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Geometric Invariants of Quantum Metrology

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Pith's one-line read This paper proves a conservation law: a quantum Fisher information matrix built on a Lie algebra of observables has eigenvalues invariant under evolution generated by that algebra.

desk verdict Correct theorem, overbroad abstract—fix the orthonormality precondition and the N=1 identity. read the letter →

arxiv 2507.06128 v1 pith:BCDHT47L submitted 2025-07-08 quant-ph physics.atom-ph

Christopher Wilson , John Drew Wilson , Luke Coffman , Shah Saad Alam , Murray J. Holland This is my paper · ORCID
classification quant-phphysics.atom-ph
keywords quantumFisherinformationmatrixUhlmanncurvatureLiealgebrametrologymetrologicalincompatibilitygeometricinvariantsspinsqueezingentanglementwitness
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves a conservation law for the quantum Fisher information matrix: when the QFIM is constructed from a Hilbert–Schmidt-orthonormal basis of observables that close under commutation (a Lie algebra $g$), its eigenvalue spectrum is invariant under every unitary evolution generated by $g$. The same invariance extends to the Uhlmann curvature tensor and, through it, to the metrological incompatibility that limits simultaneous multiparameter estimation. Each Lie algebra thus endows any quantum state with a fixed 'budget' of metrological sensitivity, a resource that, like optical squeezing, cannot be amplified by symmetry-preserving operations. This partitions the space of density operators into equivalence classes (the $G$-orbits of states) on which statistical distances, volumes, and curvatures are all conserved — a metrological analogue of Liouville's theorem — and it turns the classical optimality criteria (A-, D-, E-optimality and the quantum Jeffrey's prior) into classifiers of whole state classes rather than individual states.

What carries the argument

The load-bearing object is the adjoint action of the Lie group $G$ on its algebra $g$, expressed in the Hilbert–Schmidt-orthonormal generator basis as an orthogonal matrix $\Lambda \in \mathrm{SO}(\dim g)$. The proof's move is to show that the symmetric logarithmic derivatives — and hence the QFIM and the Uhlmann curvature — inherit exactly this transformation, so that $F_g[U\rho U^\dagger] = \Lambda^T F_g[\rho]\Lambda$ becomes a similarity transformation that leaves the spectrum fixed. The orbit $\mathcal{M}_\rho = \{U\rho U^\dagger : U \in G\}$ is the geometric carrier of the result: its dimension equals the number of nonzero QFIM eigenvalues, and on it distances, the quantum Jeffrey's prior volume form, and the raised-index curvature spectrum are all adjoint-invariant.

What would settle it

Compute the full QFIM spectrum of a mixed state $\rho$ and of $U\rho U^\dagger$ for $U = e^{-i\theta G_\mu}$ scanned over $\theta$, using a Hilbert–Schmidt-orthonormal basis of $g$: any value of $\theta$ at which the spectra differ refutes the theorem, and the paper's own numerics test only pure states, so the mixed-state case is the sharpest open check. As a control, repeat the computation with one generator rescaled so the basis is not orthonormal, where the predicted invariance should fail even though the underlying metric operator is unchanged.

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Extended reading notes

Core claim

The central result, stated as the theorem of Section II, is that the eigenvalues of a quantum Fisher information matrix $F_g[\rho]$ constructed from an orthonormal (under the Hilbert–Schmidt inner product) basis of a Hermitian Lie algebra $g$ are invariant under any unitary $U \in G$ generated by $g$. The proof supplies an explicit transformation law: the symmetric logarithmic derivatives transform under the adjoint action as $U^\dagger \tilde{L}_\mu U = \Lambda^\alpha{}_\mu L_\alpha$, which carries the QFIM to $F_g[U\rho U^\dagger] = \Lambda^T F_g[\rho]\Lambda$, and the orthonormality condition forces $\Lambda \in \mathrm{SO}(\dim g)$, making this an orthogonal similarity that preserves the spectrum. The companion lemma shows the Uhlmann curvature transforms covariantly, $U_g[U\rho U^\dagger] = \Lambda^T U_g[\rho]\Lambda$, so the mutual curvature between corresponding eigenvectors and, through the raised-index tensor defined with the Moore–Penrose inverse of the QFIM, the metrological incompatibility parameter $\gamma$ are invariant on the whole orbit. The authors frame the result as a conservation law: each Lie algebra assigns a state an irreducible budget of sensitivity that symmetry-preserving operations cannot change, and their spin-1 ensemble example shows the hierarchy in action — a hyperfine rotation generated by $\mathrm{su}(3)$ cannot alter the $\mathrm{su}(3)$ QFIM spectrum, yet it does change the spectrum of the $\mathrm{su}(2)$ dipole sub-algebra.

Load-bearing premise

The proof requires the generator basis $\{G_\mu\}$ to be orthonormal under the Hilbert–Schmidt inner product, $\mathrm{Tr}(G_\mu G_\nu) = C\delta_{\mu\nu}$, since that is what makes the adjoint action an orthogonal matrix; the paper itself notes that on infinite-dimensional Hilbert spaces this condition is lost and the invariance is significantly modified, with the basis-independent formulation deferred to a companion manuscript.

