pith. sign in

arxiv: 2505.13634 · v2 · submitted 2025-05-19 · 🪐 quant-ph

OH molecule as a quantum probe to jointly estimate electric and magnetic fields

Luca Previdi , Francesco Albarelli , Matteo G. A. Paris This is my paper

Pith reviewed 2026-05-22 13:53 UTC · model grok-4.3

classification 🪐 quant-ph
keywords OH moleculequantum probejoint field estimationmultiparameter quantum estimationStark-Zeeman Hamiltonianmeasurement incompatibilitythermal probessequential control
0
0 comments X

The pith

The OH molecule serves as a quantum probe for jointly estimating electric and magnetic fields while accounting for measurement incompatibility.

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

The paper establishes the hydroxyl radical as a natural quantum probe for simultaneous estimation of electric and magnetic fields due to its dual dipole moments and simple diatomic model. It applies multiparameter quantum estimation theory to optimize both stationary strategies using ground and thermal states of the Stark-Zeeman Hamiltonian and dynamical strategies starting from pure or thermal states. A key result is that increasing temperature in thermal probes can lower the total estimation error by weakening correlations between the two field parameters. The work also demonstrates how an optimal sequential control protocol mitigates the performance loss from noncommuting observables and assesses its robustness.

Core claim

The OH molecule carries both electric and magnetic dipole moments and admits a simple diatomic model, making it suitable for joint estimation of electric and magnetic fields. Stationary strategies identify optimal operating points for ground and thermal states, with the finding that higher temperature reduces overall error by weakening parameter correlations. Dynamical strategies with pure and thermal initial states illustrate incompatibility effects for mixed probes, yet an optimal sequential control protocol overcomes noncommutativity limitations with robustness in the multiparameter setting.

What carries the argument

The Stark-Zeeman Hamiltonian of the OH molecule combined with the quantum Fisher information matrix from multiparameter quantum estimation theory, used to quantify and minimize the trade-off from incompatible measurements of the two fields.

If this is right

  • Optimal operating points exist for stationary estimation using ground and thermal states of the Stark-Zeeman Hamiltonian.
  • Increasing temperature can reduce overall estimation error for thermal probes by weakening correlations between the fields.
  • Sequential control protocols can overcome limitations from noncommutativity in the dynamical regime.
  • The control protocol maintains robustness when extended to the full multiparameter estimation setting.

Where Pith is reading between the lines

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

  • The temperature-induced error reduction may appear in other multiparameter sensing tasks where parameter correlations dominate the precision limit.
  • Similar molecular probes with dual dipoles could be tested for field estimation in settings like astrophysical environments or lab plasmas.
  • Experimental tests could compare the predicted quantum bounds against actual measurement outcomes using trapped or beam OH molecules.

Load-bearing premise

The OH molecule can be modeled accurately as a diatomic with both electric and magnetic dipole moments, permitting direct use of multiparameter quantum estimation theory without major unmodeled effects.

What would settle it

An experiment that prepares OH molecules in thermal states at different temperatures, applies controlled electric and magnetic fields, performs joint estimation, and checks whether the total error decreases with rising temperature as predicted.

Figures

Figures reproduced from arXiv: 2505.13634 by Francesco Albarelli, Luca Previdi, Matteo G. A. Paris.

Figure 1
Figure 1. Figure 1: FIG. 1. Pictorial representation of the OHM with static elec [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Scalar bound on the total estimation error [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Scalar bound on the total estimation error [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Scalar bound on the total estimation error [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Scalar bound on the total estimation error [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Plot of the asymptotic incompatibility parameter [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Pictorial representation of a sequential feedback [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Scalar bound on the total estimation error [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
read the original abstract

