Pith. sign in

REVIEW 3 major objections 6 minor 73 references

On Identification of Heat and Work in Quantum Many-Body Systems with Local Operations and Classical Communication

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proposes that heat in LOCC-based quantum energy extraction is the shortfall of Bob's extracted local energy relative to the optimal protocol, and that this heat obeys a generalized Clausius inequality with an effective temperature

desk verdict The heat definition is a good idea, but the generalized Clausius inequality collapses to an identity because the reference state is the initial B state. read the letter →

arxiv 2608.00993 v1 pith:UOROV6XD submitted 2026-08-02 quant-ph cond-mat.stat-mech

Hiroaki Matsueda This is my paper · ORCID
classification quant-phcond-mat.stat-mech
keywords quantumthermodynamicsheatandworkidentificationenergyteleportationLOCCdaemonicergotropyClausiusinequalityeffectivetemperatureKitaevmodel
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tackles a known ambiguity in quantum thermodynamics: the standard split of energy change into heat (from density-matrix change) and work (from Hamiltonian change) is not unique. It argues that in the quantum energy teleportation protocol, the ambiguity disappears if heat and work are measured relative to the optimal LOCC: optimized extraction is work (daemonic ergotropy), and the gap between actual and optimal extraction is heat, $Q = \Delta E_{B,B} - \Delta E^{*}_{B,B}$. This heat lives in nonlocal correlations between Alice and Bob that Bob cannot access, matching the traditional idea of heat as uncontrollable energy. A clear split matters for designing quantum batteries and long-distance energy transfer with minimal waste heat. The paper then derives generalized Clausius inequalities and shows in a Kitaev-like chain that the bound can be tight, with an effective temperature fixed by the reference state $\sigma_B = e^{-\beta_{\mathrm{eff}} H_B}/Z_B$.

What carries the argument

The object that carries the argument is the reference thermal state $\sigma_B = e^{-\beta_{\mathrm{eff}} H_B}/Z_B$, with $\beta_{\mathrm{eff}}$ fixed by the tightness condition $\sigma_B = \bar{\rho}^{m\times}_B$. Around this, the paper builds two formulae: the type-I identity $\beta_{\mathrm{eff}} Q + \Delta S_B = \Delta D$, where $\Delta D$ is a difference of quantum relative entropies, and the type-II inequality $-Q \le \Pi$, where $\Pi$ is an upper bound expressible through final local-state energies. The optimization protocol is understood through daemonic ergotropy, so any deviation from optimal LOCC acquires a thermodynamic meaning. In the model, the identity $k C_{AR} = h(C_{AB} - D_

What would settle it

In the Kitaev-like model, compute or measure the two-point correlators $C_{AB}$ and $D_{AB}$ and Bob's local energy change under a deliberately non-optimal LOCC; if $Q$ does not equal $\Delta E_{B,B}-\Delta E^{*}_{B,B}$ with the predicted correlator expressions, or if $-Q$ exceeds $\Pi$ for those parameters, the central definition fails. A sharper test: independently measure the population ratio of Bob's reduced state and compare it with $e^{-2\beta_{\mathrm{eff}} h}$; a mismatch would show $\beta_{\mathrm{eff}}$ is only a fitting parameter.

Watch

Extended reading notes

Core claim

The paper's central claim is that in the quantum energy teleportation protocol, heat and work become unambiguous when measured against the optimal LOCC. The extracted energy under a given protocol is work when the protocol is optimal, because it then equals daemonic ergotropy, and the shortfall from optimality is heat. Concretely, heat is $Q = \Delta E_{B,B} - \Delta E^{*}_{B,B}$, and $Q^{*}=0$ at the optimal protocol. This $Q$ is derived from a Clausius relation without referencing a change in the reduced density matrix, which is the paper's answer to the known ambiguity of the standard $\mathrm{tr}(d\rho H)$ vs $\mathrm{tr}(\rho dH)$ split. The paper further derives the type-I formula $\be

Load-bearing premise

The load-bearing premise is that the effective inverse temperature $\beta_{\mathrm{eff}}$, fixed by the reference state $\sigma_B = e^{-\beta_{\mathrm{eff}} H_B}/Z_B$ through the tightness condition, is a genuine temperature; if that effective temperature has no independent physical meaning, the Clausius inequality collapses to an identity.

