Pith. sign in

REVIEW 3 major objections 6 minor 55 references

Linear dependence kills quadratic precision in multiparameter quantum sensing

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · glm-5.2

2026-07-08 06:31 UTC pith:RLWF2ZBF

load-bearing objection Clean no-go theorem for loss of t^{-2} scaling in multiparameter quantum metrology; the Gram-matrix diagnostic is the genuinely useful new contribution. the 3 major comments →

arxiv 2607.06410 v1 pith:RLWF2ZBF submitted 2026-07-07 quant-ph

Geometric obstructions to quadratic time scaling in multiparameter quantum estimation

Eoin O'Connor , Jiayu He , Matteo G. A. Paris , Marco G. Genoni This is my paper
classification quant-ph
keywords multiparameter quantum estimationquantum Fisher informationHeisenberg scalinggeometric obstructionHamiltonian parameter estimationmeasurement incompatibilityquantum metrologydiagonal decomposition
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper proves that when multiple parameters are encoded into a quantum system through a time-independent Hamiltonian, simultaneous quadratic-in-time precision (the gold standard of quantum metrology) can fail for a purely geometric reason. The authors decompose each Hamiltonian derivative into a part that commutes with the system Hamiltonian (the diagonal part) and a part that does not (the off-diagonal part). They show that if these diagonal parts are linearly dependent — meaning one parameter direction is redundant with respect to the energy spectrum — then along that direction the local generator loses its term proportional to evolution time t, leaving only a bounded off-diagonal contribution. The quantum Fisher information along this slow direction stays constant in time rather than growing as t^2, capping the achievable precision at O(t^0) no matter what probe state or entanglement strategy is used. The authors provide a simple diagnostic — the Gram matrix of the diagonal generators — whose determinant is zero precisely when this obstruction occurs. They demonstrate the mechanism in collective spin magnetometry and a quantum harmonic oscillator, and show a contrasting case (the Lipkin–Meshkov–Glick model) where diagonal generators remain independent and quadratic scaling survives. A key secondary result is that measurement incompatibility between fast and slow directions decays as 1/t, so the standard precision bound becomes saturable in the long-time limit even though the slow direction remains bottlenecked.

Core claim

The central object is the decomposition of each Hamiltonian derivative ∂_μ H into a diagonal part D_μ (commuting with H) and an off-diagonal part O_μ (not commuting with H). The local generator H_μ(t) then splits into a term -t·D_μ that grows linearly with time and a bounded term -K_μ(t) that stays O(t^0). When the traceless diagonal generators {D_μ} are linearly dependent, there exists a direction w in parameter space where the t-proportional term vanishes identically, leaving only the bounded K_μ(t) contribution. This forces the smallest eigenvalue of the quantum Fisher information matrix to remain O(t^0), which in turn caps the total estimation precision. The authors prove this holds for

What carries the argument

diagonal/off-diagonal decomposition of Hamiltonian derivatives; Gram matrix G_μν = Tr(D_μ D_ν) as computable diagnostic; fast/slow basis rotation in parameter space; dimensional bound: at most d-1 parameters can achieve O(t^{-2}) in a d-dimensional system

Load-bearing premise

The proof requires that the off-diagonal generators K_μ(t) remain bounded (O(t^0)) as time grows. This is guaranteed for finite-dimensional systems but must be checked case by case for infinite-dimensional systems like the quantum harmonic oscillator, where the relevant operators are unbounded. If K_μ(t) were to grow with t in some infinite-dimensional setting, the O(t^0) bottleneck would not hold.