Editorial extensions

If this is right

  • Within a $G$-orbit, no unitary generated by $g$ can amplify the QFIM spectrum: a state's metrological budget is fixed by the Lie algebra alone, so symmetry-preserving protocols such as spin squeezing redistribute sensitivity rather than create it within that algebra.
  • The metrological incompatibility parameter is conserved on each orbit, so the attainable precision for simultaneous estimation of several parameters is a property of the whole equivalence class, not of the particular state prepared.
  • Variational and machine-learning searches that optimize A-, D-, or E-optimality can be restricted to one representative per orbit; the equivalence class manifold $\mathcal{M}_\rho$ is the true search space, with dimension equal to the number of nonzero QFIM eigenvalues.
  • For $g = \mathrm{su}(d)$, the traced QFIM is bounded by $2N(N+d)(d-1)/d$ with equality for states of vanishing generator expectation values, and the coherent-state value $2N(d-1)$ yields a trace-based entanglement witness, while the pseudo-determinant defines the quantum Jeffrey's prior on each orbit.
  • When $g_1 \subset g_2$, operations in $g_2$ can modify the resources resolved by the $g_1$ QFIM but never the $g_2$ QFIM spectrum, so entanglement and usefulness for sensing are relative to the chosen algebra of observables.

Reading between the lines

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

  • My reading: the interpretive step the authors do not spell out is that any ranking of states for sensing should be built from orbit invariants; quantities that depend on a particular representative will spuriously vary under free $G$-reparametrizations, so the foliation acts as a gauge-like structure for metrology.
  • A testable extension: the theorem claims to hold for mixed as well as pure states, but the paper's numerics simulate only pure states; a depolarized-ensemble Ramsey experiment that compares QFIM spectra before and after a $g$-rotation would settle the mixed-state case directly.
  • The paper's own infinite-dimensional caveat suggests that continuous-variable metrology — squeezed light, harmonic-oscillator displacements — should exhibit a modified, rather than exact, conservation; checking whether a finite-dimensional truncation restores approximate invariance would connect this result to continuous-variable sensing platforms.
  • Because every functional of the QFIM spectrum is conserved, the orbit invariants could serve as a coordinate-free descriptor for resource theories of asymmetry, where the QFIM itself is used as a resource measure.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper claims that when the Quantum Fisher Information Matrix (QFIM) is constructed from a set of observables closed under commutation (a Lie algebra g), its spectrum is invariant under unitary evolution generated by g, and that the Uhlmann curvature tensor and the metrological incompatibility parameter inherit the same invariance. The proof uses the adjoint action of the group on an orthonormal generator basis to show that the QFIM transforms by an orthogonal similarity transformation. The authors then use these invariants to define equivalence-class manifolds of states with fixed metrological resource, derive interpretations in terms of A- and D-optimality, formulate an su(d) entanglement witness, and illustrate the result numerically for N spin-1 atoms under one-axis twisting and hyperfine rotations with g1=su(2), g2=su(3), and g3=u(Hsym). The abstract and introduction, however, omit the Hilbert-Schmidt orthonormality condition that is essential to the theorem.

Significance. If the statement is corrected, this is a useful and clean result. The central proof is elementary but the consequence is non-obvious: it identifies foliations of state space on which the QFIM spectrum, the raised-index Uhlmann curvature spectrum, and all spectral functionals are constant. The paper gives explicit spectra for coherent and N00N states under su(d), provides an entanglement witness, and supports the theory with a numerical simulation, which is a concrete strength. The main caveat is that the advertised 'closed under commutation' claim is false without the HS-orthonormality condition; the authors themselves have the right language in Appendix A, where the basis-independent invariant is g^{-1}F. With a corrected abstract and theorem statement, the paper would be a solid contribution to quantum metrology and quantum resource theory.

