a quantum field prepared in a squeezed thermal state, where explicit calculations confirm the positivity of entropy production and reveal its dependence on both time interval and position.

Core claim

By defining an effective Hamiltonian for the unitarily transformed thermal state, we rigorously establish a generalized Landauer inequality that reduces to the standard case for an ordinary thermal reservoir. This yields a consistent definition of entropy production with a proof of its non-negativity. The utility is illustrated by computing entropy production for an arbitrarily moving Unruh-DeWitt detector coupled to a quantum field in a squeezed thermal state, confirming positivity and dependence on proper time and spacetime position due to symmetry breaking.

What carries the argument

The effective Hamiltonian for the unitarily transformed thermal state, which enables the generalized Landauer inequality and non-negative entropy production.

If this is right

The generalized Landauer inequality holds for any unitarily transformed thermal state and reduces to the original when the transformation is trivial.
A consistent definition of entropy production is provided, and its non-negativity is proven.
For a moving Unruh-DeWitt detector in a squeezed thermal state, entropy production is positive and depends on the proper time interval and the absolute spacetime position.
The apparent violation of Landauer's principle with squeezed thermal states is resolved.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar effective Hamiltonian definitions could be used to generalize other thermodynamic inequalities to non-standard reservoirs.
The position dependence of entropy production may lead to new considerations in relativistic quantum thermodynamics.
This framework could inform the design of quantum thermal machines operating with squeezed or other transformed states.

Load-bearing premise

The choice of an effective Hamiltonian for the unitarily transformed thermal state is sufficient to restore the standard Landauer bound and ensure non-negative entropy production.

What would settle it

A calculation showing that entropy production becomes negative for some unitarily transformed thermal state under this definition, or that the inequality does not reduce to the standard Landauer principle when the unitary is the identity.

read the original abstract

Landauer's principle, a cornerstone of quantum information and thermodynamics, appears to be violated when the thermal reservoir is replaced by a squeezed thermal state (STS), owing to the additional thermodynamic resources inherently present in the squeezed state. We introduce a formal extension of the principle to such unitarily transformed thermal states. By defining an effective Hamiltonian, we rigorously establish a generalized Landauer inequality, which naturally reduces to the standard case for an ordinary thermal reservoir as a special instance. The framework further yields a consistent definition of entropy production and a proof of its non-negativity. We illustrate its utility by studying an arbitrarily moving Unruh-DeWitt detector coupled to a quantum field initially prepared in the STS. Using perturbation theory, we compute the entropy production explicitly, confirming its positivity. As a result of the symmetry breaking induced by the unitary transformation, it depends on both the proper time interval and the absolute spacetime position. Our work resolves the apparent violation of Landauer's principle with STSs. It also provides a robust tool for analyzing quantum thermodynamics in non-equilibrium and relativistic settings, with potential implications for quantum thermal machines and information protocols.

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 gives a usable extension of Landauer's principle to squeezed baths via an effective Hamiltonian and shows position-dependent entropy production for a moving detector, but the Hamiltonian step looks like it may be introduced to enforce the bound rather than derived from the unitary.

read the letter

The key point is that this work claims to restore Landauer's inequality for unitarily transformed thermal states like squeezed ones by defining an effective Hamiltonian, then applies the result to an Unruh-DeWitt detector in motion and gets explicit entropy production that varies with both proper time and absolute position. That relativistic example is the part that stands out as concrete and potentially useful for people already working on detector models in quantum fields. The reduction to the ordinary thermal case is shown, and the non-negativity of entropy production follows once the effective Hamiltonian is in place. The perturbation-theory calculation for the detector seems straightforward and confirms positivity as advertised. The citation list hits the expected references on Landauer's principle and Unruh-DeWitt detectors without obvious gaps. The soft spot is the status of the effective Hamiltonian itself. The abstract presents it as a definition that lets the generalized inequality hold, but it is not clear from the given material whether this Hamiltonian follows directly from applying the unitary to the original thermal state or is chosen so the inequality is recovered. If the latter, the proof of non-negative entropy production becomes close to tautological, and the claim that the apparent violation is resolved rests on that choice. The stress-test note raises exactly this issue, and the low soundness score in the reader's report matches the lack of visible derivation steps or independent checks. This is aimed at specialists in quantum thermodynamics and relativistic quantum information. Someone already thinking about non-equilibrium baths or information flow in curved spacetime could extract the detector calculation and the position dependence without needing the full foundational claim to be airtight. It is coherent enough on its own terms to deserve referee time rather than a desk reject, though any review should press on whether the effective Hamiltonian is independently motivated or simply fitted to restore the bound.

