Pith. sign in

REVIEW 4 major objections 4 minor 64 references

Information geometry of bosonic Gaussian thermal states

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read This paper derives exact Fisher–Bures and Kubo–Mori information matrices for bosonic Gaussian thermal states and turns them into explicit quantum Cramér–Rao bounds for estimating mean vectors and Hamiltonians.

desk verdict The central closed-form formulas in Theorem 1 are too large by a factor of two; the paper needs a straightforward correction but deserves a real referee. read the letter →

arxiv 2411.18268 v3 pith:BR7VIR2A submitted 2024-11-27 quant-ph cs.IThep-thmath-phmath.ITmath.MP

Zixin Huang , Mark M. Wilde This is my paper · ORCID
classification quant-phcs.IThep-thmath-phmath.ITmath.MP MSC 81P4581P5062B10
keywords bosonicGaussianthermalstatesFisher-BuresinformationmatrixKubo-MoriquantumCramer-Raoboundsymmetriclogarithmicderivativegeometryparameterestimationnaturalgradientdescent
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

Bosonic Gaussian thermal states—states of the form $\rho(\mu,H)=\exp(-\frac12(\hat x-\mu)^T H(\hat x-\mu))/Z$ that cover every full-rank continuous-variable Gaussian state—are central to quantum optics and quantum information. This paper asks how the information geometry of these states depends on their two natural parameters, the mean vector $\mu$ and the Hamiltonian matrix $H$, and derives exact integral formulas for two quantum versions of the classical Fisher information matrix: the Fisher–Bures metric (tied to fidelity distance) and the Kubo–Mori metric (tied to relative entropy). It also obtains explicit closed forms for the derivative and the symmetric logarithmic derivative of such a state. These formulas convert the quantum Cramér–Rao bound into concrete precision limits for estimating $\mu$ and $H$, and they supply the gradient and metric ingredients needed for gradient-descent and natural-gradient-descent optimization over Gaussian thermal ansätze.

What carries the argument

The load-bearing identity is the thermal-state derivative formula (Proposition 1): for $\rho(\theta)=e^{-G(\theta)}/Z$, $\partial_j\rho=-\frac12\{\Phi_\theta(\partial_jG),\rho\}+\rho\langle\partial_jG\rangle$, where $\Phi_\theta(X)=\int dt\,p(t)e^{iGt}Xe^{-iGt}$ is a dephasing channel whose kernel is the high-peak tent density $p(t)=\frac{2}{\pi}\ln|\coth(\pi t/2)|$. Proposition 3 rewrites the Fisher–Bures metric through the convolved kernel $q(t)=\int d\tau\,p(\tau)p(t+\tau)$, which is why $q(t)$ appears in Theorem 1. For the quadratic $G$ of a Gaussian thermal state, the conjugation $e^{iGt}\hat{x}^c_k e^{-iGt}=\sum_\ell (e^{\Omega Ht})_{k,\ell}\hat{x}^c_\ell$ reduces every metric element to Gaussian expectation values of quadrature polynomials. The Kubo–Mori version follows from the same calculation with $p(t)$ in place of $q(t)$.

What would settle it

Take a single-mode displaced thermal state with diagonal $H=\mathrm{diag}(\omega,\omega)$ and nonzero $\mu$, evaluate the Fisher–Bures element $I^{\mathrm{FB}}_{1,1}$ from the integral (49), and compare it with the value from the known Fisher–Bures formula in the mean-and-covariance parameterization via the Jacobian congruence; Theorem 1 requires agreement for all $\omega>0$ and all $\mu$.