major comments (2)
  1. [Abstract; Section II; Appendix A] The advertised central claim is broader than the theorem supports. The abstract and introduction state that the QFIM spectrum is invariant whenever the observables are closed under commutation, but the proof in Section II requires the generator basis to be Hilbert-Schmidt orthonormal, Tr(G_mu G_nu)=C delta_mu_nu (stated as the third condition in the proof and in Eq. (A2)). This condition is load-bearing: it makes the adjoint-action matrix Lambda orthogonal, so F[rho~]=Lambda^T F[rho] Lambda is an orthogonal similarity. Without it, only a congruence is obtained and the raw spectrum is not conserved. A concrete counterexample is g=su(2) with non-orthonormal generators {sigma_x, sigma_y, sigma_z/2}, rho=|0><0| and U=exp(-i pi/4 sigma_y) in G: the raw QFIM for rho is diag(4,4,0), while for U rho U^dagger it is diag(0,4,1), so the spectrum changes. Appendix A itself defines the basis-independent object g^{-1}F, whose spectrum {2,2,0} is the true invariant in this example. The abstract, introduction, and theorem statement should be amended to either require an HS-orthonormal generator basis or state the invariant as the spectrum of F^#=g^{-1}F.
  2. [Section I, paragraph after Eq. (3)] The orthonormality of {G_mu} is introduced as optional ('we may also constrain the set...'), but the theorem requires it as a necessary condition. This is not a harmless phrasing: if the basis is not HS-orthonormal, the adjoint map Lambda is only an invertible linear map, and the congruence F -> Lambda^T F Lambda does not preserve eigenvalues. The formalism should be rewritten so that either all QFIMs in the paper are defined on an HS-orthonormal basis of g, or the basis-independent object F^#=g^{-1}F of Appendix A is promoted to the central object whose spectrum is invariant.
minor comments (5)
  1. [Section III.B, around Eq. (24)] The sentence 'When N=1, pure states have a fixed sum_mu <G_mu>^2 = 2(d-1)/d' is inconsistent with the normalization G_mu = sigma_mu^(j)/2 and with Eq. (B9); under that normalization, the correct value is sum_mu <G_mu>^2 = C(d-1)/d = (d-1)/(2d).
  2. [Throughout] Please fix typographical errors: 'meteorological incompatibility' (Section II) should be 'metrological incompatibility', 'incompatibiltiy' in the lemma heading should be 'incompatibility', 'PVOM' in Eq. (29) should be 'POVM', 'Jeffrey's' should be 'Jeffreys', and 'Noon' should be 'N00N' for consistency.
  3. [Section II, Eq. (8)] The transformation law for the SLD is justified by a heuristic sentence; please expand the proof to show explicitly that, with the parameterization fixed, U^dagger L~_mu U = Lambda^alpha_mu L_alpha follows from cyclicity of the trace and linearity of the SLD equation.
  4. [Table I, T-optimality row] The entry 'geometrically invariant between states which co-evolve' is unclear; please clarify what 'co-evolve' means here and how T-optimality is invariant under it.
  5. [Section III.A, Eq. (21)] In a reducible representation, the equality Tr_g(F_g[rho]) = 4 zeta - 4 sum_mu <G_mu>^2 holds only for pure states in the highest-weight subspace; please either state this condition explicitly or replace the equality with an inequality when the representation is reducible.

Circularity Check

0 steps flagged · score 1.0 of 10

Main invariance proof is self-contained and not circular; only minor non-load-bearing self-citations appear, with a stated orthonormality precondition that is a precision issue rather than circularity.

full rationale

The central claim, Section II, is proven directly from the definitions of the SLD and QFIM. The transformation law F[ρ~]=ΛᵀF[ρ]Λ follows from the adjoint action of the Lie group, and the Hilbert-Schmidt orthonormality condition Tr(GμGν)=Cδμν makes Λ orthogonal, so the congruence preserves the spectrum. No fitted constant, empirical input, or external prediction is used to obtain the theorem; the derivation is self-contained. Self-citations are present but not load-bearing: Ref. [7] supplies normalization conventions and Ref. [55] is a deferred extension that is explicitly not needed for the main proof. The abstract's wording omits the orthonormality precondition, and for a non-orthonormal basis the raw matrix spectrum can indeed change, as the paper itself acknowledges by defining the basis-independent object in Appendix A. This is an overbreadth or correctness concern, not circular reasoning. The numerical example is a demonstration of the theorem, not a source of the theorem. No step in the derivation reduces to its own input, and no self-citation chain forces the conclusion.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard QFIM/SLD definitions plus the HS-orthonormal basis condition. No parameters are fitted to data; example simulation parameters (N, α, β) are chosen values, not fits. No new particles, forces, or dimensions are introduced.

assumptions (3)
  • domain assumption Hilbert-Schmidt orthonormality of the generator basis, Tr(GμGν)=Cδμν.
    Introduced in Section II and Appendix A. It makes the adjoint action matrix Λ orthogonal, which turns the QFIM transformation into an orthogonal similarity that preserves the spectrum. Without this condition, the matrix spectrum is not G-invariant.
  • domain assumption Unitary parameter encoding only, with SLDs defined by ∂ρ/∂θμ = -i[Gμ,ρ].
    Equation (2) fixes the encoding model. The conservation law is not claimed for dissipative or non-unitary encodings, and the paper explicitly limits itself to unitary evolution.
  • standard math Standard Lie group and Lie algebra facts: the adjoint action of G on g, and the covariance of SLDs under concurrent unitary transformation of state and generators.
    Used in the proof of Eq (8). These are standard background results, not introduced ad hoc for this paper.

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Pith. "Pith review of Geometric Invariants of Quantum Metrology." pith.science (2026). https://pith.science/paper/BCDHT47L

@misc{pith2026250706128,
  author       = {Pith},
  title        = {Pith review of: Geometric Invariants of Quantum Metrology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BCDHT47L}},
  note         = {Machine review of arXiv:2507.06128}
}
read the original abstract

We establish a previously unexplored conservation law for the Quantum Fisher Information Matrix (QFIM) expressed as follows; when the QFIM is constructed from a set of observables closed under commutation, i.e., a Lie algebra, the spectrum of the QFIM is invariant under unitary dynamics generated by these same operators. Each Lie algebra therefore endows any quantum state with a fixed "budget" of metrological sensitivity -- an intrinsic resource that we show, like optical squeezing in interferometry, cannot be amplified by symmetry-preserving operations. The Uhlmann curvature tensor (UCT) naturally inherits the same symmetry group, and so quantum incompatibility is similarly fixed. As a result, a metrological analog to Liouville's theorem appears; statistical distances, volumes, and curvatures are invariant under the evolution generated by the Lie algebra. We discuss this as it relates to the quantum analogs of classical optimality criteria. This enables one to efficiently classify useful classes of quantum states at the level of Lie algebras through geometric invariants.

