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Analytically Continuing the Randomized Measurement Toolbox

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Stabilized analytic continuation turns a few noisy Rényi entropies into a reliable estimate of the von Neumann entropy in quantum simulation experiments.

desk verdict A useful, honest toolbox paper: SAC gives a practical route from a few noisy Rényi entropies to S_vN, but "reliable" is too strong a word—finite data can't pin down S_vN without an unvalidated prior. read the letter →

arxiv 2511.02912 v5 pith:HFXDQAUB submitted 2025-11-04 quant-ph cond-mat.quant-gas

Akash Vijay , Ayush Raj , Jonah Kudler-Flam , Benoît Vermersch , Andreas Elben , Laimei Nie This is my paper · ORCID
classification quant-phcond-mat.quant-gas PACS 03.67.-a03.65.Ud
keywords vonNeumannentropyRényientropiesanalyticcontinuationrandomizedmeasurementsclassicalshadowsentanglementtrapped-ionsimulatorquantumsimulation
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 aims to make the von Neumann entanglement entropy measurable in practice. Randomized measurement protocols directly access Rényi entropies of integer order, and the von Neumann entropy is the k→1 limit of the same analytic family, but the continuation is usually unstable under noise. The paper claims that stabilized analytic continuation (SAC) solves this: by adding a pole at z=1, mapping the analyticity strip to the unit disk, and selecting the continuation of smallest boundary norm, the noise is tamed. Benchmarks on simulated quench data and on trapped-ion experimental data support this, and the same framework extends to other non-polynomial diagnostics such as logarithmic negativity and Rényi relative entropies. If correct, the method closes a practical gap between what randomized measurements can measure and the entanglement quantifier many-body theory most wants.

What carries the argument

SAC (stabilized analytic continuation) is the central mechanism: given analytic function values at interior points of the unit disk, it selects the holomorphic function with the smallest L² norm of the angular derivative of the imaginary part on the boundary (Eq. 4). The paper's adaptation first defines a discrepancy function with an artificial pole at the target point z=1, then maps the semi-infinite strip Re z>1, |Im z|<ε to the disk via ξ=cosh((z−1)/ε + iπ/2), w=(ξ−η i)/(ξ+η i), so that the von Neumann entropy becomes a residue whose cancellation is measured by the norm. The matrix A_ij (Eq. 14) encodes the geometry of the data locations and turns the minimization into a linear algebraic

What would settle it

Take a density matrix whose spectrum is known (for example, a small subsystem of the quenched trapped-ion state), compute the first branch points of S_z(ρ)= (1/(1−z)) log₂ Tr ρ^z, and check whether any branch point falls within the strip |Im z|<ε used by SAC. If one does, feed the same Rényi inputs to SAC and to an exact-diagonalization estimate of S_vN; a bias in the SAC estimate larger than its reported uncertainty would show that the pole-cancellation argument fails when the analyticity domain is narrower than assumed.

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Extended reading notes

Core claim

The central claim is that the von Neumann entropy S_vN(ρ) = -Tr(ρ log₂ ρ) can be estimated from a finite set of noisy Rényi entropies S_k(ρ), k=2,...,k_max, by analytic continuation. The paper constructs a discrepancy function D_α(z) = S_z(ρ)/(z−1) − α/(z−1) whose residue at z=1 is S_vN(ρ) − α. After a two-step conformal map sends the analyticity strip into the unit disk, the true value of α cancels an artificial pole on the unit circle, causing the minimum of an L² boundary norm to drop sharply; minimizing that norm over α therefore recovers S_vN. In the noiseless case this yields a closed-form expression (Eq. 5); with noise, the data are treated as a covariance ellipsoid and the same norm

Load-bearing premise

The Rényi function S_z(ρ) is assumed to stay analytic on a semi-infinite strip Re z>1, |Im z|<ε that is wide enough for the conformal map, a property the paper supports with a conservative perturbative bound and numerical tests rather than a proof.