Referee Report

2 major / 2 minor

Summary. The paper claims to generalize Landauer's principle to unitarily transformed thermal states (e.g., squeezed thermal states) by defining an effective Hamiltonian H_eff such that the inequality Tr[ρ(H_eff - T S)] ≥ 0 holds, with the standard Landauer bound recovered as a special case. It further defines entropy production and proves its non-negativity. The framework is applied to an arbitrarily moving Unruh-DeWitt detector coupled to a quantum field prepared in a squeezed thermal state, where perturbation theory yields an explicit expression for entropy production that depends on both proper time and absolute spacetime position due to symmetry breaking.

Significance. If the effective Hamiltonian is independently derived rather than chosen to enforce the bound, and if the perturbation-theory computation includes verifiable error estimates, the result would resolve apparent violations of Landauer's principle for non-standard reservoirs and supply a practical tool for relativistic quantum thermodynamics. This could inform analyses of quantum thermal machines and information protocols in non-equilibrium or curved-spacetime settings. The explicit Unruh-DeWitt example demonstrates potential utility beyond abstract thermodynamics.

major comments (2)

[Abstract and effective-Hamiltonian definition] Abstract and the section introducing the effective Hamiltonian: the definition of H_eff for a unitarily transformed thermal state is presented without an independent derivation from the unitary operator U acting on the original thermal state ρ_th = exp(-β H)/Z or from the system-reservoir interaction. The subsequent claim that the generalized inequality holds and that entropy production is non-negative therefore risks being tautological by construction rather than emergent from physical principles.
[Abstract and perturbation-theory computation] Abstract and the perturbation-theory section: the abstract asserts a 'rigorous proof' and 'explicit computation via perturbation theory' for the Unruh-DeWitt detector, yet supplies no derivation steps, error bounds, or verification that the effective Hamiltonian is not tuned to force positivity. Without these, the explicit entropy-production result and its claimed dependence on proper time and spacetime position cannot be checked.

minor comments (2)

[Introduction or effective-Hamiltonian section] The reduction of the generalized inequality to the ordinary Landauer bound is stated but not shown explicitly with the limiting form of H_eff; adding this short calculation would improve clarity.
[Throughout] Notation for the effective Hamiltonian, entropy production, and the Unruh-DeWitt coupling should be introduced with numbered equations at first use to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below and outline the revisions we will make to strengthen the presentation.

read point-by-point responses

Referee: [Abstract and effective-Hamiltonian definition] Abstract and the section introducing the effective Hamiltonian: the definition of H_eff for a unitarily transformed thermal state is presented without an independent derivation from the unitary operator U acting on the original thermal state ρ_th = exp(-β H)/Z or from the system-reservoir interaction. The subsequent claim that the generalized inequality holds and that entropy production is non-negative therefore risks being tautological by construction rather than emergent from physical principles.

Authors: We agree that the motivation for H_eff requires clearer justification. The effective Hamiltonian is defined by the requirement that the unitarily transformed state ρ = U ρ_th U† satisfies ρ = exp(−β H_eff)/Z_eff, which is the unique operator that restores the thermal form after the unitary map. This choice is not arbitrary: it is the generator of the modular automorphism group for the transformed state and ensures that the generalized Landauer inequality reduces exactly to the standard form when U is the identity. We will add an explicit derivation subsection showing how H_eff is obtained directly from the spectral decomposition of U ρ_th U† and demonstrating that the non-negativity of Tr[ρ(H_eff − T S)] follows from the standard Landauer principle applied to the effective thermal state rather than being imposed by hand. This will also clarify the link to the system-reservoir interaction Hamiltonian. revision: yes
Referee: [Abstract and perturbation-theory computation] Abstract and the perturbation-theory section: the abstract asserts a 'rigorous proof' and 'explicit computation via perturbation theory' for the Unruh-DeWitt detector, yet supplies no derivation steps, error bounds, or verification that the effective Hamiltonian is not tuned to force positivity. Without these, the explicit entropy-production result and its claimed dependence on proper time and spacetime position cannot be checked.

Authors: We accept that the abstract and main text are too terse on the perturbative calculation. In the revised manuscript we will expand the relevant section to include: (i) the full sequence of perturbative steps up to second order in the coupling, (ii) explicit error bounds obtained by controlling the remainder term in the Dyson series, and (iii) a direct verification that positivity of the entropy production follows from the general theorem proved earlier in the paper and is independent of the particular numerical values chosen for H_eff. We will also show how the symmetry breaking induced by the squeezing unitary produces the explicit dependence on both proper time and absolute spacetime coordinates. These additions will allow the computation to be reproduced and checked. revision: yes

Circularity Check

1 steps flagged

Effective Hamiltonian defined to enforce generalized Landauer bound by construction

specific steps

self definitional [Abstract / central construction]
"By defining an effective Hamiltonian, we rigorously establish a generalized Landauer inequality, which naturally reduces to the standard case for an ordinary thermal reservoir as a special instance. The framework further yields a consistent definition of entropy production and a proof of its non-negativity."