Editorial extensions

If this is right

  • At the optimal LOCC, $Q=0$: the extracted energy is work in the sense of daemonic ergotropy, with no leftover heat.
  • Heat defined this way can be positive or negative, so it can represent heat leaving Bob's subsystem through nonlocal correlation even when no bulk energy crosses the boundary.
  • The generalized Clausius inequality can run in the reversed direction compared with the standard entropy-production form, because the protocol consumes the ground-state entanglement resource.
  • Maximizing Bob's local energy extraction does not maximize total extraction; in the Kitaev-like model it produces a large negative $\Delta E_{B,R}$ and fails to extract net energy, which the paper reads as unavoidable heat generation.
  • The effective inverse temperature $\beta_{\mathrm{eff}}$ is not an input but is fixed by the tightness condition; in the model it is $\beta_{\mathrm{eff}} = (1/2h)\log\left((h-\epsilon_B)/(h+\epsilon_B)\right) \ge 0$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same 'gap from optimal feedback' construction could be applied to other measurement-and-feedback engines, where heat would be defined operationally by comparing actual and optimal local operations rather than by tracing over environments.
  • The paper leaves open whether Alice's projective measurement generates heat; a testable extension is to fix Bob's unitary at the optimal value and vary Alice's measurement axis, checking whether $Q$ shifts in the predicted direction.
  • If $Q$ is adopted as the heat definition, the type-II bound $\Pi$ becomes a resource quantifier: the maximum heat is set by how far the final state under general LOCC sits from the final state under optimal LOCC, which could be measured in a spin-chain simulator.
  • The status of $\beta_{\mathrm{eff}}$ is the vulnerable point: because it is chosen to make the inequality tight, an independent check against a locally measured temperature would say whether the Clausius inequality carries thermodynamic content or is an identity in disguise.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper addresses the identification of heat and work in quantum many-body systems, focusing on a quantum energy teleportation (QET) protocol with local operations and classical communication (LOCC). The proposed central idea is that optimized LOCC yields daemonic ergotropy and should be attributed to work, while the shortfall from optimization is the seed of heat. After separating Bob's local contribution from the interaction term, the author defines proper heat as Q = ΔE_{B,B} − ΔE*_{B,B} (Eq. 17). The paper then derives a type-I relation β_eff Q + ΔS_B = ΔD (Eq. 16) and a type-II bound −Q ≤ Π (Eq. 24), with equality when the reference thermal state satisfies σ_B = \barρ^m_B. This is applied to a Kitaev-like model, where the maximum-heat protocol and effective inverse temperature are computed. The central technical concern is that, with the chosen reference state, the type-II 'inequality' becomes an exact identity for all LOCC protocols, so the claimed generalized Clausius inequality is not a nontrivial thermodynamic constraint.

Significance. The proposed definition of heat as the deviation from optimal local energy extraction is conceptually interesting and operationally well-defined. The model calculations are explicit and provide concrete expressions for Q_max and (−Q)_max, and the algebraic consistency check in Eq. (57) is verifiable. However, the main thermodynamic result—the generalized Clausius inequality—is currently empty because the reference state is fixed to be the protocol-independent outcome-averaged post-measurement state. This makes the 'tightness' automatic and strips the effective temperature of independent physical content. The manuscript can potentially be repaired by choosing β_eff from independent physical considerations or by reframing the relation as an exact identity, but as presented the central claim of a second-law-like inequality is not supported.