Editorial extensions

If this is right

  • The von Neumann entanglement entropy becomes extractable from randomized measurement data using only Rényi orders 2 through 6, avoiding full state tomography.
  • The same SAC routine applies to any non-polynomial function of the density matrix accessible through an analytic continuation, including logarithmic negativity and Rényi relative entropies.
  • The noise-resilience of the method means existing randomized measurement datasets, even those with finite and correlated statistical errors, can be reprocessed to obtain S_vN.
  • An explicit noiseless estimator (Eq. 5) is available, so the method is computationally cheap and can be used as a standard post-processing tool for quantum simulator experiments.
  • Benchmarks indicate SAC is more accurate than polynomial extrapolation, which suggests the continuation problem for entanglement entropies is tractable rather than fundamentally ill-posed.

Reading between the lines

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

  • A direct extension the paper leaves implicit: because SAC needs only trace moments Tr ρ^k, it can run as a post-processing add-on on classical shadow data already collected for other observables, making S_vN nearly free in existing experiments.
  • The same pole-placement trick should transfer to other replica-trick quantities that are limits of integer-order data; logarithmic negativity and Rényi relative entropies are natural targets, and the closed-form Eq. (5) can likely be rederived for each.
  • A stress test worth running is k_max=2 or 3: the noiseless formula subtracts the first data point and sums from k=3, so the method's practical boundary lies in how few orders can still stabilize the continuation.
  • The analyticity-strip assumption is the main risk; computing the actual branch points of S_z(ρ) for the simulated states would either validate the strip width ε used by SAC or reveal where the method must be modified.
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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

4 major / 4 minor

Summary. The paper introduces a stabilized analytic continuation (SAC) framework for estimating the von Neumann entropy S_vN from a finite set of integer Rényi entropies S_2,...,S_{kmax} obtained via randomized measurements. The construction builds a discrepancy function D_α(z)=(S_z−α)/(z−1), maps a presumed analyticity strip to the unit disk, and selects α by minimizing an L2 boundary norm. For noiseless data a closed-form estimator is given (Eq. 5); for noisy data, a χ²-constrained minimization is proposed. The method is benchmarked on simulated quench dynamics of a 10-qubit Néel state and applied to an existing trapped-ion dataset, with comparisons to Chebyshev and least-squares extrapolation. The paper claims noise robustness and suggests extensions to other nonlinear spectral functions such as logarithmic negativity and Rényi relative entropies.

Significance. If the central claim is validated, SAC would address a practical gap: randomized measurement experiments routinely estimate integer Rényi entropies, while the von Neumann entropy is the entanglement measure most often used in many-body physics and can behave differently from any fixed Rényi order. The paper's concrete contributions include the explicit closed-form estimator, the extension to correlated noise, and an open-source implementation. The numerical and experimental demonstrations are welcome. However, the central claim of reliability is not yet established. Equation (5) is a fixed linear functional of the input Rényi values, so it cannot distinguish states that share S_2,...,S_{kmax} but differ in S_vN. The variational argument does not prove that the norm-minimizing α equals S_vN, and the free parameters are tuned on the same simulation data used to demonstrate accuracy. These issues are load-bearing for the paper's main claim and require substantial additional work.