The effective Hamiltonian is introduced to make the generalized inequality hold and to ensure ΔS ≥ 0. Once H_eff is defined precisely for this purpose, the claimed 'rigorous establishment' and reduction to the ordinary case become automatic by construction rather than emergent from the unitary transformation on ρ_th = exp(-βH)/Z.

full rationale

The paper's central step introduces an effective Hamiltonian for the unitarily transformed thermal state specifically to restore the inequality Tr[ρ(H_eff - TS)] ≥ 0 and guarantee non-negative entropy production. This definition is not derived from the unitary action on the original thermal state or from first principles of the interaction; instead, it is chosen so the bound holds and reduces to the standard Landauer case automatically. The subsequent proof of positivity is therefore tautological once H_eff is fixed to enforce the desired relation. No independent external benchmark or derivation from the unitary U is provided in the abstract or reader's summary to justify the form of H_eff outside the inequality itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the introduction of an effective Hamiltonian whose construction is not justified by prior literature or independent evidence in the abstract; no free parameters or invented entities are explicitly named, but the effective Hamiltonian functions as an ad-hoc construct.

axioms (1)

ad hoc to paper An effective Hamiltonian can be defined for any unitarily transformed thermal state such that the Landauer inequality is restored.
This definition is introduced to establish the generalized inequality and is not derived from first principles or external benchmarks in the abstract.

pith-pipeline@v0.9.0 · 5487 in / 1406 out tokens · 30639 ms · 2026-05-16T07:37:56.885021+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 unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

By defining an effective Hamiltonian, we rigorously establish a generalized Landauer inequality... βΔHeff := tr[Heff(ρ'R − ρR)] = ΔS + I(S':R') + D(ρ'R ∥ ρR)
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

ρR = O e^{-β HR} O† / tr[...] ; Heff := O HR O†

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

Works this paper leans on

33 extracted references · 33 canonical work pages

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Xu and S

H. Xu and S. Y. Chen,Entropy production and correlation spreading in the interaction between particle detector and thermal baths,Eur. Phys. J. Plus.137, no.7, 821 (2022)

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J. Goold, M. Paternostro, K. Modi,Nonequilibrium Quan- tum Landauer Principle,Phys. Rev. Lett.114, 060602 (2015)

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T. V. Vu, K. Saito,Finite-Time Quantum Landauer Principle and Quantum Coherence,Phys. Rev. Lett.128, 010602 (2022)

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W. G. Unruh,What happens when an accelerating ob- server detects a Rindler particle,Phys. Rev. D.29, 1047- 1056 (1984)

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B. S. DeWitt, inGeneral Relativity: An Einstein Cen- tenary Survey, edited by S. W. Hawking and W. Israel, Cambridge University Press, Cambridge, England, pp. 680–745 (1979)

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See Supplemental Material for further information

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This is equivalent to confining the field in a cavity with suit- able boundary conditions; the continuum description is recovered in the appropriate limit

For simplicity and to facilitate the analysis of individ- ual momentum modes, we discretize the QFT. This is equivalent to confining the field in a cavity with suit- able boundary conditions; the continuum description is recovered in the appropriate limit

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E. T. Jaynes and F. W. Cummings,Comparison of quan- tum and semiclassical radiation theories with application 6 to the beam maser,Proc. IEEE51, 89–109 (1963)

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Hammad, A

F. Hammad, A. Landry and D. Dijamco,Influence of the dispersion relation on the Unruh effect according to the relativistic Doppler shift method,Phys. Rev. D103, 085010 (2021)

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Xu,Momentum-resolved probing of Lorentz-violating dispersion relations via Unruh-DeWitt detector,Phys

H. Xu,Momentum-resolved probing of Lorentz-violating dispersion relations via Unruh-DeWitt detector,Phys. Lett. B868, 139760 (2025)

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Barbara ˇSoda, Vivishek Sudhir, and Achim Kempf, Acceleration-Induced Effects in Stimulated Light-Matter Interactions, Phys. Rev. Lett.128, 163603 (2022)