Watch

Extended reading notes

Core claim

The paper's central claim is that for the family $\rho(\mu,H)=\exp(-\frac12(\hat x-\mu)^T H(\hat x-\mu))/Z(\mu,H)$, the Fisher–Bures and Kubo–Mori information matrices have exact closed forms in terms of the covariance matrix $V$, the symplectic form $\Omega$, and the one-parameter symplectic rotation $S(t)=e^{-\Omega Ht}$. The $\mu$–$\mu$ block is $2\int dt\, q(t)[H V S(t)H]_{m_1,m_2}$ with the convolved kernel $q(t)$; the $\mu$–$H$ block vanishes; and the $H$–$H$ block is an integral of a specific quadrature fourth-moment combination minus $\frac14 V_{k_1,l_1}V_{k_2,l_2}$. The Kubo–Mori matrix is identical with $q(t)$ replaced by the high-peak tent density $p(t)$. The same derivative machinery yields Theorems 2 and 3, expressing derivatives and symmetric logarithmic derivatives of the state as anticommutator integrals against $p(t)$.

Load-bearing premise

The proof of Proposition 1, in Appendix A, assumes that the unbounded quadratic operator $G(\theta)$ can be handled through its spectral decomposition with the $t$-integral and the trace interchanged freely; if that functional-analytic step fails on the infinite-dimensional Fock space, the derivative formula and the information-matrix theorems built on it would not hold.

Editorial extensions

If this is right

  • For any unbiased estimator of the mean vector $\mu$ and Hamiltonian matrix $H$ from $n$ copies of $\rho(\mu,H)$, the weighted quantum Cramér–Rao bound becomes explicit once (49)–(51) are inserted, giving concrete precision limits.
  • Because the $\mu$–$H$ cross-block of the Fisher–Bures matrix vanishes, the multiparameter bound block-diagonalizes: estimation of $\mu$ and $H$ is asymptotically decoupled at the level of the metric.
  • The symmetric logarithmic derivative formulas in Theorem 3 specify the optimal single-parameter measurements for each component of $\mu$ and $H$, assuming the observable can be implemented.
  • The derivative formulas in Theorem 2 give a direct way to compute gradients of expectation values $\mathrm{Tr}[G\rho(\mu,H)]$ for variational optimization with Gaussian thermal ansätze.
  • The Kubo–Mori information matrix, obtained here for this parameterization, supplies the metric for natural gradient descent based on relative-entropy geometry.

Reading between the lines

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

  • The same substitution pattern suggests that the entire family of monotone quantum information metrics on Gaussian thermal states could be obtained by replacing the kernel $p(t)$ or $q(t)$ with the appropriate operator-monotone weight function; the paper stops at the Fisher–Bures and Kubo–Mori endpoints.
  • Because the Fisher–Bures matrix has zero $\mu$–$H$ cross-block, the overall estimation cost separates into a sum of mean-estimation and Hamiltonian-estimation costs; whether sequential measurements attain the joint quantum Cramér–Rao bound would require checking compatibility of the optimal measurements, which the paper does not do.
  • The integrals over $p(t)$ and $q(t)$ may admit closed-form evaluations in terms of the symplectic eigenvalues of $H$ in the commuting case, which would turn the bounds into elementary metrology formulas.
  • A natural numerical test is to use (49)–(51) inside a natural-gradient optimization loop for a few-mode photonic ansatz and compare behavior against the covariance-matrix parameterization; the paper proposes the application but does not benchmark it.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies the information geometry of n-mode bosonic Gaussian thermal states parameterized by a mean vector mu and a positive-definite Hamiltonian matrix H. Starting from recent general formulas for derivatives and information matrices of thermal states, the authors derive closed-form integral expressions for the Fisher-Bures and Kubo-Mori information matrix elements with respect to mu and H, and they state formulas for the derivative and the symmetric logarithmic derivative of such states. The stated motivation is to obtain quantum Cramer-Rao bounds for estimating mu and H and to enable natural-gradient and gradient-descent methods for bosonic Gaussian ansatze.