Figures

Figures reproduced from arXiv: 2507.06128 by the authors.

Figure 1
Figure 1. FIG. 1. The QFIM, [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. A representation of the changing incompatibility [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. A representation of two equivalence classes with dif [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) A cartoon of the example. (b) The atomic level [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) The largest eigenvalue of [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a) [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (a) A cartoon of the illustrative example. (b) The [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

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Works this paper leans on

103 extracted references · 66 canonical work pages

  1. [55]

    J. D. Wilson, C. Wilson, L. Coffman, and M. J. Holland, Manuscript in preparation (2025), (to be published)

    work page 2025

  2. [1]

    Noether, Invariant variation problems, Transport the- ory and statistical physics 1, 186 (1971)

    E. Noether, Invariant variation problems, Transport the- ory and statistical physics 1, 186 (1971)

    1971

  3. [2]

    J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum Fisher information matrix and multiparameter estima- tion, Journal of Physics A: Mathematical and Theoretical 53, 023001 (2019)

    2019

  4. [3]

    Hyllus, W

    P. Hyllus, W. Laskowski, R. Krischek, C. Schwem- mer, W. Wieczorek, H. Weinfurter, L. Pezz` e, and A. Smerzi, Fisher information and multiparticle entan- glement, Phys. Rev. A 85, 022321 (2012)

    2012

  5. [4]

    K. C. Tan, V. Narasimhachar, and B. Regula, Fisher in- formation universally identifies quantum resources, Phys. Rev. Lett. 127, 200402 (2021)

    2021

  6. [5]

    Kudo and H

    D. Kudo and H. Tajima, Fisher information matrix as a resource measure in the resource theory of asymmetry with general connected-lie-group symmetry, Phys. Rev. A 107, 062418 (2023)

    2023

  7. [6]

    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)

    2018

  8. [7]

    J. T. Reilly, J. D. Wilson, S. B. J¨ ager, C. Wilson, and M. J. Holland, Optimal generators for quantum sensing, Physical Review Letters 131, 150802 (2023)

    2023

Show all 103 references
  1. [8]

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

    1994

  2. [9]

    W. K. Wootters, Statistical distance and hilbert space, Phys. Rev. D 23, 357 (1981)

    1981

  3. [10]

    Campos Venuti and P

    L. Campos Venuti and P. Zanardi, Quantum critical scal- ing of the geometric tensors, Phys. Rev. Lett. 99, 095701 (2007)

    2007

  4. [11]

    Wang, L.-N

    T.-L. Wang, L.-N. Wu, W. Yang, G.-R. Jin, N. Lambert, and F. Nori, Quantum Fisher information as a signature of the superradiant quantum phase transition, New Jour- nal of Physics 16, 063039 (2014)

    2014

  5. [12]

    Wu and J.-B

    W. Wu and J.-B. Xu, Geometric phase, quantum Fisher information, geometric quantum correlation and quantum phase transition in the cavity-Bose–Einstein- condensate system, Quantum Information Processing 15, 3695 (2016)

    2016

  6. [13]

    H. Song, S. Luo, and S. Fu, Quantum criticality from Fisher information, Quantum Information Processing16, 1 (2017)

    2017

  7. [14]

    Lucchesi and M

    L. Lucchesi and M. L. Chiofalo, Many-body entangle- ment in short-range interacting Fermi gases for metrol- ogy, Phys. Rev. Lett. 123, 060406 (2019)

    2019

  8. [15]

    Cie´ sli´ nski, P

    P. Cie´ sli´ nski, P. Kurzy´ nski, T. Sowi´ nski, W. K lobus, and W. Laskowski, Exploring many-body interactions through quantum Fisher information, Phys. Rev. A 110, 012407 (2024)

    2024

  9. [16]

    Y. Fang, M. Mahankali, Y. Wang, L. Chen, H. Hu, S. Paschen, and Q. Si, Amplified multipartite entangle- ment witnessed in a quantum critical metal, Nature Com- munications 16, 2498 (2025)

    2025

  10. [17]

    Petz and C

    D. Petz and C. Sud´ ar, Extending the Fisher metric to density matrices, Geometry in Present Day Science 21 (1999)

    1999

  11. [18]

    J. S. Sidhu and P. Kok, Geometric perspective on quan- tum parameter estimation, A VS Quantum Science 2, 014701 (2020)

    2020

  12. [19]

    T¨ orm¨ a, Essay: Where can quantum geometry lead us?, Phys

    P. T¨ orm¨ a, Essay: Where can quantum geometry lead us?, Phys. Rev. Lett. 131, 240001 (2023)

    2023

  13. [20]

    Carollo, B

    A. Carollo, B. Spagnolo, A. A. Dubkov, and D. Valenti, On quantumness in multi-parameter quantum estima- tion, Journal of Statistical Mechanics: Theory and Ex- periment 2019, 094010 (2019)

    2019

  14. [21]