major comments (3)
  1. [§IV.C, §V.E, Eqs. (24), (58)] The equality condition of Eq. (24) is stated to be σ_B = \barρ^m_B, and §V.E fixes β_eff by σ_B(β_eff) = \barρ^{m×}_B. However, \barρ^m_B defined in Eq. (14) is, for any projective measurement P_A(n) on A, simply the initial reduced state of B: summing over n before tracing out A gives tr_{\bar B}(Σ_n P_A(n)ρP_A(n)) = tr_{\bar B}ρ, independent of the measurement direction r and of Bob's later operation. Since σ_B is set to this protocol-independent state, D(\barρ^m_B||σ_B)=0 for every LOCC protocol, not only for the maximum-heat protocol. Substituting this into Eqs. (16) and (19) makes the type-II 'inequality' (24) an exact equality −Q = Π for all protocols. The tightness at (−Q)_max is therefore automatic, not a nontrivial property of the model. Consequently β_eff carries no independent thermodynamic content beyond re-encoding the initial population ratio of Bob's reduced state. This is
  2. [§IV.B, Eqs. (13)–(19)] The derivation of the central relation (16) is not shown. The text states that substituting H_B = −(log σ_B + log z_B)/β_eff into Eq. (9) leads to Eq. (13), and then says the right-hand side is transformed using entropy and KL divergence to arrive at Eq. (16). No algebraic steps are given for either transformation. Because Eq. (16) is the basis for both the type-I and type-II formulae, this derivation must be supplied explicitly, either in the main text or in an appendix.
  3. [§V.C, Eqs. (45)–(54)] The maximization of −Q is only sketched. The transformation of ΔE_{B,B} into W + √(W²+X²)cos(2θ+δ) is correct, but the passage from Eq. (50) to the optimized parameters in Eq. (52) relies on 'clearly' and 'we find' rather than a complete global optimization. In particular, the minimization over θ is performed, but the simultaneous maximization of |W| and |X| over the unit vectors r and s is only asserted, and the sign condition αβ>0 is not discussed. Since Eq. (51) is used to claim consistency with Π_min, a fully derived optimization is needed.
minor comments (6)
  1. [Eq. (3)] The notation |ψ_A(t)⟩ appears to be a typo; it should be |ψ_A(n)⟩.
  2. [After Eq. (1)] The expression 'P n(n)' should be 'P_A(n)'.
  3. [§IV.B] There are typographical errors: 'authers' should be 'authors', 'Kullback-Leiber' should be 'Kullback-Leibler', and 'reset of the whole system' should be 'rest of the whole system'.
  4. [§IV.B] The claim that alternative decompositions of H_R violate spectral positivity and therefore make Q unique is unsupported. Please provide a proof or soften the statement to a conjecture.
  5. [§IV.C and §V.E] The notation σ_B(β_eff) = \barρ^{m×}_B is potentially confusing: it is a choice of reference state, not a dynamical equilibrium condition. Clarify this in the text.
  6. [Figure 2] The information-geometric foliation and the m- and e-geodesics are described qualitatively. A more precise definition of the leaves L_j and the projection conditions would help the reader follow the argument.

Circularity Check

2 steps flagged · score 8.0 of 10

Effective-temperature choice makes the generalized Clausius inequality an identity: σ_B = ρ̄^{m×}_B equals the initial B state, so D(ρ̄^m_B||σ_B)=0 for every LOCC and −Q=Π exactly.

  1. self definitional [Sec. IV.C (Maximal Heat), Eq. (24)-(27), Eq. (58)]
    "The equality condition of Eq. (24). This is given by σ_B = ¯ρ^m_B. Since ¯ρ^m_B is already in thermal equilibrium, we can say that the QET protocol is the process of moving away from it. In this case, the upper bound of Eq. (24) becomes tight, and we obtain −Q = Π. ... Thus, σ_B = ¯ρ^m_B is exactly the minimization condition of Π, and it also determines the effective inverse temperature, σ_B(β_eff) = ¯ρ^{m×}_B."

    With σ_B fixed this way, Eq. (24) is not a bound. Since P_A(n) acts only on A and sums to the identity, ρ̄^m_B = tr_{A,C1,C2}(Σ_n P_A(n)|ψ⟩⟨ψ|P_A(n)) equals the initial reduced state of B, independent of r, s, θ. The paper's own Eq. (58) gives ρ̄^{m×}_B = 1/2(I_B + (ϵ_B/h)σ^z_B) = ρ_B^i. Therefore D(ρ̄^m_B||σ_B)=0 for every LOCC, not only at maximum heat. This is the nonnegative KL term whose removal produced the type-II inequality; with it identically zero, Eq. (24) becomes −Q=Π exactly for all protocols. β_eff merely re-encodes the initial population ratio of Bob's reduced state, so the generalized Clausius inequality is an identity fixed by the choice of σ_B.

  2. fitted input called prediction [Sec. V.D, Eq. (57)]
    "Thus, the tightness of the upper bound of the Clausius inequality in Eq. (24) is correct for σ_B(β_eff) = ¯ρ^{m×}_B."