What would settle it

Find a multiparameter estimation problem where the diagonal generators are linearly dependent but the smallest eigenvalue of the QFIM still grows as t^2 — this would require the off-diagonal generators K_μ(t) to themselves grow with t, which the paper argues does not happen in finite dimensions.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Any multiparameter quantum metrology protocol using time-independent Hamiltonian encoding can be cheaply diagnosed: compute the Gram matrix of diagonal generators, and if its determinant is zero, at least one parameter direction is stuck at O(t^0) precision regardless of probe engineering.
  • In a d-dimensional quantum system, attempting to estimate d or more parameters simultaneously will always trigger this obstruction, setting a hard dimensional ceiling on the number of parameters that can enjoy Heisenberg-like scaling.
  • The 1/t decay of measurement incompatibility means that even in obstructed cases, the gap between the SLD bound and the tighter Holevo bound vanishes asymptotically — the bottleneck is purely geometric, not a measurement incompatibility problem.
  • Adaptive quantum control that cancels the system Hamiltonian can restore t^2 scaling by making all derivatives commute with the (zero) total Hamiltonian, but requires prior knowledge of the true parameter values.
  • Sequential measurement strategies with finite interrogation times can recover O(T^{-1}) scaling for slow directions over total experimental time T, providing a practical workaround when adaptive control is unavailable.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This manuscript establishes a general geometric obstruction to simultaneous O(t^{-2}) scaling in multiparameter quantum estimation under time-independent Hamiltonian dynamics. The central result (Sec. III) is a no-go theorem: when the traceless diagonal components of the Hamiltonian derivatives, {D̃_μ}, are linearly dependent, there exists a 'slow' parameter direction along which the quantum Fisher information remains O(t^0), bottlenecking the total estimation precision. The authors derive a computable diagnostic via the Gram matrix of diagonal generators, prove that ancilla-assisted strategies cannot bypass the obstruction, and show that the measurement incompatibility penalty decays as O(t^{-1}). The framework is illustrated through three examples: collective spin magnetometry (Sec. IV), the quantum harmonic oscillator (Sec. V), and the Lipkin-Meshkov-Glick model (Sec. VI, where the obstruction is absent). The paper also discusses circumvention via nuisance parameter relegation and adaptive quantum control (Sec. VII).

Significance. The central derivation in Sec. III is mathematically sound and logically clean. The decomposition of the local generator into a t-proportional diagonal part and a bounded off-diagonal part K_μ(t) (Eq. 9-10) is standard but effectively deployed. The key argument—that linear dependence of {D̃_μ} (Eq. 13) eliminates the t-proportional term along direction w, leaving only the bounded K_μ(t) contribution (Eq. 14-15)—correctly establishes λ_min ≤ O(t^0) and hence Tr(F^{-1}) ≥ O(1). The dimensional bound (at most d-1 parameters with O(t^{-2}) scaling in a d-dimensional system) is a clean, falsifiable consequence. The Uhlmann curvature analysis (Sec. VII.B) is also correct: det(F) = O(t^2) and U_fs = O(t) yield ||F^{-1}UF^{-1}||_1 = O(t^{-1}). The framework provides a parameter-free diagnostic (the Gram matrix determinant) with no ad-hoc assumptions or fitted constants. The three examples are well-chosen and illustrate both the obstructed and unobstructed cases. The infinite-dimensional caveat for continuous-variable systems (Sec. V) is explicitly acknowledged and verified for the QHO example where scalar prefactors in Eq. (48) are indeed bounded.