    Uhlmann, Density operators as an arena for differen- tial geometry, Rep

    A. Uhlmann, Density operators as an arena for differen- tial geometry, Rep. Math. Phys 33, 253 (1993)

    1993

  15. [22]

    Carollo, B

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

    2018

  16. [23]

    Viyuela, A

    O. Viyuela, A. Rivas, and M. A. Martin-Delgado, Uhlmann phase as a topological measure for one- dimensional fermion systems, Phys. Rev. Lett. 112, 130401 (2014)

    2014

  17. [24]

    Viyuela, A

    O. Viyuela, A. Rivas, and M. A. Martin-Delgado, Two- dimensional density-matrix topological fermionic phases: Topological Uhlmann numbers, Phys. Rev. Lett. 113, 076408 (2014)

    2014

  18. [25]

    X.-Y. Hou, H. Guo, and C.-C. Chien, Finite-temperature topological phase transitions of spin- j systems in Uhlmann processes: General formalism and experimental protocols, Phys. Rev. A 104, 023303 (2021)

    2021

  19. [26]

    Morachis Galindo, F

    D. Morachis Galindo, F. Rojas, and J. A. Maytorena, Topological Uhlmann phase transitions for a spin-j parti- cle in a magnetic field, Phys. Rev. A 103, 042221 (2021)

    2021

  20. [27]

    Yao, X.-G

    W.-M. Yao, X.-G. Fan, X.-K. Song, D. Wang, and L. Ye, Research on quantum correlations and Uhlmann phase in two-fermion mixed state system, Physica Scripta 99, 075109 (2024)

    2024

  21. [28]

    W. Guo, W. Zhong, X.-X. Jing, L.-B. Fu, and X. Wang, Berry curvature as a lower bound for multiparameter es- timation, Phys. Rev. A 93, 042115 (2016)

    2016

  22. [29]

    Nielsen and I

    M. Nielsen and I. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition(Cam- bridge University Press, 2010)

    2010

  23. [30]

    P. W. Shor, Polynomial-time algorithms for prime factor- ization and discrete logarithms on a quantum computer, SIAM Review 41, 303 (1999)

    1999

  24. [31]

    V. M. Kendon and W. J. Munro, Entanglement and its role in Shor’s algorithm, Quantum Info. Comput. 6, 630–640 (2006)

    2006

  25. [32]

    C. H. Bennett, G. Brassard, C. Cr´ epeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky- Rosen channels, Phys. Rev. Lett. 70, 1895 (1993)

    1993

  26. [33]

    Yin, Y.-H

    J. Yin, Y.-H. Li, S.-K. Liao, M. Yang, Y. Cao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, S.-L. Li, R. Shu, Y.- M. Huang, L. Deng, L. Li, Q. Zhang, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, X.-B. Wang, F. Xu, J.-Y. Wang, C.-Z. Peng, A. K. Ekert, and J.-W. Pan, Entanglement-based secure...

    2020

  27. [34]

    M. J. Holland and K. Burnett, Interferometric detection of optical phase shifts at the Heisenberg limit, Phys. Rev. Lett. 71, 1355 (1993)

    1993

  28. [35]

    Ye and P

    J. Ye and P. Zoller, Essay: Quantum sensing with atomic, molecular, and optical platforms for fundamental physics, Phys. Rev. Lett. 132, 190001 (2024)

    2024

  29. [36]

    Buscemi, E

    F. Buscemi, E. Chitambar, and W. Zhou, Complete re- source theory of quantum incompatibility as quantum programmability, Phys. Rev. Lett. 124, 120401 (2020)

    2020

  30. [37]

    Cerezo, A

    M. Cerezo, A. Sone, J. L. Beckey, and P. J. Coles, Sub-quantum Fisher information, Quantum Science and Technology 6, 035008 (2021)

    2021

  31. [38]

    J. L. Beckey, M. Cerezo, A. Sone, and P. J. Coles, Vari- ational quantum algorithm for estimating the quantum Fisher information, Physical Review Research 4, 013083 (2022)

    2022

  32. [39]

    M. Kang, S. Kim, Y. Qian, P. M. Neves, L. Ye, J. Jung, D. Puntel, F. Mazzola, S. Fang, C. Jozwiak, et al., Mea- surements of the quantum geometric tensor in solids, Na- ture Physics 21, 110 (2025)

    2025

  33. [40]

    L. P. Barnes, Y. Han, and A. ¨Ozg¨ ur, A geometric charac- terization of Fisher information from quantized samples with applications to distributed statistical estimation, in 2018 56th Annual Allerton Conference on Communica- tion, Control, and Computing (Allerton) (IEEE, 2018...