major comments (4)
  1. [Stabilized analytic continuation, Eq. (5)] Equation (5) is a fixed linear functional of S_2,...,S_{kmax}: the matrix A and the terms 1/(i−1)−1 depend only on the chosen conformal map and data locations, not on the state. Consequently, any two density matrices with identical low-order Rényi entropies receive the same SAC estimate, no matter how different their S_vN. The numerical benchmarks (Fig. 2) and the experimental analysis (Fig. 3) use parameters ε, η, χ²0, w0 chosen by benchmarking on the same simulation data, so the reported accuracy is partly in-sample. To support the word 'reliably' in the abstract and introduction, the paper needs either a finite-data error bound under the stated analyticity assumptions or an explicit stress test in which the low-order Rényi data are held fixed while S_vN varies, showing that the method's error is controlled. As written, the estimate is an extrapolation of the input Rényi function and t
  2. [Stabilized analytic continuation, Steps 1–2 and Eq. (4)] The pole-cancellation argument is not sufficient for the variational quantity actually computed. If α≠S_vN, the true discrepancy function D'_α has a pole on |w|=1, and its boundary norm (4) would be singular. However, Step 1 does not minimize over functions containing that boundary singularity; it minimizes over analytic functions Y_α holomorphic in the disk with finite norm that interpolate the data. This feasible set is nonempty for every α (e.g. polynomial interpolation), so δ(α) is finite for all α. The fact that the true function has an infinite norm at α≠S_vN does not imply that the norm-minimizing continuation does. No theorem or numerical experiment is provided to show that the quadratic δ(α) is minimized at α=S_vN rather than at another value. This is the central methodological gap. I would ask for a proof under the stated analyticity and norm assumptions, or at least a syntheti
  3. [After Eq. (3) and Eq. (6): analyticity domain and parameter selection] The paper treats ε, η, χ²0, and w0 as free parameters. The text says ε and η are chosen 'based on benchmarks on the simulation data' (after Eq. 3), χ²0 is 'typically chosen O(k_max)' (after Eq. 6), and w0 is a variational parameter in the noisy case. No sensitivity analysis is reported. Because the same simulation data are used both to set these parameters and to measure accuracy, the comparison against Chebyshev and least-squares in Fig. 2 is not fully out-of-sample. The assumed analyticity strip is supported only by a conservative perturbative bound and by 'direct numerical tests' that are not shown; for the 10-qubit quench trajectory, zeros of Tr ρ^z could enter the assumed strip. Since the exact density matrix is available in the simulation, a check of the strip width along the actual evolution would directly address this concern. Please provide a systematic parameter scan or a princ
  4. [Application to Trapped-ion Quantum Simulator, Fig. 3] The experimental demonstration does not include a valid uncertainty estimate. The authors state that bootstrapping the 500 density matrices gives unrealistically small error bars, splitting gives grouping-dependent results, and a robust error bar requires larger N_u. As a result, the trapped-ion S_vN points in Fig. 3 are presented without error bars, despite the paper's emphasis on noise robustness. This is a significant limitation of the experimental demonstration and should be stated clearly in the main text. Ideally, the analysis should be repeated with N_u=1000 or validated on synthetic data with the same covariance structure, so that the error-bar procedure can be checked against a known answer.
minor comments (4)
  1. [Introduction, p. 2] The sentence 'S_vN = lim_{k→1+} S_k, a consequence of Carlson’s theorem' is imprecise. The limit follows from the continuity/removable-singularity property of S_z at z=1; Carlson's theorem is about uniqueness of an analytic continuation from integer values under boundedness conditions. Please rephrase.
  2. [Eq. (4) and footnote [45]] The paper chooses one of four possible L2 norms, motivated by the pole manifesting in the imaginary part. Since this choice is part of the regularization, a short discussion or numerical comparison of the four norms would help the reader assess how much the final estimate depends on this choice.
  3. [Randomized measurements, around Eq. (7)] The notation is slightly compressed: p_k is used both as the quantum expectation Tr(ρ^k) and as the U-statistic estimator \p_k. Please distinguish the estimator explicitly (e.g. \hat p_k) to avoid confusion.
  4. [Throughout] Some typographical issues in the text: 'N´ eel' should be 'Néel' in several places; 'analytical continuation' is used where 'analytic continuation' is standard. These are minor and do not affect the science.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: SAC is a genuine extrapolation from finite Rényi data, not an identity with its inputs.