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Simidzija and E

P. Simidzija and E. Martin-Martinez,All coherent field states entangle equally,Phys. Rev. D.96, no.2, 025020 (2017)

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Simidzija and E

P. Simidzija and E. Martin-Martinez,Non-perturbative analysis of entanglement harvesting from coherent field states,Phys. Rev. D.96, no.6, 065008 (2017)

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H. Xu, Y. C. Ong, and M. H. Yung,Landauer’s principle in qubit-cavity quantum-field-theory interaction in vacuum and thermal states,Phys. Rev. A105, 012430 (2022)

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Xu,Energy change and Landauer’s principle in the interaction between qubit and quantum field theory,Eur

H. Xu,Energy change and Landauer’s principle in the interaction between qubit and quantum field theory,Eur. Phys. J. C84, 1057 (2024). 1 SUPPLEMENTAL MATERIAL Appendix A: Proof of Theorem 1 Proof.Usingρ SR =ρ S ⊗ρ R, the unitarity of ˆU, and the identity ˆOe−βHR ˆO† = e−β ˆOHR ˆO† = e−βHeff , we have ∆S+I(S ′ :R ′) =S(ρ S)−S(ρ ′ S) +I(S ′ :R ′) =S(ρ ′ R)−...

work page 2024

[1] [1]

Marian and T

P. Marian and T. A. Marian,Squeezed states with thermal noise. I. Photon-number statistics,Phys. Rev. A.47, 4474 (1993)

work page 1993

[2] [2]

M. S. Kim, F. A. de Oliveira and P. L. Knight,Properties of squeezed number states and STSs,Phys. Rev. A.40, 2494 (1989)

work page 1989

[3] [3]

Roßnagel, O

J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer, and E. LutzNanoscale Heat Engine Beyond the Carnot Limit, Phys. Rev. Lett..112, 030602 (2014)

work page 2014

[4] [4]

Klaers, S

J. Klaers, S. Faelt, A. Imamoglu, and E. Togan,Squeezed Thermal Reservoirs as a Resource for a Nanomechanical Engine,Phys. Rev. X.7, 031044 (2017)

work page 2017

[5] [5]

Breuer and F

H. Breuer and F. Petruccione,The Theory of Open Quan- tum Systems, Oxford University Press, Oxford (2002) [ISBN: 9780198520634]

work page 2002

[6] [6]

Landauer,Irreversibility and Heat Generation in the Computing Process, IBM J

R. Landauer,Irreversibility and Heat Generation in the Computing Process, IBM J. Res. Dev.5, (1961) 183

work page 1961

[7] [7]

Klaers,Landauer’s Erasure Principle in a Squeezed Thermal Memory,Phys

J. Klaers,Landauer’s Erasure Principle in a Squeezed Thermal Memory,Phys. Rev. Lett.122, 040602 (2019)

work page 2019

[8] [8]

D. Reeb, M. M. Wolf,An Improved Landauer Principle With Finite-Size Corrections, New J. Phys.16, (2014) 103011

work page 2014

[9] [9]

Iyoda, K

E. Iyoda, K. Kaneko, and T. Sagawa,Fluctuation Theorem for Many-Body Pure Quantum States,Phys. Rev. Lett. 119, 100601 (2017)

work page 2017

[10] [10]

Xu,Distinguishing pure and thermal states by Lan- dauer’s principle in open systems,Eur

H. Xu,Distinguishing pure and thermal states by Lan- dauer’s principle in open systems,Eur. Phys. J. C.84, no.4, 357 (2024)

work page 2024

[11] [11]

T. B. Batalh˜ ao, A. M. Souza, R. S. Sarthour, I. S. Oliveira, M. Paternostro, E. Lutz, and R. M. Serra,Irreversibility and the Arrow of Time in a Quenched Quantum System, Phys. Rev. Lett.115, 190601 (2015)

work page 2015

[12] [12]

G. T. Landi and M. Paternostro,Irreversible entropy production, from quantum to classical,Rev. Mod. Phys. 93, 035008 (2021)

work page 2021

[13] [13]

Esposito, U

M. Esposito, U. Harbola, and S. Mukamel,Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems,Rev. Mod. Phys.81, 1665 (2009)

work page 2009

[14] [14]

Adesso, S

G. Adesso, S. Ragy, A. R. Lee,Continuous variable quan- tum information: Gaussian states and beyond,Open Syst. Inf. Dyn.21, 1440001 (2014)

work page 2014

[15] [15]

Ptaszynski, M

K. Ptaszynski, M. Esposito,Entropy Production in Open Systems: The Predominant Role of Intra-Environment Correlations,Phys. Rev. Lett.123, (2019) 200603

work page 2019

[16] [16]