The hydroxyl radical, hereafter referred to as the OH molecule (OHM), carries both electric and magnetic dipole moments and, as a diatomic molecule, admits a comparatively simple and accurate model. This makes it a natural quantum probe for the joint estimation of electric and magnetic fields. Here we study simultaneous estimation of both fields using the tools of multiparameter quantum estimation theory, explicitly accounting for the performance loss caused by measurement incompatibility. We analyze and optimize both stationary and dynamical estimation strategies. In the stationary regime we consider ground and thermal states of the Stark-Zeeman Hamiltonian and identify optimal operating points. For thermal probes we find a nontrivial multiparameter effect: increasing the temperature can reduce the overall estimation error by weakening parameter correlations. In the dynamical regime we study both pure and thermal initial states, illustrating nontrivial manifestations of incompatibility for mixed probes. Finally, we show that an optimal sequential control protocol can overcome limitations due to noncommutativity, and we assess its robustness in the multiparameter setting.

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

2 major / 2 minor

Summary. The manuscript proposes the OH molecule as a natural quantum probe for joint estimation of electric and magnetic fields, leveraging its electric and magnetic dipole moments within a diatomic model. It applies multiparameter quantum estimation theory to analyze and optimize both stationary strategies (using ground and thermal states of the Stark-Zeeman Hamiltonian, identifying optimal operating points and a nontrivial temperature effect that can reduce error by weakening correlations) and dynamical strategies (with pure and thermal initial states, including an optimal sequential control protocol to mitigate incompatibility effects).

Significance. If the central claims hold, the work offers a concrete molecular platform for multiparameter quantum sensing with explicit accounting for measurement incompatibility, which is a load-bearing practical issue. The reported temperature dependence for thermal probes and the sequential control protocol represent potentially useful insights for designing robust estimation strategies in the presence of noncommuting generators.

major comments (2)
  1. [Model section / Stark-Zeeman Hamiltonian] Model section (Stark-Zeeman Hamiltonian): The analysis rests on a comparatively simple rigid-rotor Stark-Zeeman Hamiltonian. The manuscript does not appear to include or bound the effects of Lambda-doubling and proton hyperfine splitting (I=1/2), whose energy scales are comparable to laboratory fields and can modify the commutator structure of the generators as well as the temperature dependence of the QFI matrix. This directly impacts the claimed nontrivial temperature effect and the optimality of the reported stationary and dynamical strategies.
  2. [Stationary regime / thermal states] Thermal-probe results (stationary regime): The claim that increasing temperature can reduce overall estimation error relies on the specific form of the QFI matrix derived from the simplified Hamiltonian. If Lambda-doubling or hyperfine terms alter the parameter correlations, the reported effect may not survive; the manuscript should provide a quantitative estimate of the regime where the two-level or rigid-rotor approximation remains valid.
minor comments (2)
  1. [Dynamical regime] Notation for the incompatibility measure and the sequential control protocol should be defined more explicitly with reference to the relevant equations to improve readability for readers unfamiliar with the specific multiparameter formalism used.
  2. [Figures] Figure captions for the error bounds or QFI plots would benefit from explicit labels indicating which curves correspond to the simplified model versus any robustness checks.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. We address each major comment below, providing clarifications and indicating the revisions we plan to implement to improve the discussion of model limitations and validity regimes.

read point-by-point responses

  1. Referee: [Model section / Stark-Zeeman Hamiltonian] Model section (Stark-Zeeman Hamiltonian): The analysis rests on a comparatively simple rigid-rotor Stark-Zeeman Hamiltonian. The manuscript does not appear to include or bound the effects of Lambda-doubling and proton hyperfine splitting (I=1/2), whose energy scales are comparable to laboratory fields and can modify the commutator structure of the generators as well as the temperature dependence of the QFI matrix. This directly impacts the claimed nontrivial temperature effect and the optimality of the reported stationary and dynamical strategies.