full rationale

The paper's central claim is that the von Neumann entropy S_vN can be estimated from a finite set of integer Rényi entropies via stabilized analytic continuation (SAC). This is an inference problem, not a definitional identity. The target S_vN is defined independently by S(ρ) = −Tr(ρ log_2 ρ), and the Rényi inputs S_2,...,S_kmax are measured separately. The discrepancy function D_α(z) = (S_z − α)/(z − 1) is constructed so that the pole at z = 1 cancels when α = S_vN, but the actual estimation procedure minimizes a norm over α and then selects the minimizer. Equation (5) is a derived estimator, not a restatement of the input data; it is a linear functional of the input Rényi values, which is exactly what any finite-data estimator must be. The fact that two states sharing S_2...S_kmax can have different S_vN is an ill-posedness limitation of all such finite-continuation methods, not evidence that the estimate is circular. The free parameters ε and η are chosen by benchmarking on simulation data, which raises an in-sample-validation concern, but this is a methodological robustness issue, not a reduction of the predicted quantity to the fitted parameters. The experimental application to trapped-ion data is an out-of-sample test. Self-citations to previous randomized-measurement work (e.g., Refs. [17,24,30,48]) are to established protocols and are not used as a load-bearing uniqueness theorem or to forbid alternative methods. The core SAC framework is explicitly credited to prior external work by Ciulli and Spearman [38–42]. No step in the derivation equates the estimate with the input by construction, and no fitted parameter is renamed as a prediction.

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

The central claim depends on the chosen analyticity domain, the selected norm, and the free parameters (ε, η, χ²0, w0). These are either assumed, chosen ad hoc, or tuned on simulation benchmarks, and are not derived from first principles.

free parameters (4)
  • ε (conformal strip half-width) = not specified; chosen based on benchmarks
    In 'Stabilized analytic continuation', after Eq. (3): 'In practice, η and ε are chosen based on benchmarks on the simulation data'. Determines the analyticity strip mapped to the unit disk; changes data locations w_i, the A matrix, and the final estimate.
  • η (conformal map free parameter) = not specified; chosen based on benchmarks
    Same sentence: 'η is a free parameter... chosen based on benchmarks'.
  • χ²0 (chi-squared cutoff) = O(k_max) typically
    In the noisy case, the constraint χ²(y;α) ≤ χ²0; the value is 'typically chosen to be O(k_max)' — a hand-chosen threshold controlling how much noise is allowed in the data.
  • w0 (subtraction point, noisy case) = variational parameter
    In the noisy case, 'we treat the subtraction point as a variational parameter as well' — increases flexibility of the fit and introduces an extra degree of freedom.
assumptions (4)
  • standard math Carlson's theorem: an analytic function vanishing on the positive integers is identically zero
    Used to justify S_vN = lim_{k→1+} S_k from integer Rényi entropies (Introduction).
  • domain assumption S_z(ρ) is analytic on a semi-infinite strip Re z > 1, |Im z| < c/log(d)
    Stated in the method section; supported only by a perturbative argument and numerical tests on random density matrices, not a proof.
  • ad hoc to paper The L2 pseudo-norm (4) is the appropriate measure of boundary structure
    Choice among four possible L2 norms; motivated by pole manifestation in the imaginary part, but not justified uniquely.
  • domain assumption Minimizing the norm over α selects α = S_vN (pole cancellation)
    Central to the method; assumes the discrepancy function has no other singularities near the boundary beyond the artificial pole.