    The check is presented as a numerical verification that Π_min equals (−Q)_max (Eq. (57) vs Eq. (51)), but this equality is a logical consequence of the imposed reference state. Once σ_B=ρ̄^{m×}_B, the identity −Q=Π holds for every LOCC, so the equality of Π_min and (−Q)_max is not a test of a nontrivial thermodynamic bound; it is the same expression evaluated on both sides of an identity. The tightness claim is therefore forced by definition, not derived.

full rationale

The type-I formula (16) is an exact algebraic rearrangement, and the type-II inequality (24) is obtained by dropping the nonnegative KL divergence D(ρ̄^m_B||σ_B). The central circularity enters through the identification of the reference state. In Sec. IV.C the reference is fixed by the tightness condition σ_B = ρ̄^{m×}_B, and in Sec. V.E β_eff is solved from this equality. But because Alice's measurement is a complete projective measurement on subsystem A only, the outcome-averaged global post-measurement state is the original ground state, so ρ̄^m_B = tr_{A,C1,C2}|ψ⟩⟨ψ| is the initial reduced state of B, independent of the LOCC parameters. The paper's own Eq. (58) shows that ρ̄^{m×}_B is exactly this initial state. Hence D(ρ̄^m_B||σ_B)=0 for every protocol, not just the maximum-heat protocol, and the 'inequality' (24) is actually the identity −Q=Π everywhere. The consistency check in Sec. V.D therefore verifies an equality already imposed by the choice of σ_B, rather than confirming a nontrivial bound. β_eff carries no independent thermodynamic content; it only parameterizes the initial population ratio of Bob's reduced state. The model calculations of ΔE_B,B and ΔE^*_B,B are algebraically correct, and the citation of the author's prior results [53,54] for the second law is not the root cause, but it does not rescue the Clausius inequality from being self-definitional. Overall the central second-law-like claim reduces by construction to an identity, so the circularity score is 8.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No new physical entities such as particles or forces are introduced. The effective temperature is a modeling parameter, listed under free_parameters. The axioms include both standard mathematical tools and the paper-specific postulates about heat definition and temperature determination.

free parameters (1)
  • Effective inverse temperature β_eff = β_eff = (1/2h) log((h-ε_B)/(h+ε_B)) in the Kitaev-like model
    Introduced via reference thermal state σ_B; chosen so the Clausius inequality gives a tight bound (Eqs. 11-12, Sec. IV.C). Not a physical temperature, a modeling parameter.
assumptions (5)
  • domain assumption Second law of information thermodynamics for ΔE_{B,B} (Eq. 11) from Refs. [53,54]
    Taken as input from the author's previous papers to fix β_eff. It is not rederived here.
  • ad hoc to paper Heat is the energy gap between actual and optimal LOCC extraction
    The central postulate of the paper (Eq. 9 and Eq. 17). It is a definitional choice, not a derived result.
  • domain assumption The reference state σ_B ∝ e^{-β_eff H_B} is a valid effective equilibrium for Bob's subsystem
    Standard form used in effective thermodynamics; introduced in Eq. (10).
  • ad hoc to paper β_eff is determined by the condition σ_B = \bar{ρ}^{m×}_B (or Eq. 12)
    This makes the Clausius bound tight by construction; used in Sec. IV.C and V.D.
  • standard math Unitary invariance and concavity of von Neumann entropy, KL divergence positivity
    Used in Eq. (21) and to derive type-II inequality.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On Identification of Heat and Work in Quantum Many-Body Systems with Local Operations and Classical Communication." pith.science (2026). https://pith.science/paper/UOROV6XD

@misc{pith2026260800993,
  author       = {Pith},
  title        = {Pith review of: On Identification of Heat and Work in Quantum Many-Body Systems with Local Operations and Classical Communication},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UOROV6XD}},
  note         = {Machine review of arXiv:2608.00993}
}
read the original abstract