Significance. If the final formulas are correct, the paper provides exact, parameter-free closed forms for the Fisher-Bures and Kubo-Mori information matrices of an important continuous-variable family, with immediate applications to multiparameter estimation and quantum machine learning. The appendices reproduce the relevant machinery in detail, including Duhamel expansions, the high-peak tent density, symplectic evolution of quadratures, and Gaussian moment formulas, and no numerical fitting is involved. However, the central theorem currently contains algebraic prefactor errors and a parameterization ambiguity that affect the numerical content of the claimed formulas and the applicability of the Cramer-Rao bound.

major comments (4)
  1. [Appendix F and Theorem 1, Eqs. (F2), (F5), (49), (51)] There is a missing factor of 1/2 in the first (anticommutator) terms of both blocks. Proposition 3, Eq. (39), states I^FB_ij = (1/2)<{∂_i G, Ψ(∂_j G)}> - <∂_i G><∂_j G>. In the H-H block, ∂G/∂h_{k,l} = (1/4){x^c_k, x^c_l}, so substitution gives (1/2)(1/4)(1/4) = 1/32 for the first coefficient, not 1/16 as written in Eq. (F2). In the mu-mu block, Eq. (F5) omits the 1/2 prefactor entirely. Tracing the two calculations through Eqs. (F9)-(F17) and (F19)-(F28), the final theorem coefficients in Eqs. (49) and (51) should be 1 and 1/4 (under the paper's all-entries convention) rather than 2 and 1/2. Since the Kubo-Mori expressions are obtained by the same substitution q(t)->p(t), they inherit the same error. This is a load-bearing algebraic error in the proof of the paper's central result and must be corrected before the formulas can be used.
  2. [Section II and Theorem 1, Eqs. (47)-(51)] The parameterization in Eq. (47) treats all 2n x 2n entries h_{k,l} as independent coordinates, even though H is symmetric and the state depends only on the symmetric part of H. The antisymmetric directions h_{k,l} - h_{l,k} have zero derivative, so the resulting Fisher-Bures information matrix is singular and the Cramer-Rao bound (3), which requires inversion of I^FB, is not directly applicable. If the authors instead intend the usual symmetric parameterization with independent entries h_{k,l} for k <= l, then Eq. (45) needs to be modified for off-diagonal entries to (1/2){x^c_k, x^c_l}, and the coefficients in Theorem 1 change accordingly. The manuscript should state explicitly which parameterization is used and how the redundancy is handled, or restrict the information matrix to the symmetric submanifold.
  3. [Section VI, Theorem 3] Theorem 3 is stated as one of the three main results, but its proof is omitted with the remark that it is very similar to the proof of Theorem 2. Since the symmetric logarithmic derivative formulas are central to the claimed single-parameter estimation application, a complete proof or at least a detailed derivation showing how Proposition 4 combines with Eqs. (45)-(46) and Lemma 1 should be provided.
  4. [Appendix A, Eqs. (A1)-(A20)] The proof of Proposition 1 applies the Duhamel expansion, the spectral decomposition of G(theta), and the Fourier representation of the high-peak tent density to the unbounded quadratic operator G(theta). The argument assumes that the infinite sums and the t-integral can be interchanged with the trace without stating functional-analytic hypotheses. For the quadratic bosonic Hamiltonian in Eq. (44) this may be justifiable, but because Proposition 1 is the foundation for Theorems 1-3, a short justification or a statement of the regularity conditions under which Eq. (34) holds should be included.
minor comments (4)
  1. [Eq. (39)] There is an unmatched parenthesis in the expression "{∂_i G(theta)), Ψ_theta(∂_j G(theta))}"; it should read "{∂_i G(theta), Ψ_theta(∂_j G(theta))}".
  2. [Appendix F, Eqs. (F17), (F28)-(F29)] The proof first uses S(t) = e^{Omega H t} and then "redefines S(t) to be S(t)^T" to obtain the theorem's convention S(t) = e^{-H Omega t}. This is mathematically consistent because S(t)^T = e^{-H Omega t}, but reusing the same symbol is confusing. A distinct notation, such as T(t) = e^{-H Omega t}, would make the final formulas easier to verify.
  3. [Theorem 1, Kubo-Mori part] The statement that the Kubo-Mori elements are 'precisely the same' with q(t) replaced by p(t) should be checked again after correcting the Fisher-Bures proof. The Kubo-Mori formula (36) has a different operator ordering and the same 1/2 prefactor, so the simplification is not literally identical until all Gaussian expectations are evaluated.
  4. [Eq. (48)] The example single-mode parameter vector lists six entries for H, but a symmetric 2x2 matrix has only three independent parameters. If the redundant parameterization is kept, the singular nature of the information matrix should be stated explicitly; if the symmetric parameterization is intended, the example should list independent entries only.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 follows by direct substitution into previously established operator identities with proofs reproduced in the appendices.