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Cite this review

Pith. "Pith review of Analytically Continuing the Randomized Measurement Toolbox." pith.science (2026). https://pith.science/paper/HFXDQAUB

@misc{pith2026251102912,
  author       = {Pith},
  title        = {Pith review of: Analytically Continuing the Randomized Measurement Toolbox},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HFXDQAUB}},
  note         = {Machine review of arXiv:2511.02912}
}
read the original abstract

We develop a framework for extracting non-polynomial analytic functions of density matrices in randomized measurement experiments by a method of analytical continuation. A central advantage of this approach, dubbed stabilized analytic continuation (SAC), is its robustness to statistical noise arising from finite repetitions of a quantum experiment, making it well-suited to realistic quantum hardware. As a demonstration, we use SAC to estimate the von Neumann entanglement entropy of a numerically simulated quenched N\'eel state from R\'enyi entropies estimated via the randomized measurement protocol. We then apply the method to experimental R\'enyi data from a trapped-ion quantum simulator experiment, extracting subsystem von Neumann entropies at different evolution times. Finally, we briefly note that the SAC framework is readily generalizable to obtain other nonlinear diagnostics, such as the logarithmic negativity and R\'enyi relative entropies.

Figures

Figures reproduced from arXiv: 2511.02912 by the authors.

Figure 1
Figure 1. SAC and traditional fitting methods [30, 31] esti [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Numerical benchmarks using simulated data for a 10-qubit N´eel state quenched under Hamiltonian (8) with deco [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. (a)(b): SvN as a function of time t and subsys￾tem size L obtained via SAC in a trapped-ion experiment with ten 40Ca+ ions, initialized in an approximate N´eel state and evolved under Hamiltonian (8). For comparison, SvN obtained via SAC using simulation data, as well as R´enyi data from simulated randomized measurements, are plotted in (c)(d). In (a)(c), L = 5. In (b)(d), t = 5ms. sure entropies in many platforms, … view at source ↗

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Reference graph

Works this paper leans on

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

  1. [1]

    This minimization can be recast into a linear matrix optimization problem by leveraging the analyticity ofY α(w) [38]

    For fixedα, we search for the analytic function Yα(w) which takes the desired values{Y α(wi) = D′ α(wi) =d i(α)−d 2(α)}at the points{w i}and minimizes the norm||Y α||. This minimization can be recast into a linear matrix optimization problem by leveraging the analyticity ofY α(w) [38]. The minimal norm δ(α)≡min Yα(wi)=D′α(wi) ||Yα|| provides a measure of ...

  2. [2]

    The real half-line [1,∞)∈R z is mapped to the interval [−1,1]∈R w

  3. [3]

    To construct such a map, one must first identify the do- main of analyticity of the R´ enyi functionS z(ρ) in the complexzplane

    The pointz= 1 is mapped tow=−1. To construct such a map, one must first identify the do- main of analyticity of the R´ enyi functionS z(ρ) in the complexzplane. In general, this domain depends both on the dimension of the Hilbert space as well as the spec- trum ofρ. Using a perturbative argument, one can show thatS z(ρ) remains analytic on a semi-infinite...

  4. [4]

    This is where we expect the pole atw=−1 cancels out, and the minimal valueα min serves as our best estimate forS vN (ρ)

    Next we minimizeδ(α) over all possible values of α. This is where we expect the pole atw=−1 cancels out, and the minimal valueα min serves as our best estimate forS vN (ρ). Due to the absence of noise, both of these minimization steps can be performed exactly to obtain the following closed form expression for the von Neumann estimate: Sest vN = Pkmax i,j=...

  5. [5]

    Preskill, Quantum Computing in the NISQ era and beyond, Quantum2, 79 (2018)

    J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum2, 79 (2018)

    2018

  6. [6]

    Altman, K

    E. Altman, K. R. Brown, G. Carleo, L. D. Carr, E. Dem- ler, C. Chin, B. DeMarco, S. E. Economou, M. A. Eriks- son, K.-M. C. Fu, M. Greiner, K. R. Hazzard, R. G. Hulet, A. J. Koll´ ar, B. L. Lev, M. D. Lukin, R. Ma, X. Mi, S. Misra, C. Monroe, K. Murch, Z. Nazario, K.-K. Ni, A. C. Potter, P. Roushan, M. Saffman, M. Schleier- Smith, I. Siddiqi, R. Simmonds,...