    Authors: We appreciate the referee highlighting this important aspect of the model. Our analysis indeed employs a simplified rigid-rotor Stark-Zeeman Hamiltonian to capture the essential physics of the OH molecule's response to external fields, as this provides an analytically tractable framework for applying multiparameter quantum estimation theory. Lambda-doubling and hyperfine interactions are neglected in the current treatment. These effects have characteristic energy scales of approximately 1.7 GHz for Lambda-doubling and tens of MHz for proton hyperfine splitting in OH. We will revise the manuscript to include a dedicated paragraph or subsection in the Model section that discusses these approximations. Specifically, we will provide order-of-magnitude estimates for the field strengths (e.g., B ≳ 0.1 T or E ≳ few kV/cm) where the Stark and Zeeman terms dominate, ensuring the commutator structure and QFI properties remain qualitatively similar. We will also note that the reported temperature effect and optimal strategies are representative within this regime, and the sequential control protocol can be generalized. This addition will bound the applicability without altering the core results. revision: yes

  2. Referee: [Stationary regime / thermal states] Thermal-probe results (stationary regime): The claim that increasing temperature can reduce overall estimation error relies on the specific form of the QFI matrix derived from the simplified Hamiltonian. If Lambda-doubling or hyperfine terms alter the parameter correlations, the reported effect may not survive; the manuscript should provide a quantitative estimate of the regime where the two-level or rigid-rotor approximation remains valid.

    Authors: We agree that the nontrivial temperature dependence observed in our thermal-probe calculations is tied to the structure of the QFI matrix in the simplified model, where increased temperature reduces parameter correlations in a beneficial way for the combined estimation error. To address the concern, we will add quantitative estimates of the validity regime in the revised version. For instance, at temperatures below 5-10 K and for field values where the interaction energies exceed the neglected splittings, the rigid-rotor approximation holds. We will include a brief analysis or reference to literature on OH spectroscopy to estimate the parameter range (e.g., magnetic fields from 0.05 to 1 T) where the effect is expected to be robust. If the additional terms modify correlations, the temperature benefit may be reduced but not necessarily eliminated; we will clarify this as a direction for more detailed modeling. This revision will strengthen the manuscript by explicitly stating the limitations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard QET to explicit Hamiltonian model

full rationale

The paper models the OH molecule with the Stark-Zeeman Hamiltonian, computes the quantum Fisher information matrix for joint E/B estimation, and optimizes stationary/thermal and dynamical strategies directly from the eigenvalues, eigenvectors, and commutator structure. The reported temperature effect on error reduction follows from the explicit dependence of the QFI matrix elements on temperature via the thermal state; no parameter is fitted to a subset of data and then renamed as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in. All steps remain self-contained against the stated model assumptions without reducing to input definitions by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, the work rests on standard assumptions from quantum estimation theory and the physical properties of the OH molecule; no explicit free parameters, ad-hoc axioms, or invented entities are identifiable from the summary.

pith-pipeline@v0.9.0 · 5705 in / 1057 out tokens · 34605 ms · 2026-05-22T13:53:20.143350+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.

Reference graph

Works this paper leans on

96 extracted references · 96 canonical work pages

  1. [1]

    , λd)T ∈R d

    Classical bound Let us suppose to have a parametric family of states ρλ dependent on a vector ofdreal parametersλ= (λ1, . . . , λd)T ∈R d. In order to estimate these param- eters, we have to perform a measurement, mathemat- ically described by a positive operator-valued measure (POVM) ˆΠ ={ ˆΠk | ˆΠk ⪰0, P k ˆΠk =1}. Then we can associate a probabilityp(k...

    work page

  2. [2]

    (2) or minimize its in- verse, i.e

    Quantum bound Clearly, the Born rule implies a dependence of the in- formation on the chosen POVM, consequently, one would like to maximize the FIM in Eq. (2) or minimize its in- verse, i.e. the right-hand side of Eq. (3), over the set of possible measurements. While these optimizations are equivalent and lead to a simple result for a single param- eter, ...

    work page

  3. [3]

    (6), we obtain: Tr h W V(λ,{ ˆΠk}) i ≥ 1 M C S(λ, W),(9) where we have introduced C S(λ, W)≡Tr[W Q −1],(10) which we call the SLD scalar bound