How we identify heat and work is a fundamental question in modern quantum thermodynamics. Usually, heat and work are attributed to changes in the density matrix and the Hamiltonian, respectively, during time-evolution processes in quantum systems. Recently, it has been recognized that this identification is ambiguous. Furthermore, quantum thermodynamics involving quantum measurement is still under development. Motivated by these on-going works, we consider a quantum many-body system from which we extract energy by local operations and classical communication (LOCC) according to the quantum energy teleportation (QET) protocol. The central idea to define heat and work unambiguously is based on a sharp insight into the optimization condition of LOCC. When LOCC is optimized, the extractable energy by QET becomes a daemonic ergotropy; thus, it can be attributed as work. On the other hand, when LOCC is not optimized, we have not squeezed out all the energy with the unitary operation. It means there is uncontrollable energy left in the system. The uncontrollable energy can be attributed as heat after careful treatment of many-body interactions. The heat term consists of nonlocal correlation due to communication between remote participants, and the correlation cannot be directly observed for the participant in the subsystem. Thus, this feature is consistent with the traditional perspective of heat as an uncontrollable energy. To deeply understand the nature of heat, we derive two types of generalized Clausius inequality in our effective quantum thermodynamics, and discuss the direction of the inequality. To justify our perspective, we examine a one-dimensional Kitaev-like model and discuss the physical meaning of effective temperature in our thermodynamics.

Figures

Figures reproduced from arXiv: 2608.00993 by the authors.

Figure 1
Figure 1. FIG. 1: Thermodynamic view of our model and protocol. The [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Geometrical relation among ¯ρ [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

73 extracted references · 64 canonical work pages

  1. [1]

    Strasberg and A

    P. Strasberg and A. Winter, Phys. Rev. X QUANTUM 2, 030202 (2021)

    work page 2021

  2. [2]

    P. P. Potts, arXiv:2406.19206

    arXiv

  3. [3]

    Ahmadi, S

    B. Ahmadi, S. Salimi, and A. S. Khorashad, Sicentific Reports13, 160 (2023)

    work page 2023

  4. [4]

    Esposito, K

    M. Esposito, K. Lindenberg, and C. Van den Broeck, New J. Phys.12, 013013 (2010)

    2010

  5. [5]

    A. E. Allahverdyan and T. M. Neiuwenhuizen, Phys. Rev. Lett.85, 1799 (2000)

    work page 2000

  6. [6]

    Hilt and E

    S. Hilt and E. Lutz, Phys. Rev. A79, 010101(R) (2009)

    work page 2009

  7. [7]

    Carrega, P

    M. Carrega, P. Solinas, M. Sassetti, and U. Weiss, Phys. Rev. Lett.116, 240403 (2016)

    work page 2016

  8. [8]

    Alipour, F

    S. Alipour, F. Benatti, F. Bakhshinezhad, M. Afsary, S. Marcantoni, and A. T. Rezakhani, Sci. Rep.6, 35568 (2016)

    2016

Show all 73 references
  1. [9]

    Y. Y. Xu, Phys. Rev. E94, 062145 (2016)

    2016

  2. [10]

    M. Bera, A. Riera, M. Lewenstein, and A. Winter, Nat. Commun.8, 2180 (2017). 10

    2017

  3. [11]

    Perarnau-Llobet, H

    M. Perarnau-Llobet, H. Wilming, A. Riera, R. Gallego, and J. Eisert, Phys. Rev. Lett.120, 120602 (2018)

    2018

  4. [12]

    Strasberg, Phys

    P. Strasberg, Phys. Rev. Lett.123, 180604 (2019)

    2019

  5. [13]

    Micadei, J

    K. Micadei, J. P. S. Peterson, A. M. Souza, R. S. Sarthour, I. S. Oliveira, G. T. Landi, T. B. Batalhao, R. M. Serra, and E. Lutz, Nature Comm.10, 2456 (2019)

    2019

  6. [14]

    Sapienza, F

    F. Sapienza, F. Cerisola, and A. J. Roncaglia, Nature Comm.10, 2492 (2019)

    2019

  7. [15]

    Dolatkhah, S

    H. Dolatkhah, S. Salimi, A. S. Khorashard, and S. Heseli, Sci. Rep.10, 9757 (2020)

    2020

  8. [16]

    J. D. Clesser and J. Anders, Phys. Rev. Lett.127, 250601 (2021)

    2021

  9. [17]

    Vallejo, A

    A. Vallejo, A. Romanelli, and R. Donangelo, Phys. Rev. E103, 042105 (2021)

    2021

  10. [18]

    Huang and W.-M

    W.-M. Huang and W.-M. Zhang, Phys. Rev. A106, 032607 (2022)