full rationale

The paper's central results are obtained by fixing G(theta) = 1/2 (xhat - mu)^T H (xhat - mu) and inserting the derivatives (45)-(46) into Propositions 2-3. Propositions 1-4 are not imported as black boxes: Appendices A-D give self-contained proofs following Duhamel's formula and the stated Fourier identity. The only self-citations are [19] and [40], co-authored by one of the present authors, but the load-bearing content (the thermal-state derivative formula, Fisher-Bures/Kubo-Mori information expressions, and SLD) is rederived in these appendices rather than assumed. The use of [40, Lemma 12] for the Fourier transform of the high-peak tent density p(t) is an explicit, parameter-free mathematical identity stated in the paper (Eq. A11) and externally checkable; it is not a fitted value, an empirical input, or an assumption whose content equals the target theorem. No parameters are fitted to data, no prediction is constructed from its own target, and no uniqueness theorem is imported to force a choice. The derivation is therefore self-contained for circularity purposes; any algebraic or functional-analytic defect would be a correctness issue, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted and no new physical entities are introduced. The derivations rest on the exponential parameterization of faithful Gaussian states, the symplectic diagonalization of covariance matrices, Gaussian moment formulas, and an unproved convergence assumption for the operator Fourier representation of unbounded quadratic Hamiltonians.

assumptions (4)
  • standard math Williamson's theorem: every covariance matrix can be symplectically diagonalized as V = S(D xor D) S^T.
    Invoked in Section II A, Eq. (13), to define symplectic eigenvalues and to normalize the thermal state.
  • domain assumption A faithful bosonic Gaussian state admits the exponential representation rho(mu,H) = exp(-1/2 (xhat-mu)^T H (xhat-mu)) / Z with V = coth(i Omega H / 2) i Omega / 2.
    Stated in Section II A, Eqs. (14)-(16), cited to Refs. [35,42-45]. This parameterization is the whole setting of the paper.
  • ad hoc to paper The high-peak tent density p(t) satisfies the Fourier transform relation R dt p(t) e^{-i omega t} = tanh(omega/2)/(omega/2), and all t-integrals over the unbounded quadratic operator G(theta) converge and can be interchanged with traces.
    Used in Appendix A, Eq. (A11) and throughout Appendices B-G. For unbounded operators like G = (1/2)(xhat-mu)^T H (xhat-mu) on Fock space, the Duhamel expansion and the interchange of the t-integral with the trace are assumed to converge; this is the paper's main technical assumption.
  • domain assumption Gaussian moment identities: third-order quadrature moments vanish, and fourth-order moments obey the formula in Eq. (E11) of Ref. [57].
    Used in Appendix F to evaluate the Fisher-Bures matrix elements: third-order moments are set to zero (citing [46]) and fourth-order moments follow from the Wick-like formula in (F15) (citing [57]).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Information geometry of bosonic Gaussian thermal states." pith.science (2026). https://pith.science/paper/BR7VIR2A

@misc{pith2026241118268,
  author       = {Pith},
  title        = {Pith review of: Information geometry of bosonic Gaussian thermal states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BR7VIR2A}},
  note         = {Machine review of arXiv:2411.18268}
}
abstract