major comments (3)
  1. Sec. III, Eqs. (12)-(16): The proof that λ_min = O(t^0) proceeds by exhibiting a specific direction w for which w^T F w = O(t^0). This establishes an upper bound on λ_min. However, the conclusion in Eq. (16) that Tr(F^{-1}) ≥ 1/λ_min ~ t^0 requires that λ_min does not accidentally vanish or scale worse than t^0 due to the structure of K_μ(t) in the slow subspace. The argument is correct when the slow-subspace QFIM block is non-singular (which is generically the case), but the manuscript should state more explicitly that the O(t^0) conclusion assumes the variance of Σ w_μ K_μ(t) is strictly positive and bounded away from zero for the chosen probe state. If this variance vanishes for a particular |ψ_0⟩, the slow direction becomes unestimable (singular QFIM), which is a different failure mode. Clarifying this genericity assumption would strengthen the no-go claim.
  2. Sec. VII.B, Eqs. (85)-(90): The incompatibility analysis is restricted to the two-parameter case. The manuscript states (end of Sec. VII.B) that 'we conjecture that for any number of parameters the O(t^0) contribution to the incompatibility is determined solely by the slow subspace.' This conjecture is load-bearing for the generality of the asymptotic saturability claim, yet it is unproven. The step from the 2×2 identity in Eq. (88) to the general p-parameter case is non-trivial because the trace norm of F^{-1}UF^{-1} for p > 2 involves a more complex singular-value structure. The authors should either (a) restrict the asymptotic saturability claim to two parameters in the abstract and conclusions, or (b) provide at least a sketch of why the conjecture is expected to hold (e.g., by block-decomposition into fast/slow subspaces and noting that the fast-fast block of U vanishes asymptotical
  3. Sec. V, Eq. (48) and surrounding text: For the QHO, K_μ(t) contains unbounded operators (ĉ^2, ĉ†^2). The manuscript correctly notes that the scalar prefactors are bounded and that one must verify boundedness case by case. However, the statement 'K_μ(t) ~ O(t^0)' in Eq. (10) is stated for finite-dimensional systems, and the extension to the QHO is justified only by the boundedness of prefactors, not of the operators themselves. The variance Var(Q̂(t)) in Eq. (56) is finite only because of the restriction to Gaussian states with fixed energy. The manuscript should clarify that the O(t^0) scaling of the QFIM in the slow direction is not a purely operator-theoretic result here but depends on the state-space restriction. This is acknowledged but the logical flow could be tighter.
minor comments (6)
  1. Sec. III, Eq. (11): The notation D̃_μ is introduced for the traceless diagonal generator, but in subsequent equations (e.g., Eq. 13, Eq. 17) the tilde is sometimes dropped or inconsistently applied. Standardize the notation throughout.
  2. Sec. IV.D, Eq. (40): The comparison with Ref. [33] is useful, but the factor (N+2)/N difference is explained somewhat informally. A brief explicit statement of how the two optimization problems differ in dimensionality would help the reader.
  3. Sec. VI, Fig. 1: The y-axis label 'N^3 Tr[G^{-1}]' and the caption could clarify what the reference value in the strong-field limit is, so the reader can assess how much the quantity varies across regimes.
  4. Sec. VII.C, Eq. (93): The condition ker G ⊆ ker W is elegant and could be highlighted more prominently, perhaps in the abstract or introduction, as it provides a concise operational criterion for when quadratic scaling is recoverable.
  5. References [47, 48] are dated 2026; verify these are correctly cited and not preprints with updated dates.
  6. Sec. II: The SLD bound is introduced with W = I, but the general weighted bound (Eq. 91) appears only in Sec. VII.C. A forward reference would help readers who wonder about general weight matrices early on.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. All three major comments identify legitimate points that warrant clarification in the manuscript. We address each below and will incorporate revisions accordingly.

read point-by-point responses

  1. Referee: Sec. III, Eqs. (12)-(16): The proof that λ_min = O(t^0) proceeds by exhibiting a specific direction w for which w^T F w = O(t^0). This establishes an upper bound on λ_min. However, the conclusion in Eq. (16) that Tr(F^{-1}) ≥ 1/λ_min ~ t^0 requires that λ_min does not accidentally vanish or scale worse than t^0 due to the structure of K_μ(t) in the slow subspace. The argument is correct when the slow-subspace QFIM block is non-singular (which is generically the case), but the manuscript should state more explicitly that the O(t^0) conclusion assumes the variance of Σ w_μ K_μ(t) is strictly positive and bounded away from zero for the chosen probe state. If this variance vanishes for a particular |ψ_0⟩, the slow direction becomes unestimable (singular QFIM), which is a different failure mode. Clarifying this genericity assumption would strengthen the no-go claim.