    SLD and Holevo–Cram´ er–Rao bounds Starting from the matrix bound Eq. (6), we obtain: Tr h W V(λ,{ ˆΠk}) i ≥ 1 M C S(λ, W),(9) where we have introduced C S(λ, W)≡Tr[W Q −1],(10) which we call the SLD scalar bound. Since in this work, we will mostly use the identity as the weight matrix, we simplify the notation asC S(λ)≡C S(λ,I). A tighter lower bound on ...

    work page

  4. [4]

    for a large number of repetitions

    Asymptotic and single-copy attainability The classical CRB is attainable in the limitM→ ∞, i.e. for a large number of repetitions. Since an inde- pendent copy of the state must be prepared for each ex- periment repetition, the whole procedure usesMcopies of the state, i.e. the overall state isρ ⊗M λ . Crucially, in the quantum case it is theoretically pos...

    work page

  5. [5]

    Upper bounds and weak commutativity Even if the HCRB can be efficiently evaluated through a semidefinite program (SDP) [48], it remains gener- ally difficult to compute and it is useful to identify the discrepancy withC S and conditions for equality. The HCRB is bounded from above as [49, 50] C S(λ, W)≤C H(λ, W)≤ C H(λ, W) ≤C S(λ, W)(1 +R),(15) where the ...

    work page

  6. [6]

    Thanks to this,Rcan be considered a bonafide quantifier of asymptotic incompatibility

    is sufficient for the equalityCS(λ, W) =C H(λ, W) for allW, it can be shown that this is also a necessary condi- tion [57]. Thanks to this,Rcan be considered a bonafide quantifier of asymptotic incompatibility. This quantity is invariant under reparametrizations of the model, and can thus be understood as a geometrical feature of the model. However, this ...

    work page

  7. [7]

    Attaining the QFI matrix Instead of focusing on scalar quantities, we now fo- cus on the conditions for the existence of a POVM{ ˆΠk} such thatF(λ,{ ˆΠk}) =Q(λ), with single-copy measure- ments. A necessary condition for the existence of such a POVM is known as partial commutativity condition (PCC) [33], and stated as follows: ⟨ϕ|[ˆLµ, ˆLν]|ψ⟩= 0∀|ψ⟩,|ϕ⟩ ...

    work page

  8. [8]

    (20) and it has been proven to be sufficient [34] for finding a POVM that attains an equality between the QFIM and FIM

    Compatibility conditions for pure-state models For pure-state models, the WCC reduces to the PCC in Eq. (20) and it has been proven to be sufficient [34] for finding a POVM that attains an equality between the QFIM and FIM. Moreover, it can also be reworked in a more constructive manner [35]. Indeed, given a projection-valued measure (PVM) con- sisting of...

    work page

  9. [9]

    short” and “long

    Exploiting Eq. (8) it is possible to explicitly evaluate the QFIM relative to λ1 andλ 2. However, it is already clear from Eq. (34) that the probe does not depend onλ 1, consequently its derivatives will vanish and with them all the entries of the QFIM relative toλ 1. This implies that the system does not carry information about the magnetic field. Never-...

    work page

  10. [10]

    Ticknor and J

    C. Ticknor and J. L. Bohn, Influence of magnetic fields on cold collisions of polar molecules, Phys. Rev. A71, 022709 (2005)

    work page 2005

  11. [11]

    A. V. Avdeenkov and J. L. Bohn, Linking ultracold polar molecules, Phys. Rev. Lett.90, 043006 (2003)

    work page 2003

  12. [12]

    B. C. Sawyer, B. K. Stuhl, D. Wang, M. Yeo, and J. Ye, Molecular beam collisions with a magnetically trapped target, Phys. Rev. Lett.101, 203203 (2008)

    work page 2008

  13. [13]

    T. V. Tscherbul, Z. Pavlovic, H. R. Sadeghpour, R. Cˆ ot´ e, and A. Dalgarno, Collisions of trapped molecules with slow beams, Phys. Rev. A82, 022704 (2010)

    work page 2010

  14. [14]