    2022

  11. [19]

    Woldsworth and R

    T. Woldsworth and R. Kawai, Phys. Rev. A106, 062604 (2022)

    2022

  12. [20]

    Elouard and C

    C. Elouard and C. L. Latune, Phys. Rev. X QUANTUM 4, 020309 (2023)

    2023

  13. [21]

    Dann and R

    R. Dann and R. Kosloff, New. J. Phys.25, 043019 (2023)

    2023

  14. [22]

    Lipka-Bartosik, G

    P. Lipka-Bartosik, G. F. Diotallevi, and P. Bakhshinezhad, Phys. Rev. Lett.132, 140402 (2024)

    2024

  15. [23]

    Ye, H.-G

    Y.-C. Ye, H.-G. Duan, and X.-T. Liang, Physica A646, 129869 (2024)

    2024

  16. [24]

    Aguilar and E

    M. Aguilar and E. Lutz, Sci. Adv.11, eadw8462 (2025)

    2025

  17. [25]

    Rivas, Phys

    ´A. Rivas, Phys. Rev. Lett.124, 160601 (2020)

    2020

  18. [26]

    Colla and H.-P

    A. Colla and H.-P. Breuer, Phys. Rev. A105, 052216 (2022)

    2022

  19. [27]

    Seegebrecht and T

    A. Seegebrecht and T. Schilling, J. Stat. Phys.191, 34 (2024)

    2024

  20. [28]

    Colla and H.-P

    A. Colla and H.-P. Breuer, Quantum Sci. Technol.10, 015047 (2025)

    2025

  21. [29]

    Rupush and O

    W. Rupush and O. Gr ˚ an¨ as, arXiv:2403.02022

    arXiv

  22. [30]

    T. Zhou, J. Pu, and X. Wu, arXiv:2501.00832

    arXiv

  23. [31]

    A. E. Allahverdyan, R. Balian, Th. M. Nieuwenhuizen, Europhys. Lett.67, 565 (2004)

    2004

  24. [32]

    Pusz and S

    W. Pusz and S. L. Woronowicz, Commun. math. Phys. 58, 273 (1978)

    1978

  25. [33]

    Skrzypczyk, A

    P. Skrzypczyk, A. J. Short, and S. Popescu, Nat. Com- mun.5, 4185 (2014)

    2014

  26. [34]

    Perarnau-Llobet, K

    M. Perarnau-Llobet, K. V. Hovhannisyan, M. Huber, P. Skrzypczyk, N. Brunner, and A. Ac ´ ın, Phys. Rev. X5, 041011 (2015)

    2015

  27. [35]

    Francica, J

    G. Francica, J. Goold, F. Plastina, M. Paternostro, npj Quant. Inf.3, 12 (2018)

    2018

  28. [36]

    Bernards, M

    F. Bernards, M. Kleinmann, O. G¨ uhne, and M. Pater- nostro, Entropy21, 771 (2019)

    2019

  29. [37]

    Francica, F

    G. Francica, F. C. Binder, G. Guarnieri, M. T. Mitchison, J. Goold, and F. Plastina, Phys. Rev. Lett.125, 180603 (2020)

    2020

  30. [38]

    Lobejko, Nat

    M. Lobejko, Nat. Commun,12, 918 (2021)

    2021

  31. [39]

    Salvia and V

    R. Salvia and V. Giovannetti, Phys. Rev. A105, 012414 (2022)

    2022

  32. [40]

    Touil, B

    A. Touil, B. Cakmak, and S. Deffner, J. Phys. A: Math. Theor.55, 025301 (2022)

    2022

  33. [41]

    Hadipour and S

    M. Hadipour and S. Haseli, Sci. Rep.14, 24876 (2024)

    2024

  34. [42]

    R. Basu, A. Chakrabortyy, H. Badhani, M. Alimuddin, and S. Bhattacharya, Phys. Rev. A111, 032416 (2025)

    2025

  35. [43]

    Hotta, Phys

    M. Hotta, Phys. Lett. A372, 5671 (2008)

    2008

  36. [44]

    Hotta, Phys

    M. Hotta, Phys. Rev. D78, 045006 (2008)

    2008

  37. [45]

    Hotta, J

    M. Hotta, J. Phys. Soc. Jpn.78, 034001 (2009)

    2009

  38. [46]