Bosonic Gaussian thermal states form a fundamental class of states in quantum information science. This paper explores the information geometry of these states, focusing on characterizing the distance between two nearby states and the geometry induced by a parameterization in terms of their mean vectors and Hamiltonian matrices. In particular, for the family of bosonic Gaussian thermal states, we derive expressions for their Fisher-Bures, Kubo-Mori, and $\alpha$-$z$ information matrices with respect to their mean vectors and Hamiltonian matrices. An important application of our formulas consists of fundamental limits on how well one can estimate these parameters. We additionally establish formulas for the derivatives and the symmetric logarithmic derivatives of bosonic Gaussian thermal states. The former could have applications in gradient descent algorithms for quantum machine learning when using bosonic Gaussian thermal states as an ansatz, and the latter in formulating optimal strategies for single parameter estimation of bosonic Gaussian thermal states. Finally, the expressions for the aforementioned information matrices could have additional applications in natural gradient descent algorithms when using bosonic Gaussian thermal states as an ansatz.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

64 extracted references · 37 canonical work pages

  1. [1]

    Proof of Equation (35) 11

    work page

  2. [2]

    Information geometry of bosonic Gaussian thermal states

    Proof of Equation (36) 12 C. Proof of Proposition 3 14 D. Proof of Proposition 4 15 E. Action of quadratic Hamiltonians on quadrature operators 15 F. Proof of Theorem 1 17 G. Proof of Theorem 2 20 I. INTRODUCTION A. Motivation Bosonic Gaussian states are foundational in quantum information science [1–4]. Not only do they generalize Gaussian probability de...

    work page Pith review arXiv 2024

  3. [3]

    formulas for the Fisher–Bures and Kubo–Mori in- formation matrices of bosonic Gaussian thermal states (Theorem 1),

    work page

  4. [4]

    formulas for the derivatives of bosonic Gaussian thermal states (Theorem 2),

    work page

  5. [5]

    Quantum-enabled super-resolution imaging

    formulas for the symmetric logarithmic derivatives of bosonic Gaussian thermal states (Theorem 3). Although there is already prior work on deriving the Fisher–Bures information matrices of bosonic Gaussian states (see Section I C), we believe our derivations are insightful, and we suspect that all of our results listed above will be useful in quantum mach...

    work page

  6. [6]

    X.-B. Wang, T. Hiroshima, A. Tomita, and M. Hayashi, Quantum information with Gaussian states, Physics Re- ports 448, 1 (2007)

    work page 2007

  7. [7]

    Adesso, S

    G. Adesso, S. Ragy, and A. R. Lee, Continuous vari- able quantum information: Gaussian states and beyond, Open Systems & Information Dynamics 21, 1440001 (2014)

    2014

  8. [8]

    Serafini, Quantum Continuous Variables: A Primer of Theoretical Methods(CRC Press, 2017)

    A. Serafini, Quantum Continuous Variables: A Primer of Theoretical Methods(CRC Press, 2017)

    2017

Show all 64 references
  1. [9]

    Weedbrook, S

    C. Weedbrook, S. Pirandola, R. Garc ´ ıa-Patr´ on, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaus- sian quantum information, Reviews of Modern Physics 84, 621 (2012)

    2012

  2. [10]

    N. Lee, H. Benichi, Y. Takeno, S. Takeda, J. Webb, E. Huntington, and A. Furusawa, Teleportation of non- classical wave packets of light, Science 332, 330 (2011)

    2011

  3. [11]

    J. Aasi, J. Abadie, B. P. Abbott, et al., Enhanced sen- sitivity of the LIGO gravitational wave detector by us- ing squeezed states of light, Nature Photonics 7, 613–619 (2013)

    2013

  4. [12]

    B. P. Abbott, R. Abbott, T. D. Abbott, et al., Obser- vation of gravitational waves from a binary black hole merger, Physical Review Letters 116, 061102 (2016)

    2016

  5. [13]

    L. S. Madsen, F. Laudenbach, M. F. Askarani, F. Rortais, T. Vincent, J. F. F. Bulmer, F. M. Miatto, L. Neuhaus, L. G. Helt, M. J. Collins, A. E. Lita, T. Gerrits, S. W. Nam, V. D. Vaidya, M. Menotti, I. Dhand, Z. Vernon, N. Quesada, and J. Lavoie, Quantum computational ad- van...