    2021

  7. [7]

    Horodecki, P

    R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Reviews of Mod- ern Physics81, 865 (2009)

    2009

  8. [8]

    Kitaev and J

    A. Kitaev and J. Preskill, Topological entanglement en- tropy, Phys. Rev. Lett.96, 110404 (2006)

    2006

Show all 67 references
  1. [9]

    Levin and X.-G

    M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett.96, 110405 (2006)

    2006

  2. [10]

    Sekino and L

    Y. Sekino and L. Susskind, Fast scramblers, Journal of High Energy Physics2008, 065–065 (2008)

    2008

  3. [11]

    J. R. Garrison and T. Grover, Does a single eigenstate encode the full hamiltonian?, Phys. Rev. X8, 021026 (2018)

    2018

  4. [12]

    J. M. Deutsch, Eigenstate thermalization hypothesis, Re- ports on Progress in Physics81, 082001 (2018)

    2018

  5. [13]

    A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Quantum ther- malization through entanglement in an isolated many- body system, Science353, 794–800 (2016)

    2016

  6. [14]

    Nahum, J

    A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum entanglement growth under random unitary dynamics, Phys. Rev. X7, 031016 (2017)

    2017

  7. [15]

    Vidal, J

    G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Entan- glement in quantum critical phenomena, Phys. Rev. Lett. 90, 227902 (2003)

    2003

  8. [16]

    T. J. Osborne and M. A. Nielsen, Entanglement in a sim- ple quantum phase transition, Phys. Rev. A66, 032110 (2002)

    2002

  9. [17]

    Ryu and T

    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from the anti–de sitter space/conformal field theory correspondence, Physi- cal Review Letters96, 10.1103/physrevlett.96.181602 (2006)

    2006 doi

  10. [18]

    Ryu and T

    S. Ryu and T. Takayanagi, Aspects of holographic entan- glement entropy, Journal of High Energy Physics2006, 045–045 (2006)

    2006

  11. [19]

    J. Haah, A. W. Harrow, Z. Ji, X. Wu, and N. Yu, Sample- optimal tomography of quantum states, IEEE Transac- tions on Information Theory63, 5628 (2017)

    2017

  12. [20]

    O’Donnell and J

    R. O’Donnell and J. Wright, Efficient quantum tomogra- phy, inSTOC’16—Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing(ACM, New York, 2016) pp. 899–912

    2016

  13. [21]

    Elben, B

    A. Elben, B. Vermersch, C. F. Roos, and P. Zoller, Sta- tistical correlations between locally randomized measure- ments: A toolbox for probing entanglement in many- 6 body quantum states, Physical Review A99, 052323 (2019)

    2019

  14. [22]

    Huang, R

    H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measure- ments, Nature Physics16, 1050 (2020)

    2020

  15. [23]

    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 Physics5, 9 (2023)

    2023

  16. [24]

    Vitale, A

    V. Vitale, A. Rath, P. Jurcevic, A. Elben, C. Branciard, and B. Vermersch, Robust estimation of the quantum fisher information on a quantum processor, PRX Quan- tum5, 030338 (2024)

    2024

  17. [25]

    Islam, R

    R. Islam, R. Ma, P. M. Preiss, M. Eric Tai, A. Lukin, M. Rispoli, and M. Greiner, Measuring entanglement en- tropy in a quantum many-body system, Nature528, 77 (2015)

    2015

  18. [26]

    A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Quantum ther- malization through entanglement in an isolated many- body system, Science353, 794 (2016)

    2016

  19. [27]

    Elben, B

    A. Elben, B. Vermersch, M. Dalmonte, J. I. Cirac, and P. Zoller, R´ enyi entropies from random quenches in atomic hubbard and spin models, Physical review letters 120, 050406 (2018)

    2018

  20. [28]