    Authors: The referee is correct that our argument establishes an upper bound on λ_min by exhibiting a direction w with w^T F w = O(t^0), and that the conclusion Tr(F^{-1}) ≥ 1/λ_min ~ t^0 requires the slow-subspace QFIM block to be non-singular. We agree that this genericity assumption should be stated explicitly. We will revise the text following Eq. (16) to clarify two points: (1) the O(t^0) scaling of the no-go bound holds when Var_{ψ_0}(Σ w_μ K_μ(t)) is strictly positive, which is the generic case for physically reasonable probe states; and (2) if this variance vanishes for a particular |ψ_0⟩, the QFIM becomes singular along the slow direction, which is a distinct (and in some sense more severe) failure mode. We emphasize that the no-go theorem is not weakened by this clarification: in both cases—whether the slow-subspace variance is positive (yielding O(t^0) scaling) or zero (yielding a singular QFIM)—simultaneous O(t^{-2}) scaling is unachievable. The genericity assumption only distinguishes which failure mode occurs, not whether failure occurs. revision: partial

  2. Referee: Sec. VII.B, Eqs. (85)-(90): The incompatibility analysis is restricted to the two-parameter case. The manuscript states (end of Sec. VII.B) that 'we conjecture that for any number of parameters the O(t^0) contribution to the incompatibility is determined solely by the slow subspace.' This conjecture is load-bearing for the generality of the asymptotic saturability claim, yet it is unproven. The step from the 2×2 identity in Eq. (88) to the general p-parameter case is non-trivial because the trace norm of F^{-1}UF^{-1} for p > 2 involves a more complex singular-value structure. The authors should either (a) restrict the asymptotic saturability claim to two parameters in the abstract and conclusions, or (b) provide at least a sketch of why the conjecture is expected to hold (e.g., by block-decomposition into fast/slow subspaces and noting that the fast-fast block of U vanishes asymptotical

    Authors: The referee correctly identifies that the asymptotic saturability result is proven only for two parameters and that the extension to p > 2 is conjectural. We agree that the conjecture should not be presented as established. We will take approach (b): we will add a paragraph sketching the expected argument for the general case. The key observation is that in the fast/slow decomposition, the Uhlmann curvature matrix U has a block structure where the fast-fast block vanishes asymptotically (since fast-direction diagonal generators commute in non-degenerate Hamiltonians, making U_{ff} = O(t^0) while F_{ff} = O(t^2)), the fast-slow block scales as O(t) (as shown in Eq. 87), and the slow-slow block is O(t^0). Since F^{-1} has blocks scaling as O(t^{-2}) (fast-fast), O(t^{-1}) (fast-slow), and O(t^0) (slow-slow), the product F^{-1} U F^{-1} has entries that are at most O(t^{-1}) from the fast-slow coupling, with the O(t^0) contribution arising solely from the slow-slow block. While a complete proof for general p requires careful analysis of the singular-value structure of the resulting matrix (which may have dimension > 2 in the slow subspace), the scaling of individual entries supports the conjecture. We will also add a qualifying clause in the abstract and conclusions noting that the O(t^{-1}) incompatibility decay is proven for two parameters and conjectured for the general case. revision: partial

  3. Referee: Sec. V, Eq. (48) and surrounding text: For the QHO, K_μ(t) contains unbounded operators (ĉ^2, ĉ†^2). The manuscript correctly notes that the scalar prefactors are bounded and that one must verify boundedness case by case. However, the statement 'K_μ(t) ~ O(t^0)' in Eq. (10) is stated for finite-dimensional systems, and the extension to the QHO is justified only by the boundedness of prefactors, not of the operators themselves. The variance Var(Q̂(t)) in Eq. (56) is finite only because of the restriction to Gaussian states with fixed energy. The manuscript should clarify that the O(t^0) scaling of the QFIM in the slow direction is not a purely operator-theoretic result here but depends on the state-space restriction. This is acknowledged but the logical flow could be tighter.

    Authors: The referee is correct that the O(t^0) scaling in the QHO example is not purely operator-theoretic but depends on the restriction to Gaussian states with fixed average energy. The unboundedness of the operators ĉ^2 and ĉ†^2 means that K_μ(t) = O(t^0) holds at the level of scalar prefactors, but the finiteness of the variance Var(Q̂(t)) in Eq. (56) requires the state-space restriction. We will tighten the logical flow in Sec. V as follows: (1) after Eq. (48), we will explicitly state that the O(t^0) designation refers to the boundedness of the time-dependent scalar prefactors, and that the physical relevance of this bound for the QFIM requires verifying that the operator variances are finite for the chosen class of probe states; (2) at the beginning of Sec. V.B, we will move the statement about the restriction to Gaussian states with fixed ⟨n̂⟩ earlier, before presenting the QFIM entries, so that the reader sees the state-space assumption before encountering Eq. (56). This reorganization will make clear that the no-go result in the CV setting is a statement about the scaling of the QFIM within a physically motivated state class, not a purely algebraic property of the operators. revision: partial