    E. R. Hudson, H. J. Lewandowski, B. C. Sawyer, and J. Ye, Cold molecule spectroscopy for constraining the evolution of the fine structure constant, Phys. Rev. Lett. 96, 143004 (2006)

    work page 2006

  15. [15]

    M. G. Kozlov, Λ-doublet spectra of diatomic radicals and their dependence on fundamental constants, Phys. Rev. A80, 022118 (2009)

    work page 2009

  16. [16]

    B. L. Lev, E. R. Meyer, E. R. Hudson, B. C. Sawyer, J. L. Bohn, and J. Ye, Oh hyperfine ground state: From precision measurement to molecular qubits, Phys. Rev. A74, 061402 (2006)

    work page 2006

  17. [17]

    M. Lara, B. L. Lev, and J. L. Bohn, Loss of molecules in magneto-electrostatic traps due to nonadiabatic tran- sitions, Phys. Rev. A78, 033433 (2008)

    work page 2008

  18. [18]

    Bhattacharya, Z

    M. Bhattacharya, Z. Howard, and M. Kleinert, Ground- state oh molecule in combined electric and magnetic fields: Analytic solution of the effective hamiltonian, Phys. Rev. A88, 012503 (2013). 14

    work page 2013

  19. [19]

    C. W. Helstrom, Quantum detection and estimation the- ory, J. Stat. Phys.1, 231 (1969)

    work page 1969

  20. [20]

    A. S. Holevo,Probabilistic and Statistical Aspects of Quantum Theory, 2nd ed. (Edizioni della Normale, Pisa, 2011)

    work page 2011

  21. [21]

    M. G. A. Paris, Quantum estimation for quantum tech- nology, Int. J. Quantum Inf.07, 125 (2009)

    work page 2009

  22. [22]

    Giovannetti, S

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

    work page 2011

  23. [23]

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

    work page 2017

  24. [24]

    Vaneph, T

    C. Vaneph, T. Tufarelli, and M. G. Genoni, Quantum estimation of a two-phase spin rotation, Quantum Meas. Quantum Metrol.1, 12 (2013)

    work page 2013

  25. [25]

    Baumgratz and A

    T. Baumgratz and A. Datta, Quantum Enhanced Es- timation of a Multidimensional Field, Phys. Rev. Lett. 116, 030801 (2016)

    work page 2016

  26. [26]

    Yuan, Sequential Feedback Scheme Outperforms the Parallel Scheme for Hamiltonian Parameter Estimation, Phys

    H. Yuan, Sequential Feedback Scheme Outperforms the Parallel Scheme for Hamiltonian Parameter Estimation, Phys. Rev. Lett.117, 160801 (2016)

    work page 2016

  27. [27]

    Z. Hou, Z. Zhang, G.-Y. Xiang, C.-F. Li, G.-C. Guo, H. Chen, L. Liu, and H. Yuan, Minimal Tradeoff and Ul- timate Precision Limit of Multiparameter Quantum Mag- netometry under the Parallel Scheme, Phys. Rev. Lett. 125, 020501 (2020)

    work page 2020

  28. [28]

    G´ orecki and R

    W. G´ orecki and R. Demkowicz-Dobrza´ nski, Multiparam- eter quantum metrology in the Heisenberg limit regime: Many-repetition scenario versus full optimization, Phys. Rev. A106, 012424 (2022)

    work page 2022

  29. [29]

    Kaubruegger, A

    R. Kaubruegger, A. Shankar, D. V. Vasilyev, and P. Zoller, Optimal and Variational Multiparameter Quantum Metrology and Vector-Field Sensing, PRX Quantum4, 020333 (2023)

    work page 2023

  30. [30]

    Baamara, M

    Y. Baamara, M. Gessner, and A. Sinatra, Quantum- enhanced multiparameter estimation and compressed sensing of a field, SciPost Phys.14, 050 (2023)

    work page 2023

  31. [31]

    Hayashi and Y

    M. Hayashi and Y. Ouyang, Finding the optimal probe state for multiparameter quantum metrology using conic programming, npj Quantum Inf10, 1 (2024)

    work page 2024

  32. [32]

