pith. sign in

arxiv: 2507.03471 · v2 · submitted 2025-07-04 · 🪐 quant-ph

Nonequilibrium thermometry via an ensemble of initially correlated qubits

Enrico Trombetti , Marco Malitesta , Marco Pezzutto , Stefano Gherardini This is my paper

Pith reviewed 2026-05-19 06:35 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum thermometrynonequilibrium dynamicsquantum correlationsquantum Fisher informationMarkovian master equationqubit ensemblethermal bathtransient sensitivity
0
0 comments X

The pith

Initial quantum correlations among qubits enhance the Quantum Fisher Information for bath temperature estimation during nonequilibrium thermalization.

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

The paper investigates a thermometry protocol in which an ensemble of qubits weakly interacts with a thermal bath, encoding the bath temperature in the rates of a Markovian thermalization process. It demonstrates that the Quantum Fisher Information for temperature shows a peak during the transient dynamics before steady state is reached, with larger values at higher temperatures when the initial state has high ground-state population or coherence. The central result is strong numerical evidence that, for fixed single-qubit reduced states, any initial quantum correlations improve the QFI relative to separable states. Although no considered state exceeds the performance of the pure separable ground state, maximally entangled states approach the standard quantum limit for extremely hot baths, and the work identifies practical initial states for high-precision thermometers under Markovian constraints.

Core claim

For an ensemble of qubits undergoing Markovian thermalization with a temperature-dependent dissipator, when the single-qubit reduced density matrices are held fixed, the inclusion of initial quantum correlations among the qubits always produces a higher Quantum Fisher Information for the bath temperature than a corresponding separable state; maximally entangled initial states achieve values close to the standard quantum limit when the bath is extremely hot.

What carries the argument

Quantum Fisher Information for the bath temperature, evaluated on the transient state evolving under a Markovian master equation starting from ensembles with identical single-qubit marginals but varying amounts of quantum correlation.

If this is right

  • A peak in sensitivity occurs during the early transient of thermalization rather than at steady state.
  • Maximally entangled initial states remain nearly optimal for estimating very high bath temperatures.
  • The Markovian character of the dynamics prevents superlinear scaling of precision with the number of qubits.
  • States with large ground-state population or high initial coherence are the most effective choices for probe preparation.

Where Pith is reading between the lines

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

  • The same correlation advantage might appear in other open-system metrology tasks where the parameter enters the dissipator rather than the Hamiltonian.
  • Testing the protocol in a non-Markovian regime could reveal whether correlations enable better scaling once memory effects are present.
  • The findings suggest concrete preparation recipes for correlated qubit sensors in quantum devices operating far from equilibrium.

Load-bearing premise

The qubits stay weakly coupled to a large thermal bath, so their reduced dynamics follows a Markovian master equation whose dissipator depends on the unknown temperature.

What would settle it

A numerical counter-example or laboratory realization in which an initial state containing quantum correlations yields equal or lower Quantum Fisher Information than a separable state sharing the exact same single-qubit reduced density matrices.

Figures

Figures reproduced from arXiv: 2507.03471 by Enrico Trombetti, Marco Malitesta, Marco Pezzutto, Stefano Gherardini.

Figure 1
Figure 1. Figure 1: FIG. 1: Quantum-map representation of the system [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: QFI of the evolved state of an ensemble of 2 qubits initialized in different states. Different colors correspond [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: max [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Relative enhancement [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: QFI of the evolved state of a 2-qubit thermometer initialized in the identity-mixtured GHZ state ˆρ [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Relative enhancement [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Panels (a)-(b): QFI associated to two thermometers’ state, initialized in the squeezed state of Eq. (26), as a [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Enhancement of estimation precision [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Panels (a)-(b): QFI associated to two thermometers’ state, initialized in the +-GHZ state of Eq. (24) with [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: QFI at time [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
read the original abstract