    2022

  6. [14]

    L. Lami, B. Regula, X. Wang, R. Nichols, A. Winter, and G. Adesso, Gaussian quantum resource theories, Physical 9 Review A 98, 022335 (2018)

    2018

  7. [15]

    Bengtsson and K

    I. Bengtsson and K. Zyczkowski, Geometry of Quan- tum States: An Introduction to Quantum Entanglement (Cambridge University Press, 2006)

    2006

  8. [16]

    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

  9. [17]

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

    2020

  10. [18]

    Jarzyna and J

    M. Jarzyna and J. Kolodynski, Geometric approach to quantum statistical inference, IEEE Journal on Selected Areas in Information Theory 1, 367 (2020)

    2020

  11. [19]

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

    2021

  12. [20]

    F. M. Sbahi, A. J. Martinez, S. Patel, D. Saberi, J. H. Yoo, G. Roeder, and G. Verdon, Provably efficient vari- ational generative modeling of quantum many-body sys- tems via quantum-probabilistic information geometry (2022), arXiv:2206.04663v1 [quant-ph]

    2022 arXiv

  13. [21]

    Scandi, P

    M. Scandi, P. Abiuso, J. Surace, and D. D. Santis, Quan- tum Fisher information and its dynamical nature (2024), arXiv:2304.14984 [quant-ph]

    2024 arXiv

  14. [22]

    N. N. ˇCencov, Algebraic foundation of mathematical statistics, Series Statistics 9, 267 (1978)

    1978

  15. [23]

    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

  16. [24]

    Patel and M

    D. Patel and M. M. Wilde, Natural gradient and parame- ter estimation for quantum Boltzmann machines (2024), arXiv:2410.24058v1 [quant-ph]

    2024

  17. [25]

    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

  18. [26]

    Carollo, B

    A. Carollo, B. Spagnolo, A. A. Dubkov, and D. Valenti, Erratum: On quantumness in multi-parameter quan- tum estimation (2019 J. Stat. Mech. 094010), Journal of Statistical Mechanics: Theory and Experiment 2020, 029902 (2020)

    2020

  19. [27]

    Tsang, F

    M. Tsang, F. Albarelli, and A. Datta, Quantum semi- parametric estimation, Physical Review X 10, 031023 (2020)

    2020

  20. [28]

    F. A. Mele, A. A. Mele, L. Bittel, J. Eisert, V. Giovan- netti, L. Lami, L. Leone, and S. F. E. Oliviero, Learning quantum states of continuous variable systems (2024), arXiv:2405.01431 [quant-ph]

    2024

  21. [29]

    Bittel, F

    L. Bittel, F. A. Mele, A. A. Mele, S. Tirone, and L. Lami, Optimal estimates of trace distance between bosonic Gaussian states and applications to learning (2024), arXiv:2411.02368v1 [quant-ph]

    2024 arXiv

  22. [30]

    Fanizza, C

    M. Fanizza, C. Rouz´ e, and D. S. Fran¸ ca, Efficient Hamil- tonian, structure and trace distance learning of Gaussian states (2024), arXiv:2411.03163 [quant-ph]

    2024 arXiv

  23. [31]

    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, Physical Review Letters 133, 040802 (2024)

    2024

  24. [32]

    Abiuso, P

    P. Abiuso, P. Sekatski, J. Calsamiglia, and M. Perarnau- Llobet, Fundamental limits of metrology at thermal equi- librium (2024), arXiv:2402.06582 [quant-ph]