    J. Guo, F. Sun, Q. He, and M. Fadel, Assisted metrology and preparation of macroscopic superpositions with split spin-squeezed states, Phys. Rev. A108, 053327 (2023)

    work page 2023

  33. [33]

    Kahn, Fast rate estimation of a unitary operation in SU(d), Phys

    J. Kahn, Fast rate estimation of a unitary operation in SU(d), Phys. Rev. A75, 022326 (2007)

    work page 2007

  34. [34]

    Imai and A

    H. Imai and A. Fujiwara, Geometry of optimal estimation scheme for SU(D) channels, J. Phys. A40, 4391 (2007)

    work page 2007

  35. [35]

    Kura and M

    N. Kura and M. Ueda, Finite-error metrological bounds on multiparameter Hamiltonian estimation, Phys. Rev. A97, 012101 (2018)

    work page 2018

  36. [36]

    T. Esat, D. Borodin, J. Oh, A. J. Heinrich, F. S. Tautz, Y. Bae, and R. Temirov, A quantum sensor for atomic- scale electric and magnetic fields, Nat. Nanotechnol.19, 1466 (2024)

    work page 2024

  37. [37]

    The symbol Π denotes Λ = 1, the projection of the electronic orbital angular momentum on the internuclear axis

    For clarity, we briefly recall the spectroscopic term- symbol notation for diatomic molecules. The symbol Π denotes Λ = 1, the projection of the electronic orbital angular momentum on the internuclear axis. The super- script 2 gives the spin multiplicity 2S+ 1 (here a doublet withS= 1/2 for an open-shell radical with one unpaired electron). The subscript ...

    work page

  38. [38]

    Albarelli, M

    F. Albarelli, M. Barbieri, M. Genoni, and I. Gianani, A perspective on multiparameter quantum metrology: From theoretical tools to applications in quantum imag- ing, Phys. Lett. A384, 126311 (2020)

    work page 2020

  39. [39]

    J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum Fisher information matrix and multiparameter estima- tion, J. Phys. A53, 023001 (2020)

    work page 2020

  40. [40]

    Demkowicz-Dobrza´ nski, W

    R. Demkowicz-Dobrza´ nski, W. G´ orecki, and M. Gut ¸˘ a, Multi-parameter estimation beyond quantum Fisher in- formation, J. Phys. A53, 363001 (2020)

    work page 2020

  41. [41]

    Razavian, M

    S. Razavian, M. G. A. Paris, and M. G. Genoni, On the quantumness of multiparameter estimation problems for qubit systems, Entropy22, 1197 (2020)

    work page 2020

  42. [42]

    J. Yang, S. Pang, Y. Zhou, and A. N. Jordan, Optimal measurements for quantum multiparameter estimation with general states, Phys. Rev. A100, 032104 (2019)

    work page 2019

  43. [43]

    Matsumoto, A new approach to the Cram´ er-Rao-type bound of the pure-state model, J

    K. Matsumoto, A new approach to the Cram´ er-Rao-type bound of the pure-state model, J. Phys. A35, 3111 (2002)

    work page 2002

  44. [44]

    Pezz` e, M

    L. Pezz` e, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, Optimal measurements for simultaneous quantum estimation of multiple phases, Phys. Rev. Lett. 119, 130504 (2017)

    work page 2017

  45. [45]

    Advances in multipa- rameter quantum sensing and metrology.arXiv preprint arXiv:2502.17396, 2025

    L. Pezz` e and A. Smerzi, Advances in multipa- rameter quantum sensing and metrology (2025), arXiv:2502.17396

    work page arXiv 2025

  46. [46]

    Cram´ er,Mathematical methods of statistics, Vol

    H. Cram´ er,Mathematical methods of statistics, Vol. 43 (Princeton university press, 1999)

    work page 1999

  47. [47]

    E. L. Lehmann and G. Casella,Theory of point estima- tion, Vol. 2nd ed. (Springer texts in statistics, 1998)

    work page 1998

  48. [48]