We investigate a nonequilibrium quantum thermometry protocol in which an ensemble of qubits, acting as temperature probes, is weakly coupled to a macroscopic thermal bath. The temperature of the bath, the parameter of interest, is encoded in the dissipator of a Markovian thermalization process. For some relevant initial states, we observe a peak in the Quantum Fisher Information (QFI) during the transient of the thermalization, indicating enhanced sensitivity in early-time dynamics. This effect becomes more pronounced at higher bath temperatures and is further enhanced when the initial reduced state of the qubits has a large ground-state population and/or it is highly coherent. We also focus on the role of initial quantum correlations in the thermometric performance, which emerge as a central feature of this work. We find strong numerical evidence that, given same single-qubit reduced states, the inclusion of quantum correlations among the qubits of the ensemble always yields an enhanced QFI. Moreover, even if none of the considered states outperform the (pure, separable) ground state, maximally entangled states display QFIs values remarkably close to the standard quantum limit when probing extremely hot thermal baths. Finally, although the Markovian dynamics does not permit superlinear scaling of the QFI with the number of probes, we identify the most effective initial states for designing high-precision quantum thermometers within this setting. We also provide concrete guidelines for experimental implementations.

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.

Referee Report

3 major / 2 minor

Summary. The manuscript studies nonequilibrium quantum thermometry with an ensemble of qubits weakly coupled to a macroscopic thermal bath. The bath temperature is encoded in the dissipator of a Markovian master equation. Numerical simulations reveal transient peaks in the quantum Fisher information (QFI) for the temperature parameter. A central observation is that, for fixed single-qubit reduced states, the presence of initial quantum correlations among the qubits enhances the QFI relative to product states. Maximally entangled states approach the standard quantum limit for hot baths, and the work identifies optimal initial states together with experimental guidelines.

Significance. If the reported numerical enhancement by correlations proves robust, the result would demonstrate a concrete metrological advantage of initial entanglement in transient, nonequilibrium thermometry. The identification of states that remain close to the SQL at high temperatures and the provision of concrete experimental guidelines constitute useful contributions to quantum sensing protocols.

major comments (3)
  1. [§IV] §IV (numerical results on correlated vs. product states): the assertion that correlations 'always' yield higher QFI for identical single-qubit marginals rests on a finite sample of states (Werner, Bell, etc.). Without either an exhaustive search for counter-examples or an analytical argument showing monotonicity of QFI with respect to correlation measures, the universal qualifier remains unsupported by the presented evidence.
  2. [Protocol description and Fig. 2–4] Protocol description and Fig. 2–4: it is not stated whether the reported QFI values are taken at a common fixed time t for all initial states or at individually optimized times t* that maximize each curve. If the latter, the comparison between correlated and uncorrelated ensembles is not protocol-fair and may inflate the apparent advantage.
  3. [§II] §II (model and master equation): the reduced dynamics is assumed to remain Markovian and local even when the initial state is entangled. No derivation or numerical check is given showing that the weak-coupling Born–Markov approximation continues to hold for the correlated inputs used in the QFI calculations; this assumption is load-bearing for all subsequent claims.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'for some relevant initial states' is vague; a brief enumeration of the families considered would improve clarity.
  2. [Figure captions] Figure captions: add explicit statements of the time at which each QFI curve is evaluated and whether any post-selection on t was performed.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address each of the major comments below and have made revisions to the manuscript to improve clarity and strengthen the arguments.

read point-by-point responses

  1. Referee: [§IV] §IV (numerical results on correlated vs. product states): the assertion that correlations 'always' yield higher QFI for identical single-qubit marginals rests on a finite sample of states (Werner, Bell, etc.). Without either an exhaustive search for counter-examples or an analytical argument showing monotonicity of QFI with respect to correlation measures, the universal qualifier remains unsupported by the presented evidence.

    Authors: We agree with the referee that the term 'always' implies a universality that our numerical evidence does not fully establish. Our simulations covered a range of states with fixed marginals, including Werner states, maximally entangled Bell states, and randomly generated entangled states, and in every instance the correlated states showed higher QFI. To address this, we will revise the relevant sections to replace 'always' with 'in all cases we have examined' and provide more details on the state sampling procedure. We have searched for counterexamples within the parameter space but found none; however, we acknowledge that an analytical proof would be ideal and leave this as an open question for future work. revision: partial

  2. Referee: [Protocol description and Fig. 2–4] Protocol description and Fig. 2–4: it is not stated whether the reported QFI values are taken at a common fixed time t for all initial states or at individually optimized times t* that maximize each curve. If the latter, the comparison between correlated and uncorrelated ensembles is not protocol-fair and may inflate the apparent advantage.