    2018

  34. [41]

    M. Yu, P. Yang, M. Gong, Q. Cao, Q. Lu, H. Liu, S. Zhang, M. B. Plenio, F. Jelezko, T. Ozawa, N. Gold- man, and J. Cai, Experimental measurement of the quan- tum geometric tensor using coupled qubits in diamond, National Science Review 7, 254 (2019)

    2019

  35. [42]

    M. Yu, Y. Liu, P. Yang, M. Gong, Q. Cao, S. Zhang, H. Liu, M. Heyl, T. Ozawa, N. Goldman, and J. Cai, Quantum Fisher information measurement and verifica- tion of the quantum cram´ er–rao bound in a solid-state qubit, npj Quantum Information 8, 56 (2022)

    2022

  36. [44]

    Kitagawa and M

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

    1993

  37. [45]

    T´ oth, C

    G. T´ oth, C. Knapp, O. G¨ uhne, and H. J. Briegel, Spin squeezing and entanglement, Phys. Rev. A 79, 042334 (2009)

    2009

  38. [46]

    G. P. Greve, C. Luo, B. Wu, and J. K. Thompson, Entanglement-enhanced matter-wave interferometry in a high-finesse cavity, Nature 610, 472 (2022)

    2022

  39. [47]

    J. J. Meyer, Fisher information in noisy intermediate- scale quantum applications, Quantum 5, 539 (2021)

    2021

  40. [48]

    J. T. Reilly, S. B. J¨ ager, J. D. Wilson, J. Cooper, S. Eg- gert, and M. J. Holland, Speeding up squeezing with a pe- riodically driven dicke model, Phys. Rev. Res. 6, 033090 (2024)

    2024

  41. [49]

    Elben, S

    A. Elben, S. T. Flammia, H.-Y. Huang, R. Kueng, J. Preskill, B. Vermersch, and P. Zoller, The random- ized measurement toolbox, Nature Reviews Physics 5, 9 (2023)

    2023

  42. [50]

    S. L. Braunstein, Squeezing as an irreducible resource, Phys. Rev. A 71, 055801 (2005)

    2005

  43. [51]

    Kiefer, Optimum experimental designs, Journal of the Royal Statistical Society: Series B (Methodological) 21, 272 (1959)

    J. Kiefer, Optimum experimental designs, Journal of the Royal Statistical Society: Series B (Methodological) 21, 272 (1959)

    1959

  44. [52]

    Liouville, Note sur la th´ eorie de la variation des con- stantes arbitraires, Journal de math´ ematiques pures et appliqu´ ees3, 342 (1838)

    J. Liouville, Note sur la th´ eorie de la variation des con- stantes arbitraires, Journal de math´ ematiques pures et appliqu´ ees3, 342 (1838)

  45. [53]

    See Supplemental Material. Here, we outline the trace- inner product and describe the normalization, show the explicit calculation of the trace and determinant for co- herent and NOON states, directly calculate the average response of the quantum state, provide a formal constr...

  46. [54]

    M. G. Paris, Quantum estimation for quantum technol- ogy, International Journal of Quantum Information 7, 125 (2009)

    2009

  47. [56]

    Petz, Monotone metrics on matrix spaces, Linear Al- gebra and its Applications 244, 81 (1996)

    D. Petz, Monotone metrics on matrix spaces, Linear Al- gebra and its Applications 244, 81 (1996)

    1996

  48. [57]

    Candeloro, S

    A. Candeloro, S. Razavian, M. Piccolini, B. Teklu, S. Olivares, and M. G. A. Paris, Quantum probes for the characterization of nonlinear media, Entropy 23, 10.3390/e23101353 (2021)

    2021 doi

  49. [58]

    Suzuki, Y

    J. Suzuki, Y. Yang, and M. Hayashi, Quantum state esti- mation with nuisance parameters, Journal of Physics A: Mathematical and Theoretical 53, 453001 (2020)

    2020

  50. [59]

    S. S. Alam, V. E. Colussi, J. D. Wilson, J. T. Reilly, M. A. Perlin, and M. J. Holland, Robust quantum sens- ing with multiparameter decorrelation, arXiv preprint arXiv:2405.07907 (2024)

    2024 arXiv

  51. [60]

    This is because a matrix, M, maps a vector to a vector; M α β V β = W α

    In index notation, we’d say that a matrix must have one upper and one lower index. This is because a matrix, M, maps a vector to a vector; M α β V β = W α

  52. [61]

    [20] corresponds to a differing convention in the definition of Eq

    Here, the differing factor of 2 from Ref. [20] corresponds to a differing convention in the definition of Eq. 10

  53. [62]

    A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, North-Holland series in statistics and probability (North-Holland Publishing Company, 1982)

    1982

  54. [63]

    C. W. Helstrom, Quantum detection and estimation the- ory, Journal of Statistical Physics 1, 231 (1969)

    1969

  55. [64]

    Hayashi, Quantum information theory (Springer, 2017)

    M. Hayashi, Quantum information theory (Springer, 2017)

    2017

  56. [65]

    Liu, H.-N

    J. Liu, H.-N. Xiong, F. Song, and X. Wang, Fidelity susceptibility and quantum Fisher information for den- sity operators with arbitrary ranks, Physica A: Statistical Mechanics and its Applications 410, 167 (2014)

    2014

  57. [66]

    We note that this is equivalent to integrating against the Haar measure on G to pick a random direction for ˆG⃗ n= ˆV (⃗ n) ˆG1 ˆV (⃗ n)† where ˆV (⃗ n) ∈ G and ˆG1 is the (arbtirarily chosen) first element of our basis for g

  58. [67]

    P. B. Slater, Quantum coin-tossing in a bayesian jeffreys framework, Physics Letters A 206, 66 (1995)

    1995

  59. [68]