    2024

  25. [33]

    Anshu, S

    A. Anshu, S. Arunachalam, T. Kuwahara, and M. Soleimanifar, Sample-efficient learning of interacting quantum systems, Nature Physics 17, 931 (2021)

    2021

  26. [34]

    S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Physical Review Letters 72, 3439 (1994)

    1994

  27. [35]

    Biamonte, P

    J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Quantum machine learning, Na- ture 549, 195 (2017)

    2017

  28. [36]

    Amari, Natural gradient works efficiently in learn- ing, Neural Computation 10, 251 (1998)

    S.-I. Amari, Natural gradient works efficiently in learn- ing, Neural Computation 10, 251 (1998)

    1998

  29. [37]

    Stokes, J

    J. Stokes, J. Izaac, N. Killoran, and G. Carleo, Quantum natural gradient, Quantum 4, 269 (2020)

    2020

  30. [38]

    Monras, Phase space formalism for quantum estima- tion of Gaussian states (2013), arXiv:1303.3682 [quant- ph]

    A. Monras, Phase space formalism for quantum estima- tion of Gaussian states (2013), arXiv:1303.3682 [quant- ph]

    2013 arXiv

  31. [39]

    Gao and H

    Y. Gao and H. Lee, Bounds on quantum multiple- parameter estimation with Gaussian state, The European Physical Journal D 68, 347 (2014)

    2014

  32. [40]

    Banchi, S

    L. Banchi, S. L. Braunstein, and S. Pirandola, Quantum fidelity for arbitrary Gaussian states, Physical Review Letters 115, 260501 (2015)

    2015

  33. [41]

    Safranek, Estimation of Gaussian quantum states, Journal of Physics A: Mathematical and Theoretical 52, 035304 (2018)

    D. Safranek, Estimation of Gaussian quantum states, Journal of Physics A: Mathematical and Theoretical 52, 035304 (2018)

    2018

  34. [42]

    Nichols, P

    R. Nichols, P. Liuzzo-Scorpo, P. A. Knott, and G. Adesso, Multiparameter Gaussian quantum metrology, Physical Review A 98, 012114 (2018)

    2018

  35. [43]

    Huang, B

    Z. Huang, B. Q. Baragiola, N. C. Menicucci, and M. M. Wilde, Limited quantum advantage for stellar interferom- etry via continuous-variable teleportation, Physical Re- view A 109, 052434 (2024)

    2024

  36. [44]

    Jiang, Quantum Fisher information for states in ex- ponential form, Physical Review A 89, 032128 (2014)

    Z. Jiang, Quantum Fisher information for states in ex- ponential form, Physical Review A 89, 032128 (2014)

    2014

  37. [45]

    Patel, D

    D. Patel, D. Koch, S. Patel, and M. M. Wilde, Quan- tum Boltzmann machine learning of ground-state ener- gies (2024), arXiv:2410.12935v2 [quant-ph]

    2024

  38. [46]

    Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, American Journal of Mathematics 58, 141 (1936)

    J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, American Journal of Mathematics 58, 141 (1936)

    1936

  39. [47]

    Chen, Gaussian relative entropy of entanglement, Physical Review A 71, 062320 (2005), arXiv:quant- ph/0402109

    X.-Y. Chen, Gaussian relative entropy of entanglement, Physical Review A 71, 062320 (2005), arXiv:quant- ph/0402109

    2005

  40. [48]

    Krueger, Quantum Information Theory with Gaussian Systems , Ph.D

    O. Krueger, Quantum Information Theory with Gaussian Systems , Ph.D. thesis, Technische Universit¨ at Braunschweig (2006), available at https://publikationsserver.tu-braunschweig. de/receive/dbbs_mods_00020741

    2006

  41. [49]