Circularity Check

0 steps flagged

No significant circularity found

full rationale

The paper derives its central no-go theorem from first principles: the decomposition of the local generator into a t-proportional diagonal part and a bounded off-diagonal part (Eq. 9-10) follows directly from the integral form of the Heisenberg evolution. The key step—linear dependence of the traceless diagonal generators (Eq. 13) causing the t-term to vanish along direction w, leaving only the bounded K_mu(t) (Eq. 14-15)—is a mathematical deduction, not a fit or a definition. The examples (spin systems, QHO, LMG) are used to illustrate the theorem, not to tune parameters. Self-citations (Refs [33, 41]) are used for comparison of bounds and prior results, not to define the core result or smuggle in an ansatz. The measurement incompatibility result (Sec. VII.B) follows cleanly from the scaling of F and U. No step reduces to its inputs by construction. The infinite-dimensional caveat is acknowledged and handled case-by-case. The derivation is self-contained against external benchmarks. The reader's score of 2 is slightly generous; the self-citations are not load-bearing for the central theorem, so the circularity score is 0-1. I assign 0 as no circularity was identified in any load-bearing step.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper is a theoretical derivation with no free parameters or invented entities. The axioms are standard results from quantum estimation theory, with the boundedness of K_mu(t) being a domain assumption that holds generally in finite dimensions.

axioms (3)
  • standard math The quantum Fisher information matrix for a pure state under unitary encoding is given by the covariance matrix of the local generators (Eq. 5).
    Standard result in quantum metrology, cited from Refs [34, 35]. Used as the starting point for the analysis.
  • standard math The local generator can be decomposed into a part commuting with the Hamiltonian (diagonal) and a non-commuting part (off-diagonal) (Eq. 6-10).
    Standard decomposition in quantum metrology, cited from Refs [8, 36, 37]. The paper builds its main result on this decomposition.
  • domain assumption The off-diagonal generator K_mu(t) is bounded as O(t^0) as t -> infinity.
    Crucial for the main result. Guaranteed in finite dimensions but requires case-by-case verification in infinite-dimensional continuous-variable systems, as noted in Sec. V.

pith-pipeline@v1.1.0-glm · 21620 in / 1662 out tokens · 218075 ms · 2026-07-08T06:31:39.820755+00:00 · methodology

0 comments
read the original abstract

Unitary encoding of a single parameter provides quadratic enhancement in precision, with the quantum Fisher information scaling quadratically with the encoding time. However, when estimating multiple parameters simultaneously, this fundamental scaling is not guaranteed. Here, we establish a universal geometric obstruction that dictates when multiparameter quantum metrology fails to achieve simultaneous $t^{-2}$ scaling. By decomposing the Hamiltonian derivatives into components that commute and do not commute with the system Hamiltonian, we prove that linear dependence among the commuting components inevitably generates a slow parameter direction whose Fisher information remains bounded as O$(t^0)$, limiting the overall estimation precision. We demonstrate this mechanism in both discrete- and continuous-variable setups, including collective spin magnetometry and a generalized quantum harmonic oscillator, and contrast it with the Lipkin--Meshkov--Glick model where $t^{-2}$ decay is preserved. Remarkably, while the slow direction fundamentally limits the achievable precision, the measurement incompatibility between fast and slow directions decays as $1/t$, rendering the symmetric logarithmic derivative bound asymptotically saturable. Our framework provides a readily computable diagnostic, given by the Gram matrix of the diagonal generators, for identifying such obstructions in arbitrary multiparameter estimation problems. We further show that the bottleneck can be circumvented by relegating slow directions to nuisance parameters or by employing adaptive quantum control.

Figures

Figures reproduced from arXiv: 2607.06410 by Eoin O'Connor, Jiayu He, Marco G. Genoni, Matteo G. A. Paris.