    P. B. Slater, The quantum jeffreys’ prior/bures metric volume element for squeezed thermal states and a uni- versal coding conjecture, Journal of Physics A: Mathe- matical and General 29, L601 (1996)

    1996

  60. [69]

    A. A. Mele, Introduction to haar measure tools in quan- tum information: A beginner’s tutorial, Quantum 8, 1340 (2024)

    2024

  61. [70]

    B. Q. Song, J. D. H. Smith, L. Luo, and J. Wang, Quan- tum liouville theorem based on haar measure, Phys. Rev. B 109, 144301 (2024)

    2024

  62. [71]

    ´Eva Sz˝ oke, A quantummechanical derivation of the eigen- values of the quadratic Casimir operator of the algebra 11 SU(n) in Young tableau representation, Physics Letters A 383, 1260 (2019)

    2019

  63. [72]

    Vitagliano, O

    G. Vitagliano, O. G¨ uhne, and G. T´ oth,su(d)-squeezing and many-body entanglement geometry in finite- dimensional systems, arXiv preprint arXiv:2406.13338 (2024)

    2024 arXiv

  64. [73]

    T´ oth, Multipartite entanglement and high-precision metrology, Phys

    G. T´ oth, Multipartite entanglement and high-precision metrology, Phys. Rev. A 85, 022322 (2012)

    2012

  65. [74]

    This could equivalently be found by the observation that su(d) has a maximal Cartan sub-algebra of dimension d − 1, which yields a counting argument that there are 2(d − 1) unique directions of rotation

  66. [75]

    Pezz` e and A

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

    2009

  67. [76]

    Combes and H

    J. Combes and H. Wiseman, States for phase estimation in quantum interferometry, Journal of Optics B: Quan- tum and Semiclassical Optics 7, 14 (2004)

    2004

  68. [77]

    ˙Zukowski, A

    M. ˙Zukowski, A. Zeilinger, M. Horne, and H. Weinfurter, Quest for GHZ states, Acta Physica Polonica A 93, 187 (1998)

    1998

  69. [78]

    Chen, X.-H

    Y.-A. Chen, X.-H. Bao, Z.-S. Yuan, S. Chen, B. Zhao, and J.-W. Pan, Heralded generation of an atomic noon state, Phys. Rev. Lett. 104, 043601 (2010)

    2010

  70. [79]

    L. H. C. Vaecairn, J. T. Reilly, J. D. Wilson, S. B. J¨ ager, and M. Holland, Engineering tunable decoherence-free subspaces with collective atom-cavity interactions, Phys. Rev. A 111, 033718 (2025)

    2025

  71. [80]

    Pi˜ neiro Orioli, J

    A. Pi˜ neiro Orioli, J. K. Thompson, and A. M. Rey, Emer- gent dark states from superradiant dynamics in multi- level atoms in a cavity, Phys. Rev. X 12, 011054 (2022)

    2022

  72. [81]

    R. E. F. Silva and J. Feist, Permutational symmetry for identical multilevel systems: A second-quantized ap- proach, Phys. Rev. A 105, 043704 (2022)

    2022

  73. [82]

    Yukawa and K

    E. Yukawa and K. Nemoto, Classification of spin and multipolar squeezing, Journal of Physics A: Mathemati- cal and Theoretical 49, 255301 (2016)

    2016

  74. [83]

    E. J. Davis, G. Bentsen, L. Homeier, T. Li, and M. H. Schleier-Smith, Photon-mediated spin-exchange dynam- ics of spin-1 atoms, Phys. Rev. Lett. 122, 010405 (2019)

    2019

  75. [84]

    Bilitewski, L

    T. Bilitewski, L. De Marco, J.-R. Li, K. Matsuda, W. G. Tobias, G. Valtolina, J. Ye, and A. M. Rey, Dynam- ical generation of spin squeezing in ultracold dipolar molecules, Phys. Rev. Lett. 126, 113401 (2021)

    2021

  76. [85]

    C. Luo, H. Zhang, V. P. Koh, J. D. Wilson, A. Chu, M. J. Holland, A. M. Rey, and J. K. Thompson, Momentum- exchange interactions in a bragg atom interferometer suppress doppler dephasing, Science 384, 551 (2024)

    2024

  77. [86]

    J. D. Wilson, J. T. Reilly, H. Zhang, C. Luo, A. Chu, J. K. Thompson, A. M. Rey, and M. J. Holland, Entan- gled matter waves for quantum enhanced sensing, Phys. Rev. A 110, L041301 (2024)

    2024

  78. [87]

    Pfeifer, The Lie algebras su(N), in The Lie Algebras su(N): An Introduction(Birkh¨ auser Basel, Basel, 2003)

    W. Pfeifer, The Lie algebras su(N), in The Lie Algebras su(N): An Introduction(Birkh¨ auser Basel, Basel, 2003)

    2003

  79. [88]

    Barnum, E

    H. Barnum, E. Knill, G. Ortiz, and L. Viola, General- izations of entanglement based on coherent states and convex sets, Phys. Rev. A 68, 032308 (2003)

    2003

  80. [89]

    Mathur, I

    M. Mathur, I. Raychowdhury, and R. Anishetty, SU(N) irreducible Schwinger bosons, Journal of Mathematical Physics 51, 093504 (2010)