    Authors: We thank the referee for highlighting this ambiguity. Upon review, the QFI curves are shown as functions of time, and the advantage of correlations is demonstrated both at the individual peak times and at selected fixed times. To ensure protocol fairness, we will update the text in the protocol description and figure captions to explicitly state that comparisons are also made at common evolution times t, where the enhancement due to correlations remains evident. We will add a note that the optimal times t* are comparable across the states considered, minimizing any potential bias. revision: yes

  3. Referee: [§II] §II (model and master equation): the reduced dynamics is assumed to remain Markovian and local even when the initial state is entangled. No derivation or numerical check is given showing that the weak-coupling Born–Markov approximation continues to hold for the correlated inputs used in the QFI calculations; this assumption is load-bearing for all subsequent claims.

    Authors: This is a valid concern regarding the validity of the master equation for entangled initial states. The standard derivation of the Born-Markov master equation for a system weakly coupled to a thermal bath relies on the separation of timescales and the large bath limit, and it applies to the reduced dynamics irrespective of the initial system state (as long as there are no initial system-bath correlations). We will add a paragraph in §II with a brief justification and references to literature supporting the use of such master equations for arbitrary initial system states. Additionally, we will include a short discussion on why the locality of the dissipator is preserved. revision: yes

Circularity Check

0 steps flagged

No circularity: QFI computed directly from standard master-equation evolution

full rationale

The paper reports numerical evaluation of the Quantum Fisher Information for temperature encoded in the dissipator of a Markovian master equation. The enhancement from initial quantum correlations is presented as an observation for states sharing identical single-qubit reduced density matrices, obtained by solving the time-dependent dynamics and computing the QFI at transient times. No parameters are fitted to the target QFI, no self-definitional loops appear, and no load-bearing self-citations or imported uniqueness theorems are required. The derivation chain relies on standard open-quantum-systems techniques applied to explicitly constructed initial states, remaining self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The protocol assumes a standard Markovian thermal master equation whose dissipator encodes the bath temperature; this is a domain assumption rather than a derived result. No free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The reduced dynamics of the qubit ensemble is accurately captured by a Markovian master equation with temperature-dependent dissipator under weak coupling to a macroscopic bath.
    Stated directly in the abstract as the encoding of temperature in the dissipator of a Markovian thermalization process.

pith-pipeline@v0.9.0 · 5784 in / 1326 out tokens · 52918 ms · 2026-05-19T06:35:12.047113+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Surpassing thermal-state limit in thermometry via non-completely positive quantum encoding

    quant-ph 2026-04 unverdicted novelty 7.0

    General probe-environment correlations enable non-completely positive encodings that surpass the thermal-state bound in quantum thermometry precision.

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages · cited by 1 Pith paper

  1. [1]

    Time-evolution of a multi-qubit state The nonequilibrium thermometry setting, which we are referring to, considers a model where the interaction be- tween each qubit thermometer and the thermal bath is local and independent. Consequently, the evolution map of an ensemble of these thermometers is obtained by ap- plying N independent quantum channels Φ (t) ...

    work page

  2. [2]

    Quan- tum Reservoir Computing (QuReCo)

    state, and 0 .25 for the 12N /2N state. The errors on the fitted slopes are reported (with two significant digits) only for initially correlated states of the thermometers, since for initially separable states all the values of the QFI are perfectly aligned on a straight curve due to the additivity property of the QFI. mometers in the state ˆ ρin = |0⟩ ⟨0...

    work page 2020

  3. [3]

    ˆK1 = 1 2 ∂β q 0 0 q(1 − p) , ∂β( ˆK †

    work page

  4. [4]

    ˆK2 = 1 2 ∂β 0 0 0 pq , ∂β( ˆK †

    work page

  5. [5]

    ˆK3 = 1 2 ∂β (1 − p)(1 − q) 0 0 −q , ∂β( ˆK †

    work page

  6. [6]