Figure 1
Figure 1. Figure 1: FIG. 1. Trace of the inverse Gram matrix Tr [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

discussion (0)

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

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages · 1 internal anchor

  1. [1]

    A full-rank weight matrix can always be decomposed asW= MT Mfor some invertible matrixM

    Full rank weight matrices IfWis full rank, strictly positive definite and time- independent, it does not change the asymptotic analysis. A full-rank weight matrix can always be decomposed asW= MT Mfor some invertible matrixM. The weighted SLD bound can then be rewritten as: Tr(WF −1)=Tr(MF −1MT )=Tr h (M −T FM −1)−1i .(92) The matrixF ′ =M −T FM −1 is pre...

    work page

  2. [2]

    fast” and “slow

    Rank-deficient matrices and nuisance parameters The situation changes whenWis not full rank. The zero- eigenvalue subspace ofWcorresponds to the presence of nui- sance parameters, i.e., parameters (or linear combinations of parameters) that we do not know but that we do not care to es- timate. As mentioned in Sec. III, the Gram matrixGdefined in Eq. (17) ...

    work page

  3. [3]

    Giovannetti, S

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

    work page 2006

  4. [4]

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

    work page 2009

  5. [5]

    Giovannetti, S

    V . Giovannetti, S. Lloyd, and L. Maccone, Advances in quan- tum metrology, Nat. Photonics5, 222 (2011)

    work page 2011

  6. [6]

    Hayashi, Quantum Information Theory: Mathematical Foundation, Graduate Texts in Physics (Springer, Berlin, Heidelberg, 2017)

    M. Hayashi, Quantum Information Theory: Mathematical Foundation, Graduate Texts in Physics (Springer, Berlin, Heidelberg, 2017)

    work page 2017

  7. [7]

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

    work page 1969

  8. [8]

    S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett.72, 3439 (1994)

    work page 1994

  9. [9]

    Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Edizioni della Normale, Pisa, 2011)

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

    work page 2011

  10. [10]

    Pang and T

    S. Pang and T. A. Brun, Quantum metrology for a general Hamiltonian parameter, Phys. Rev. A90, 022117 (2014)

    work page 2014

  11. [11]

    A. Das, W. G ´orecki, and R. Demkowicz-Dobrza´nski, Universal time scalings of sensitivity in markovian quantum metrology, Phys. Rev. A111, L020403 (2025)

    work page 2025

  12. [12]

    J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum Fisher in- 12 formation matrix and multiparameter estimation, J. Phys. A53, 023001 (2020)

    work page 2020

  13. [13]

    Albarelli, M

    F. Albarelli, M. Barbieri, M. G. Genoni, and I. Gianani, A per- spective on multiparameter quantum metrology: From theoreti- cal tools to applications in quantum imaging, Phys. Lett. A384, 126311 (2020)

    work page 2020

  14. [14]

    Demkowicz-Dobrza ´nski, W

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

    work page 2020

  15. [15]

    M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, Optimal estimation of joint parameters in phase space, Phys. Rev. A87, 012107 (2013)

    work page 2013

  16. [16]

    M. D. Vidrighin, G. Donati, M. G. Genoni, X.-M. Jin, W. S. Kolthammer, M. S. Kim, A. Datta, M. Barbieri, and I. A. Walm- sley, Joint estimation of phase and phase diffusion for quantum metrology, Nat. Commun.5, 3532 (2014)

    work page 2014

  17. [17]

    S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrza ´nski, Compat- ibility in multiparameter quantum metrology, Phys. Rev. A94, 052108 (2016)

    work page 2016

  18. [18]

    Roccia, I

    E. Roccia, I. Gianani, L. Mancino, M. Sbroscia, F. Somma, M. G. Genoni, and M. Barbieri, Entangling measurements for multiparameter estimation with two qubits, Quantum Sci. Tech- nol.3, 01LT01 (2017)

    work page 2017

  19. [19]

    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

  20. [20]

    Carollo, B

    A. Carollo, B. Spagnolo, A. A. Dubkov, and D. Valenti, On quantumness in multi-parameter quantum estimation, J. Stat. Mech.: Theory Exp.2019(9), 094010

    work page 2019

  21. [21]