    C. W. Helstrom, Minimum mean-squared error of es- timates in quantum statistics, Phys. Lett. A25, 101 (1967)

    work page 1967

  49. [49]

    A. S. Holevo, Commutation superoperator of a state and its applications to the noncommutative statistics, Rep. Math. Phys.12, 251 (1977)

    work page 1977

  50. [50]

    Nagaoka, A New Approach to Cram´ er-Rao Bounds for Quantum State Estimation, IEICE Tech

    H. Nagaoka, A New Approach to Cram´ er-Rao Bounds for Quantum State Estimation, IEICE Tech. Rep.IT 89-42, 9 (1989)

    work page 1989

  51. [51]

    Yamagata, A

    K. Yamagata, A. Fujiwara, and R. D. Gill, Quantum local asymptotic normality based on a new quantum likelihood ratio, Ann. Stat.41, 2197 (2013)

    work page 2013

  52. [52]

    Hayashi and K

    M. Hayashi and K. Matsumoto, Asymptotic performance of optimal state estimation in qubit system, J. Math. Phys.49, 102101 (2008)

    work page 2008

  53. [53]

    Y. Yang, G. Chiribella, and M. Hayashi, Attaining the Ultimate Precision Limit in Quantum State Estimation, Commun. Math. Phys.368, 223 (2019)

    work page 2019

  54. [54]

    Hayashi and Y

    M. Hayashi and Y. Ouyang, Tight Cram´ er-Rao type bounds for multiparameter quantum metrology through conic programming, Quantum7, 1094 (2023)

    work page 2023

  55. [55]

    L. O. Conlon, J. Suzuki, P. K. Lam, and S. M. Assad, Efficient computation of the Nagaoka–Hayashi bound for multiparameter estimation with separable measure- ments, Npj Quantum Inf.7, 110 (2021)

    work page 2021

  56. [56]

    Belliardo and V

    F. Belliardo and V. Giovannetti, Incompatibility in quan- tum parameter estimation, New J. Phys.23, 063055 (2021)

    work page 2021

  57. [57]

    Albarelli, J

    F. Albarelli, J. F. Friel, and A. Datta, Evaluating the Holevo Cram´ er-Rao Bound for Multiparameter Quantum Metrology, Phys. Rev. Lett.123, 200503 (2019)

    work page 2019

  58. [58]

    Carollo, B

    A. Carollo, B. Spagnolo, A. A. Dubkov, and D. Valenti, On quantumness in multi-parameter quantum estima- tion, J. Stat. Mech. Theory Exp.2019, 094010 (2019). 15

    work page 2019

  59. [59]

    Tsang, F

    M. Tsang, F. Albarelli, and A. Datta, Quantum Semi- parametric Estimation, Phys. Rev. X10, 031023 (2020)

    work page 2020

  60. [60]

    Candeloro, M

    A. Candeloro, M. G. A. Paris, and M. G. Genoni, On the properties of the asymptotic incompatibility measure in multiparameter quantum estimation, J. Phys. A54, 485301 (2021)

    work page 2021

  61. [61]

    Candeloro, Z

    A. Candeloro, Z. Pazhotan, and M. G. A. Paris, Dimen- sion matters: Precision and incompatibility in multi- parameter quantum estimation models, Quantum Sci. Technol.9, 045045 (2024)

    work page 2024

  62. [62]

    quantum holon- omy

    A. Uhlmann, Parallel transport and “quantum holon- omy” along density operators, Rep. Math. Phys.24, 229 (1986)

    work page 1986

  63. [63]

    Dittmann, The scalar curvature of the Bures metric on the space of density matrices, J

    J. Dittmann, The scalar curvature of the Bures metric on the space of density matrices, J. Geom. Phys.31, 16 (1999)

    work page 1999

  64. [64]

    Carollo, B

    A. Carollo, B. Spagnolo, and D. Valenti, Uhlmann cur- vature in dissipative phase transitions, Sci. Rep.8, 9852 (2018)

    work page 2018

  65. [65]