    A. S. Holevo, The entropy gain of infinite-dimensional quantum evolutions, Doklady Mathematics 82, 730 (2010), arXiv:1003.5765

    2010 arXiv

  42. [50]

    A. S. Holevo, Quantum Systems, Channels, Informa- tion: A Mathematical Introduction, 2nd ed. (Walter de Gruyter, 2019)

    2019

  43. [51]

    M. M. Wilde, M. Tomamichel, S. Lloyd, and M. Berta, Gaussian hypothesis testing and quantum illumination, Physical Review Letters 119, 120501 (2017)

    2017

  44. [52]

    D. Bures, An extension of Kakutani’s theorem on infi- nite product measures to the tensor product of semifinite W *-algebras, Transactions of the American Mathemati- cal Society 135, 199 (1969)

    1969

  45. [53]

    transition probability

    A. Uhlmann, The “transition probability” in the state space of a *-algebra, Reports on Mathematical Physics 9, 273 (1976)

    1976

  46. [54]

    Umegaki, Conditional expectation in an operator alge- bra, IV (entropy and information), Kodai Mathematical 10 Journal 14, 59 (1962)

    H. Umegaki, Conditional expectation in an operator alge- bra, IV (entropy and information), Kodai Mathematical 10 Journal 14, 59 (1962)

    1962

  47. [55]

    Lindblad, Completely positive maps and entropy in- equalities, Communications in Mathematical Physics 40, 147 (1975)

    G. Lindblad, Completely positive maps and entropy in- equalities, Communications in Mathematical Physics 40, 147 (1975)

    1975

  48. [56]

    Hiai and D

    F. Hiai and D. Petz, The proper formula for relative en- tropy and its asymptotics in quantum probability, Com- munications in Mathematical Physics 143, 99 (1991)

    1991

  49. [57]

    Nagaoka and T

    H. Nagaoka and T. Ogawa, Strong converse and Stein’s lemma in quantum hypothesis testing, IEEE Transac- tions on Information Theory 46, 2428 (2000)

    2000

  50. [58]

    M. B. Hastings, Quantum belief propagation: An algo- rithm for thermal quantum systems, Physical Review B 76, 201102 (2007)

    2007

  51. [59]

    Anshu, S

    A. Anshu, S. Arunachalam, T. Kuwahara, and M. Soleimanifar, Sample-efficient learning of quantum many-body systems (2020), arXiv:2004.07266v1 [quant- ph]

    2020 arXiv

  52. [60]

    Coopmans and M

    L. Coopmans and M. Benedetti, On the sample complex- ity of quantum Boltzmann machine learning, Communi- cations Physics 7, 274 (2024)

    2024

  53. [61]

    NISO, Credit – contributor roles taxonomy, https:// credit.niso.org/, Accessed 2024-10-28

    2024

  54. [62]

    Monras and F

    A. Monras and F. Illuminati, Information geometry of Gaussian channels, Physical Review A81, 062326 (2010). Appendix A: Proof of Proposition 1 Here we follow the proof of [40, Lemma 10] quite closely, with the main change being that Gj gets replaced by ∂jG(θ). We provide a det...

    2010

  55. [63]

    We provide a detailed proof for completeness

    Proof of Equation (35) Here we follow the proof of [19, Theorem 1] quite closely, with the main change being that Gi gets replaced by ∂iG(θ). We provide a detailed proof for completeness. Plugging (34) into (24), we find that we should consider the following term: ⟨k|∂iρ(θ)|ℓ⟩...

  56. [64]

    We provide a detailed proof for completeness

    Proof of Equation (36) Here we follow the proof of [19, Theorem 2] quite closely, with the main change being that Gi gets replaced by ∂iG(θ). We provide a detailed proof for completeness. Consider that I KM ij (θ) = X k,ℓ ln λk − ln λℓ λk − λℓ ⟨k|∂iρ(θ)|ℓ⟩ ⟨ℓ|∂jρ(θ)|k⟩ (B13) 1...

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.