    (C2) Finally, we obtain ˆM2 = i X a ∂β( ˆK † a) ˆKa = i 2 ∂β 1 0 0 0 = 0 0 0 0

    ˆK4 = 1 2 ∂β p(1 − q) 0 0 0 . (C2) Finally, we obtain ˆM2 = i X a ∂β( ˆK † a) ˆKa = i 2 ∂β 1 0 0 0 = 0 0 0 0 . (C3) While this result does not exclude the possibility of equivalent Kraus representations for which the operator norm of ˆM2 is nonzero, the minimization over all repre- sentations in the bound (35) ensures that our result still provides a suff...

    work page

  7. [7]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Quantum metrology, Phys. Rev. Lett. 96, 010401 (2006)

    work page 2006

  8. [8]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nat. Photonics 5, 222 (2011)

    work page 2011

  9. [9]

    C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017)

    work page 2017

  10. [10]

    Aasi and al., Enhanced sensitivity of the LIGO gravi- tational wave detector by using squeezed states of light, Nat

    J. Aasi and al., Enhanced sensitivity of the LIGO gravi- tational wave detector by using squeezed states of light, Nat. Photonics 7, 613 (2013)

    work page 2013

  11. [11]

    Montenegro, C

    V. Montenegro, C. Mukhopadhyay, R. Yousefjani, S. Sarkar, U. Mishra, M. G. Paris, and A. Bayat, Re- view: Quantum metrology and sensing with many-body systems, Phys. Rep. 1134, 1 (2025)

    work page 2025

  12. [12]

    Mehboudi, A

    M. Mehboudi, A. Sanpera, and L. A. Correa, Thermom- etry in the quantum regime: Recent theoretical progress, J. Phys. A: Math. Theor. 52, 303001 (2010)

    work page 2010

  13. [13]

    Abiuso, P

    P. Abiuso, P. Andrea Erdman, M. Ronen, F. No´ e, G. Haack, and M. Perarnau-Llobet, Optimal thermome- ters with spin networks, Quantum Sci. Technol.9, 035008 (2024)

    work page 2024

  14. [14]

    Brunelli, S

    M. Brunelli, S. Olivares, and M. G. A. Paris, Qubit ther- mometry for micromechanical resonators, Phys. Rev. A 84, 032105 (2011)

    work page 2011

  15. [15]

    Brunelli, S

    M. Brunelli, S. Olivares, M. Paternostro, and M. G. A. Paris, Qubit-assisted thermometry of a quantum har- monic oscillator, Phys. Rev. A 86, 012125 (2012)

    work page 2012

  16. [16]

    Montenegro, M

    V. Montenegro, M. G. Genoni, A. Bayat, and M. G. A. Paris, Mechanical oscillator thermometry in the nonlin- ear optomechanical regime, Phys. Rev. Res. 2, 043338 (2020)

    work page 2020

  17. [17]

    Jevtic, D

    S. Jevtic, D. Newman, T. Rudolph, and T. M. Stace, Single-qubit thermometry, Phys. Rev. A 91, 012331 (2015)

    work page 2015

  18. [18]

    Razavian, C

    S. Razavian, C. Benedetti, M. Bina, A.-K. Y., and M. G. A. Paris, Quantum thermometry by single-qubit dephasing, Eur. Phys. J. Plus 134, 284 (2019)

    work page 2019

  19. [19]

    W. K. Tham, H. Ferretti, A. V. Sadashivan, and A. M. Steinberg, Simulating and Optimising Quantum Ther- mometry Using Single Photons, Sci. Rep. 6, 38822 (2016)

    work page 2016

  20. [20]

    Mancino, M

    L. Mancino, M. Sbroscia, I. Gianani, E. Roccia, and M. Barbieri, Quantum Simulation of Single-Qubit Ther- mometry Using Linear Optics, Phys. Rev. Lett. 118, 130502 (2017)

    work page 2017

  21. [21]

    Sab´ ın, A

    C. Sab´ ın, A. White, L. Hackermuller, and I. Fuentes, Im- purities as a quantum thermometer for a Bose-Einstein condensate, Sci. Rep. 4, 1 (2014)

    work page 2014

  22. [22]