    Albarelli and R

    F. Albarelli and R. Demkowicz-Dobrza´ nski, Probe In- compatibility in Multiparameter Noisy Quantum Metrol- ogy, Phys. Rev. X12, 011039 (2022)

    work page 2022

  66. [66]

    S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrza´ n ski, Compatibility in multiparameter quantum metrology, Phys. Rev. A94, 10.1103/physreva.94.052108 (2016)

    work page doi:10.1103/physreva.94.052108 2016

  67. [67]

    Mihailescu, A

    G. Mihailescu, A. Bayat, S. Campbell, and A. K. Mitchell, Multiparameter critical quantum metrology with impurity probes, Quantum Science and Technology 9, 035033 (2024)

    work page 2024

  68. [68]

    P. F. Bernath,Spectra of Atoms and Molecules, fourth edition ed. (Oxford University Press, New York, 2020)

    work page 2020

  69. [69]

    J. M. Brown and A. Carrington,Rotational Spectroscopy of Diatomic Molecules, 1st ed. (Cambridge University Press, 2003)

    work page 2003

  70. [70]

    B. K. Stuhl, M. Yeo, B. C. Sawyer, M. T. Hummon, and J. Ye, Microwave state transfer and adiabatic dynamics of magnetically trapped polar molecules, Phys. Rev. A 85, 033427 (2012)

    work page 2012

  71. [71]

    Troiani and M

    F. Troiani and M. G. A. Paris, Universal Quantum Mag- netometry with Spin States at Equilibrium, Phys. Rev. Lett.120, 260503 (2018)

    work page 2018

  72. [72]

    L. P. Garc´ ıa-Pintos, K. Bharti, J. Bringewatt, H. De- hghani, A. Ehrenberg, N. Yunger Halpern, and A. V. Gorshkov, Estimation of Hamiltonian Parameters from Thermal States, Phys. Rev. Lett.133, 040802 (2024)

    work page 2024

  73. [73]

    Abiuso, P

    P. Abiuso, P. Sekatski, J. Calsamiglia, and M. Perarnau- Llobet, Fundamental Limits of Metrology at Thermal Equilibrium, Phys. Rev. Lett.134, 010801 (2025)

    work page 2025

  74. [74]

    Qiu and L

    D. Qiu and L. Li, Minimum-error discrimination of quan- tum states: Bounds and comparisons, Phys. Rev. A81, 042329 (2010)

    work page 2010

  75. [75]

    R. P. Feynman, Forces in molecules, Phys. Rev.56, 340 (1939)

    work page 1939

  76. [76]

    Jiang, Quantum Fisher information for states in ex- ponential form, Phys

    Z. Jiang, Quantum Fisher information for states in ex- ponential form, Phys. Rev. A89, 032128 (2014)

    work page 2014

  77. [77]

    III, this low-temperature expansion should be interpreted as an expansion within the effec- tive Hamiltonian

    As discussed in Sec. III, this low-temperature expansion should be interpreted as an expansion within the effec- tive Hamiltonian. The finite-temperature numerical re- sults shown in Fig. 4, including the curves atT= 10 mK, are above the quoted 5 mK validity scale

    work page

  78. [78]

    Mihailescu, U

    G. Mihailescu, U. Alushi, R. D. Candia, S. Felicetti, and K. Gietka, Critical Quantum Sensing: A tutorial on parameter estimation near quantum phase transitions (2025), arXiv:2510.02035

    work page arXiv 2025

  79. [79]

    L. R. B. Picard, A. J. Park, G. E. Patenotte, S. Gebret- sadkan, D. Wellnitz, A. M. Rey, and K.-K. Ni, Entangle- ment and iSWAP gate between molecular qubits, Nature 637, 821 (2025)

    work page 2025

  80. [80]

    C. M. Holland, Y. Lu, and L. W. Cheuk, On-demand entanglement of molecules in a reconfigurable optical tweezer array, Science382, 1143 (2023)

    work page 2023

Showing first 80 references.