    2010

  81. [90]

    Tsang and C

    M. Tsang and C. M. Caves, Evading quantum mechan- ics: Engineering a classical subsystem within a quantum environment, Phys. Rev. X 2, 031016 (2012)

    2012

  82. [91]

    Zhang and D

    D.-J. Zhang and D. M. Tong, Krylov shadow tomogra- phy: Efficient estimation of quantum Fisher information, Phys. Rev. Lett. 134, 110802 (2025)

    2025

  83. [92]

    S. Imai, J. Yang, and L. Pezz` e, Semi-classical geomet- ric tensor in multiparameter quantum information, arXiv preprint arXiv:2504.06812 (2025)

    2025 arXiv

  84. [93]

    C. H. Baldwin, B. J. Bjork, J. P. Gaebler, D. Hayes, and D. Stack, Subspace benchmarking high-fidelity en- tangling operations with trapped ions, Phys. Rev. Res. 2, 013317 (2020)

    2020

  85. [94]

    Nemoto, Generalized coherent states for SU(n) sys- tems, Journal of Physics A: Mathematical and General 33, 3493 (2000)

    K. Nemoto, Generalized coherent states for SU(n) sys- tems, Journal of Physics A: Mathematical and General 33, 3493 (2000)

    2000

  86. [95]

    S. B. J¨ ager, T. Schmit, G. Morigi, M. J. Holland, and R. Betzholz, Lindblad master equations for quantum sys- tems coupled to dissipative bosonic modes, Phys. Rev. Lett. 129, 063601 (2022). 12 In appendix A we outline the trace-inner product and describe the normalization, C...

    2022

  87. [96]

    x”-like and “y

    g = su(d) Now, we will construct the Lie algebra of the general N particle representation of su( d). Here, we will con- struct the N particle representation of these operators according to ˆOµ ≡ NX j=1 ˆσ(j) µ 2 (B3) where ˆσ(j) µ is, once again, the d × d Gell-Mann operator f...

  88. [97]

    The representation of this algebra is spanned by three operators, ˆSx, ˆSy and ˆSz

    g1 = su(2) Here, we explicitly construct the dipole operators con- sidered in the example of the main text: g1 = su(2), or the operators which generate collective spin-1 rotations. The representation of this algebra is spanned by three operators, ˆSx, ˆSy and ˆSz. These are gi...

  89. [98]

    g2 = su(3) The second algebra are the collective hyperfine tran- sitions for the three levels, corresponding to g2 = su(3). To build the su(3) algebra, we can once again use sums over over the single particle operators: ˆQµ ≡ NX j=1 ˆσ(j) µ /2, (B15) where we use σ here to evo...

  90. [99]

    The norm used for g2 is the same norm outlined in Ref

    corresponds to the fact that the standard quantum limit and Heisenberg limit for g1 are 2 N and 4 N 2 respectively, whereas they are N and N 2 respectively for g2. The norm used for g2 is the same norm outlined in Ref. [7] and is, in some sense, canonical by fixing the standar...

  91. [100]

    particle number

    g3 = u(Hsym) Now we consider the third Lie algebra from the exam- ple in the main text. This is the set of all possible observ- ables on the symmetric subspace, denoted g3 = u(Hsym) where no representation is needed. In the limit that d = D and N = 1 this matches Appendix B 1,...

  92. [101]

    North pole

    The Coherent Spin State A generalized Coherent Spin State (CSS) is given by |C⟩ = |ψ⟩⊗N where |ψ⟩ is any single particle pure state [94]. We want to classify the trace and determinant of the QFIM of this state, to do so we will use the obser- vation made in Ref. [88]. Namely, ...

  93. [102]

    We can use the same labeling argument as above to identify the state |ψ⟩ = |1⟩ and |⊥⟩ = |2⟩

    The Noon State In the text, we give the definition for the generalzied Noon state, which we construct as the sum of two or- thogonal CSS: |N ⟩= (|ψ⟩⊗N + |⊥⟩⊗N )/ √ 2 (C5) where |ψ⟩ is a single particle state, just like in |C⟩, and |⊥⟩ is any perpendicular single particle state...

  94. [103]

    T race as an Entanglement Witness We have that Trg(Fg[C]) = 2N (d − 1) for a CSS. First, we can verify that no separable state can surpass this 16 through the following, where ρ is pure: Trg(Fg[ρ]) = 4 X µ ⟨ ˆG2 µ⟩ρ − ⟨ˆGµ⟩2 ρ = 4⟨ ˆG2⟩ρ − 4 X µ ⟨ ˆGµ⟩2 ρ (C12) where ˆG∈ is th...

  95. [104]

    ˆS+ acts as the effective pump terms to the cavity elimination. We solve ∂ ˆα ∂t = −i∆s[ ˆSz, ˆα] − i∆c ˆα − κˆα − iη − i g√ 2 ˆS+ (E4) whereupon the effective evolution is given by ∂ρ ∂t = Leff ρ = −i[ ˆHa, ρ] + D[√κˆα]ρ, (E5) with the effective Hamiltonian being ˆHS = ˆHsys ...

Pith tools

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