    Razavian, M

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

    work page 2020

  22. [22]

    Candeloro, M

    A. Candeloro, M. G. A. Paris, and M. G. Genoni, On the prop- erties of the asymptotic incompatibility measure in multiparam- eter quantum estimation, J. Phys. A54, 485301 (2021)

    work page 2021

  23. [23]

    L. O. Conlon, J. Suzuki, P. K. Lam, and S. M. Assad, Efficient computation of the nagaoka–hayashi bound for multiparameter estimation with separable measurements, npj Quantum Inf.7, 110 (2021)

    work page 2021

  24. [24]

    Albarelli and R

    F. Albarelli and R. Demkowicz-Dobrza ´nski, Probe Incompati- bility in Multiparameter Noisy Quantum Metrology, Phys. Rev. X12, 011039 (2022)

    work page 2022

  25. [25]

    Candeloro, Z

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

    work page 2024

  26. [26]

    J. He, G. Fazio, and M. G. A. Paris, Weight-dependent and weight-independent measures of quantum incompatibility in multiparameter estimation (2025), arXiv:2510.18864 [quant- ph]

    work page arXiv 2025

  27. [27]

    Gessner, L

    M. Gessner, L. Pezz `e, and A. Smerzi, Sensitivity Bounds for Multiparameter Quantum Metrology, Phys. Rev. Lett.121, 130503 (2018)

    work page 2018

  28. [28]

    Frigerio and M

    M. Frigerio and M. G. A. Paris, Overcoming sloppiness for enhanced metrology in a Mach–Zehnder interferometer, Int. J. Quantum Inf. , 2540001 (2025)

    work page 2025

  29. [29]

    Sharma, S

    P. Sharma, S. Olivares, D. K. Mishra, and M. G. A. Paris, Mit- igating sloppiness in joint estimation of successive squeezing parameters, New J. Phys.27, 104511 (2025)

    work page 2025

  30. [30]

    He and M

    J. He and M. G. A. Paris, Scrambling for precision: Optimizing multiparameter qubit estimation in the face of sloppiness and incompatibility, J. Phys. A58, 325301 (2025)

    work page 2025

  31. [31]

    A. Z. Goldberg, J. L. Romero, ´A. S. Sanz, and L. L. S ´anchez- Soto, Taming singularities of the quantum Fisher information, Int. J. Quantum Inf.19, 2140004 (2021)

    work page 2021

  32. [32]

    Y . Yang, V . Montenegro, and A. Bayat, Overcoming Quantum Metrology Singularity through Sequential Measurements, Phys. Rev. Lett.135, 010401 (2025)

    work page 2025

  33. [33]

    Mihailescu, S

    G. Mihailescu, S. Sarkar, A. Bayat, S. Campbell, and A. K. Mitchell, Metrological symmetries in singular quantum multi- parameter estimation, Quantum Sci. Technol.11, 015006 (2025)

    work page 2025

  34. [34]

    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

  35. [35]

    Z. Hou, Z. Zhang, G.-Y . Xiang, C.-F. Li, G.-C. Guo, H. Chen, L. Liu, and H. Yuan, Minimal Tradeoffand Ultimate Preci- sion Limit of Multiparameter Quantum Magnetometry under the Parallel Scheme, Phys. Rev. Lett.125, 020501 (2020)

    work page 2020

  36. [36]

    Boixo, S

    S. Boixo, S. T. Flammia, C. M. Caves, and JM. Geremia, Gener- alized Limits for Single-Parameter Quantum Estimation, Phys. Rev. Lett.98, 090401 (2007)

    work page 2007

  37. [37]

    Liu, X.-X

    J. Liu, X.-X. Jing, and X. Wang, Quantum metrology with uni- tary parametrization processes, Sci. Rep.5, 8565 (2015)

    work page 2015

  38. [38]

    J. S. Sidhu and P. Kok, Quantum Fisher information for general spatial deformations of quantum emitters (2018), arXiv:1802.01601 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  39. [39]