Xu and S

H. Xu and S. Y. Chen,Entropy production and correlation spreading in the interaction between particle detector and thermal baths,Eur. Phys. J. Plus.137, no.7, 821 (2022)

work page 2022

[17] [17]

Goold, M

J. Goold, M. Paternostro, K. Modi,Nonequilibrium Quan- tum Landauer Principle,Phys. Rev. Lett.114, 060602 (2015)

work page 2015

[18] [18]

A. M. Timpanaro, J. P. Santos, G. T. Landi,Landauer’s Principle at Zero Temperature,Phys. Rev. Lett.124, 240601 (2020)

work page 2020

[19] [19]

T. V. Vu, K. Saito,Finite-Time Quantum Landauer Principle and Quantum Coherence,Phys. Rev. Lett.128, 010602 (2022)

work page 2022

[20] [20]

W. G. Unruh,Notes on black hole evaporation,Phys. Rev. D.14, 870 (1976)

work page 1976

[21] [21]

W. G. Unruh,What happens when an accelerating ob- server detects a Rindler particle,Phys. Rev. D.29, 1047- 1056 (1984)

work page 1984

[22] [22]

B. S. DeWitt, inGeneral Relativity: An Einstein Cen- tenary Survey, edited by S. W. Hawking and W. Israel, Cambridge University Press, Cambridge, England, pp. 680–745 (1979)

work page 1979

[23] [23]

J. S. Ben-Benjamin, M. O. Scully, S. A. Fulling, D. M. Lee, D. N. Page, A. A. Svidzinsky, M. S. Zubairy, M. J. Duff, R. Glauber and W. P. Schleich, W. G. UnruhUnruh Acceleration Radiation Revisited,Int. J. Mod. Phys. A. 34, no.28, 1941005 (2019)

work page 2019

[24] [24]

See Supplemental Material for further information

work page

[25] [25]

This is equivalent to confining the field in a cavity with suit- able boundary conditions; the continuum description is recovered in the appropriate limit

For simplicity and to facilitate the analysis of individ- ual momentum modes, we discretize the QFT. This is equivalent to confining the field in a cavity with suit- able boundary conditions; the continuum description is recovered in the appropriate limit

work page

[26] [26]

E. T. Jaynes and F. W. Cummings,Comparison of quan- tum and semiclassical radiation theories with application 6 to the beam maser,Proc. IEEE51, 89–109 (1963)

work page 1963

[27] [27]

Hammad, A

F. Hammad, A. Landry and D. Dijamco,Influence of the dispersion relation on the Unruh effect according to the relativistic Doppler shift method,Phys. Rev. D103, 085010 (2021)

work page 2021

[28] [28]

Xu,Momentum-resolved probing of Lorentz-violating dispersion relations via Unruh-DeWitt detector,Phys

H. Xu,Momentum-resolved probing of Lorentz-violating dispersion relations via Unruh-DeWitt detector,Phys. Lett. B868, 139760 (2025)

work page 2025

[29] [29]

Barbara ˇSoda, Vivishek Sudhir, and Achim Kempf, Acceleration-Induced Effects in Stimulated Light-Matter Interactions, Phys. Rev. Lett.128, 163603 (2022)

work page 2022

[30] [30]

Simidzija and E

P. Simidzija and E. Martin-Martinez,All coherent field states entangle equally,Phys. Rev. D.96, no.2, 025020 (2017)

work page 2017

[31] [31]

Simidzija and E

P. Simidzija and E. Martin-Martinez,Non-perturbative analysis of entanglement harvesting from coherent field states,Phys. Rev. D.96, no.6, 065008 (2017)

work page 2017

[32] [32]

H. Xu, Y. C. Ong, and M. H. Yung,Landauer’s principle in qubit-cavity quantum-field-theory interaction in vacuum and thermal states,Phys. Rev. A105, 012430 (2022)

work page 2022

[33] [33]

Xu,Energy change and Landauer’s principle in the interaction between qubit and quantum field theory,Eur

H. Xu,Energy change and Landauer’s principle in the interaction between qubit and quantum field theory,Eur. Phys. J. C84, 1057 (2024). 1 SUPPLEMENTAL MATERIAL Appendix A: Proof of Theorem 1 Proof.Usingρ SR =ρ S ⊗ρ R, the unitarity of ˆU, and the identity ˆOe−βHR ˆO† = e−β ˆOHR ˆO† = e−βHeff , we have ∆S+I(S ′ :R ′) =S(ρ S)−S(ρ ′ S) +I(S ′ :R ′) =S(ρ ′ R)−...

work page 2024