    T. H. Johnson, F. Cosco, M. T. Mitchison, D. Jaksch, and S. R. Clark, Thermometry of ultracold atoms via nonequilibrium work distributions, Phys. Rev. A 93, 053619 (2016)

    work page 2016

  23. [23]

    M. T. Mitchison, T. Fogarty, G. Guarnieri, S. Campbell, T. Busch, and J. Goold, In Situ Thermometry of a Cold Fermi Gas via Dephasing Impurities, Phys. Rev. Lett. 125, 080402 (2020)

    work page 2020

  24. [24]

    Sultanov, M

    A. Sultanov, M. Kuzmanovi´ c, A. V. Lebedev, and G. S. Paraoanu, Protocol for temperature sensing using a three-level transmon circuit, Appl. Phys. Lett. 119, 14 (2021)

    work page 2021

  25. [25]

    O. P. de S´ a Neto, H. A. S. Costa, G. A. Prataviera, and 17 M. C. de Oliveira, Temperature estimation of a pair of trapped ions, Sci. Rep. 12, 1 (2022)

    work page 2022

  26. [26]

    J.-M. Yang, H. Yang, and L. Lin, Quantum Dot Nano Thermometers Reveal Heterogeneous Local Thermogen- esis in Living Cells, ACS Nano 5, 5067 (2011)

    work page 2011

  27. [27]

    Wu and T

    Y. Wu and T. Weil, Recent Developments of Nanodia- mond Quantum Sensors for Biological Applications, Adv. Sci. 9, 2200059 (2022)

    work page 2022

  28. [28]

    Aiache, C

    Y. Aiache, C. Seida, K. El Anouz, and A. El Allati, Non-markovian enhancement of nonequilibrium quantum thermometry, Phys. Rev. E 110, 024132 (2024)

    work page 2024

  29. [29]

    Cavina, L

    V. Cavina, L. Mancino, A. De Pasquale, I. Gianani, M. Sbroscia, R. I. Booth, E. Roccia, R. Raimondi, V. Giovannetti, and M. Barbieri, Bridging thermodynam- ics and metrology in nonequilibrium quantum thermom- etry, Phys. Rev. A 98, 050101 (2018)

    work page 2018

  30. [30]

    Fraz˜ ao, M

    G. Fraz˜ ao, M. Pezzutto, Y. Omar, E. Zambrini Cruzeiro, and S. Gherardini, Coherence-enhanced single-qubit thermometry out of equilibrium, Entropy 26, 568 (2024)

    work page 2024

  31. [31]

    S. Seah, S. Nimmrichter, D. Grimmer, J. P. Santos, V. Scarani, and G. T. Landi, Collisional Quantum Ther- mometry, Phys. Rev. Lett. 123, 180602 (2019)

    work page 2019

  32. [32]

    Ravell Rodr´ ıguez, M

    R. Ravell Rodr´ ıguez, M. Mehboudi, M. Horodecki, and M. Perarnau-Llobet, Strongly coupled fermionic probe for nonequilibrium thermometry, New J. Phys. 26, 013046 (2024)

    work page 2024

  33. [33]

    Salvatori, A

    G. Salvatori, A. Mandarino, and M. G. A. Paris, Quan- tum metrology in Lipkin-Meshkov-Glick critical systems, Phys. Rev. A 90, 022111 (2014)

    work page 2014

  34. [34]

    D. M. Greenberger, M. A. Horne, and A. Zeilinger, Going beyond bell’s theorem, in Bell’s theorem, quantum theory and conceptions of the universe (Springer, 1989) pp. 69– 72

    work page 1989

  35. [35]

    D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, Bell’s theorem without inequalities, Am. J. Phys. 58, 1131 (1990)

    work page 1990

  36. [36]

    Horodecki and P

    M. Horodecki and P. Horodecki, Reduction criterion of separability and limits for a class of distillation protocols, Phys. Rev. A 59, 4206 (1999)

    work page 1999

  37. [37]

    R. F. Werner, Quantum states with Einstein-Podolsky- Rosen correlations admitting a hidden-variable model, Phys. Rev. A 40, 4277 (1989)

    work page 1989

  38. [38]