    J. S. Sidhu and P. Kok, Geometric perspective on quantum pa- rameter estimation, A VS Quantum Sci.2, 014701 (2020)

    work page 2020

  40. [40]

    Demkowicz-Dobrza ´nski and L

    R. Demkowicz-Dobrza ´nski and L. Maccone, Using entangle- ment against noise in quantum metrology, Phys. Rev. Lett.113, 250801 (2014)

    work page 2014

  41. [41]

    Huang, C

    Z. Huang, C. Macchiavello, and L. Maccone, Noise-dependent optimal strategies for quantum metrology, Phys. Rev. A97, 032333 (2018)

    work page 2018

  42. [42]

    Sbroscia, I

    M. Sbroscia, I. Gianani, L. Mancino, E. Roccia, Z. Huang, L. Maccone, C. Macchiavello, and M. Barbieri, Experimental ancilla-assisted phase estimation in a noisy channel, Phys. Rev. A97, 032305 (2018)

    work page 2018

  43. [43]

    Hou, J.-F

    Z. Hou, J.-F. Tang, H. Chen, H. Yuan, G.-Y . Xiang, C.-F. Li, and G.-C. Guo, Zero–trade-offmultiparameter quantum estimation via simultaneously saturating multiple Heisenberg uncertainty relations, Sci. Adv.7, eabd2986 (2021)

    work page 2021

  44. [44]

    X. Song, F. Salvati, C. Gaikwad, N. Yunger Halpern, D. R. Arvidsson-Shukur, and K. Murch, Agnostic phase estimation, Phys. Rev. Lett.132, 260801 (2024)

    work page 2024

  45. [45]

    Kolenderski and R

    P. Kolenderski and R. Demkowicz-Dobrzanski, Optimal state for keeping reference frames aligned and the platonic solids, Phys. Rev. A78, 052333 (2008)

    work page 2008

  46. [46]

    X.-X. Jing, J. Liu, H.-N. Xiong, and X. Wang, Maximal quan- tum Fisher information for general su(2) parametrization pro- cesses, Phys. Rev. A92, 012312 (2015)

    work page 2015

  47. [47]

    Baumgratz and A

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

    work page 2016

  48. [48]

    Y . Yang, S. Ru, M. An, Y . Wang, F. Wang, P. Zhang, and F. Li, Multiparameter simultaneous optimal estimation with an su (2) coding unitary evolution, Phys. Rev. A105, 022406 (2022)

    work page 2022

  49. [49]

    Mihailescu, U

    G. Mihailescu, U. Alushi, R. Di Candia, S. Felicetti, and K. Gi- etka, Critical quantum sensing: A tutorial on parameter estima- tion near quantum phase transitions, PRX Quantum7, 020201 (2026)

    work page 2026

  50. [50]

    Mihailescu and K

    G. Mihailescu and K. Gietka, Anti-Critical Quantum Metrology (2026), arXiv:2602.03675 [quant-ph]

    work page arXiv 2026

  51. [51]

    Pang and A

    S. Pang and A. N. Jordan, Optimal adaptive control for quantum metrology with time-dependent Hamiltonians, Nat. Commun. 8, 14695 (2017)

    work page 2017

  52. [52]

    Wei and S

    Q. Wei and S. Pang, Efficient adaptive control strategy for multi-parameter quantum metrology in two-dimensional sys- tems, arXiv 10.48550/arXiv.2510.14811 (2025). 13

    work page doi:10.48550/arxiv.2510.14811 2025

  53. [53]

    Burgarth, V

    D. Burgarth, V . Giovannetti, A. N. Kato, and K. Yuasa, Quan- tum estimation via sequential measurements, New J. Phys.17, 113055 (2015)

    work page 2015

  54. [54]

    Radaelli, G

    M. Radaelli, G. T. Landi, K. Modi, and F. C. Binder, Fisher information of correlated stochastic processes, New J. Phys.25, 053037 (2023)

    work page 2023

  55. [55]

    O’Connor, S

    E. O’Connor, S. Campbell, and G. T. Landi, Fisher information rates in sequentially measured quantum systems, New J. Phys. 26, 033048 (2024)

    work page 2024