    Hotta, Phys

    M. Hotta, Phys. Lett. A374, 3416 (2010)

    2010

  39. [47]

    M. Frey, K. Funo, and M. Hotta, Phys. Rev. E90, 012127 (2014)

    2014

  40. [48]

    Trevison and M

    J. Trevison and M. Hotta, J. Phys. A: Math. Theor.48, 175302 (2015)

    2015

  41. [49]

    N. A. Rodriguez-Briones, H. Katiyar, E. Martin- Martinez, and R. Laflamme., Phys. Rev. Lett.130, 110801 (2023)

    2023

  42. [50]

    Ikeda, Phys

    K. Ikeda, Phys. Rev. Appl.20, 024051 (2023)

    2023

  43. [51]

    Wang and S

    J. Wang and S. Yao, Quantum8, 1564 (2024)

    2024

  44. [52]

    Hotta and K

    M. Hotta and K. Ikeda, Quant. Inf. Proc.24, 186 (2025)

    2025

  45. [53]

    Matsueda, Y

    H. Matsueda, Y. Masaki, K. Itoh, A. Ono, and J. Nasu, Phys. Rev. Res.7, 033137 (2025)

    2025

  46. [54]

    K. Itoh, Y. Masaki, and H. Matsueda, Phys. Rev. E113, 024108 (2026)

    2026

  47. [55]

    Sagawa and M

    T. Sagawa and M. Ueda, Phys. Rev. Lett.100, 080403 (2008)

    2008

  48. [56]

    Tajima, Phys

    H. Tajima, Phys. Rev. E88, 042143 (2013)

    2013

  49. [57]

    J. J. Park, K.-H. Kim, T. Sagawa, and S. W. Kim, Phys. Rev. Lett.111, 230402 (2013)

    2013

  50. [58]

    K. Funo, Y. Watanabe, and M. Ueda, Phys. Rev. A88, 052319 (2013)

    2013

  51. [59]

    Manzano, F

    G. Manzano, F. Plastina, and R. Zambrini, Phys. Rev. Lett.121, 120602 (2018)

    2018

  52. [60]

    Minagawa, M

    S. Minagawa, M. H. Mohammadt, K. Sakai, K. Kato, and F. Buscemi, npj Quant. Inf.11, 18 (2025)

    2025

  53. [61]

    Abe and S

    S. Abe and S. Okuyama, Phys. Rev. E83, 021121 (2011)

    2011

  54. [62]

    Abe, Phys

    S. Abe, Phys. Rev. E83, 041117 (2011)

    2011

  55. [63]

    Abe and S

    S. Abe and S. Okuyama, Phys. Rev. E85, 011104 (2012)

    2012

  56. [64]

    C. M. Bender, D. C. Brody, and B. K. Meister, J. Phys. A: Math. Gen.33, 4427-4436 (2000)

    2000

  57. [65]

    Elouard, D

    C. Elouard, D. A. Herrera-Mart ´ ı, M. Clusel, and A. Auff` eves, npj Quant. Inf.3, 9 (2017)

    2017

  58. [66]

    Yamamoto and Y

    T. Yamamoto and Y. Tokura, Phys. Rev. Res.6, 013300 (2024)

    2024

  59. [67]

    Cafaro and S

    C. Cafaro and S. Mancini, Physica A391, 1610 (2012)

    2012

  60. [68]

    Tan Van Vu and Keiji Saito, Phys. Rev. X13, 011013 (2023)

    2023

  61. [69]

    Nakazato and S

    M. Nakazato and S. Ito, Phys. Rev. Res.3, 043093 (2021)

    2021

  62. [70]

    Dechant and S

    A. Dechant and S. Sasa, J. Stat. Mech.: Theor. Exp., 063202, (2021)

    2021

  63. [71]

    Kitaev, Ann

    A. Kitaev, Ann. Phys. (NY)321, 2 (2006)

    2006

  64. [72]

    Nussinov and J

    Z. Nussinov and J. van den Brink, Rev. Mod. Phys.87, 1 (2015)

    2015

  65. [73]

    Hermanns, I

    M. Hermanns, I. Kimchi, and J. Knolle, Annu. Rev. Con- dens. Matter Phys.9, 17 (2018)

    2018

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.