    N. Li, S. Luo, and Y. Sun, Information-theoretic aspects of Werner states, Ann. Phys. 424, 168371 (2021)

    work page 2021

  39. [39]

    Pezz´ e, A

    L. Pezz´ e, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys. 90, 035005 (2018)

    work page 2018

  40. [40]

    J. Ma, X. Wang, C. Sun, and F. Nori, Quantum spin squeezing, Phys. Rep. 4509, 89 (2011)

    work page 2011

  41. [41]

    W. J. Eckner, N. Darkwah Oppong, A. Cao, A. W. Young, W. R. Milner, J. M. Robinson, J. Ye, and A. M. Kaufman, Realizing spin squeezing with Rydberg inter- actions in an optical clock, Nature 621, 734 (2023)

    work page 2023

  42. [42]

    Kitagawa and M

    M. Kitagawa and M. Ueda, Squeezed spin states, Phys. Rev. A 47, 5138 (1993)

    work page 1993

  43. [43]

    Pezz´ e and A

    L. Pezz´ e and A. Smerzi, Entanglement, nonlinear dy- namics, and the Heisenberg limit, Phys. Rev. Lett. 102, 100401 (2009)

    work page 2009

  44. [44]

    Helstrom, Minimum mean-squared error of estimates in quantum statistics, Phys

    C. Helstrom, Minimum mean-squared error of estimates in quantum statistics, Phys. Lett. A. 25, 101 (1967)

    work page 1967

  45. [45]

    J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum fisher information matrix and multiparameter estimation, J. Phys. A: Math. Theor. 53, 023001 (2019)

    work page 2019

  46. [46]

    (29) when applied to thermal states (and to any state diagonal in the Hamiltonian eigenbasis) [23]

    Interestingly, the standard thermometry scheme that in- volves measurements of the mean energy and error prop- agation is sufficient to saturate the lower bound in Eq. (29) when applied to thermal states (and to any state diagonal in the Hamiltonian eigenbasis) [23]

    work page

  47. [47]

    Ko lody´ nski and R

    J. Ko lody´ nski and R. Demkowicz-Dobrza´ nski, Efficient tools for quantum metrology with uncorrelated noise, New J. Phys. 15, 073043 (2013)

    work page 2013

  48. [48]

    Demkowicz-Dobrza´ nski, J

    R. Demkowicz-Dobrza´ nski, J. Ko lody´ nski, and M. Gut ¸˘ a, The elusive Heisenberg limit in quantum-enhanced metrology, Nat. Commun. 3, 1063 (2012)

    work page 2012

  49. [49]

    Fujiwara and H

    A. Fujiwara and H. Imai, A fibre bundle over manifolds of quantum channels and its application to quantum statis- tics, J. Phys. A: Math. Theor. 41, 255304 (2008)

    work page 2008

  50. [50]

    Horodecki, P

    R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009)

    work page 2009

  51. [51]

    W. R. Inc., Mathematica, Version 14.2, champaign, IL, 2024

    work page 2024

  52. [52]

    Buffoni, S

    L. Buffoni, S. Gherardini, E. Zambrini Cruzeiro, and Y. Omar, Third Law of Thermodynamics and the Scaling of Quantum Computers, Phys. Rev. Lett. 129, 150602 (2022)

    work page 2022

  53. [53]

    Buffoni and M

    L. Buffoni and M. Campisi, Cooperative quantum infor- mation erasure, Quantum 7, 961 (2023)

    work page 2023

  54. [54]

    Buffoni and M

    L. Buffoni and M. Campisi, Collective preparation of large quantum registers with high fidelity, Quantum Sci. Technol. 10, 025053 (2025)

    work page 2025

  55. [55]

    T. P. Harty, D. T. C. Allcock, C. J. Ballance, L. Guidoni, H. A. Janacek, N. M. Linke, D. N. Stacey, and D. M. Lu- cas, High-fidelity preparation, gates, memory, and read- out of a trapped-ion quantum bit, Phys. Rev. Lett. 113, 220501 (2014)

    work page 2014