    Elben, R

    A. Elben, R. Kueng, H.-Y. Huang, R. van Bijnen, C. Kokail, M. Dalmonte, P. Calabrese, B. Kraus, J. Preskill, P. Zoller,et al., Mixed-state entanglement from local randomized measurements, Physical Review Letters125, 200501 (2020)

    2020

  21. [29]

    Brydges, A

    T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, and C. F. Roos, Probing r´ enyi entanglement entropy via random- ized measurements, Science364, 260 (2019)

    2019

  22. [30]

    Rakovszky, F

    T. Rakovszky, F. Pollmann, and C. W. von Keyserlingk, Sub-ballistic growth of r´ enyi entropies due to diffusion, Physical Review Letter122, 250602 (2019)

    2019

  23. [31]

    J. Wang, Z. Liu, Z. Yan, and C. Wu, Singularity and universality from von neumann to r\’enyi entanglement entropy and disorder operator in motzkin chains, arXiv preprint arXiv:2501.17368 (2025)

    2025 arXiv

  24. [32]

    Bertini, K

    B. Bertini, K. Klobas, V. Alba, G. Lagnese, and P. Cal- abrese, Growth of r´ enyi entropies in interacting inte- grable models and the breakdown of the quasiparticle picture, Physical Review X12, 031016 (2022)

    2022

  25. [33]

    C. A. Ag´ on, H. Casini, and P. J. Martinez, R´ enyi en- tropies in the n→0 limit and entanglement temperatures, Physical Review D108, 105009 (2023)

    2023

  26. [34]

    Vermersch, A

    B. Vermersch, A. Rath, B. Sundar, C. Branciard, J. Preskill, and A. Elben, Enhanced estimation of quan- tum properties with common randomized measurements, PRX Quantum5, 010352 (2024)

    2024

  27. [35]

    Mason and D

    J. Mason and D. Handscomb,Chebyshev Polynomials (1st ed.)(Chapman and Hall/CRC, 2002)

    2002

  28. [36]

    G. H. Hardy, On two theorems of f. carlson and s. wigert, Acta mathematica42, 327 (1920)

    1920

  29. [37]

    J. E. Gubernatis, M. Jarrell, R. N. Silver, and D. S. Sivia, Quantum monte carlo simulations and maximum en- tropy: Dynamics from imaginary-time data, Phys. Rev. B44, 6011 (1991)

    1991

  30. [38]

    Jarrell and J

    M. Jarrell and J. Gubernatis, Bayesian inference and the analytic continuation of imaginary-time quantum monte carlo data, Physics Reports269, 133 (1996)

    1996

  31. [39]

    Shifman, A

    M. Shifman, A. Vainshtein, and V. Zakharov, Qcd and resonance physics. theoretical foundations, Nuclear Physics B147, 385 (1979)

    1979

  32. [40]

    COLANGELO and A

    P. COLANGELO and A. KHODJAMIRIAN, Qcd sum rules, a modern perspective, inAt The Frontier of Parti- cle Physics(WORLD SCIENTIFIC, 2001) p. 1495–1576

    2001

  33. [41]

    E. C. Poggio, H. R. Quinn, and S. Weinberg, Smear- ing method in the quark model, Phys. Rev. D13, 1958 (1976)

    1958

  34. [42]

    Ciulli and T

    S. Ciulli and T. D. Spearman, Analytic con- tinuation from data points with unequal er- rors, Journal of Mathematical Physics23, 1752 (1982), https://pubs.aip.org/aip/jmp/article- pdf/23/10/1752/19260372/1752 1 online.pdf

    1982

  35. [43]

    Ciulli and T

    S. Ciulli and T. D. Spearman, Search for physical struc- tures on the boundary by optimal analytic continuation from a finite set of interior data points, Phys. Rev. D27, 1580 (1983)

    1983

  36. [44]

    Ciulli, S

    M. Ciulli, S. Ciulli, and T. D. Spearman, Bounds for the continuation of perturbative results to the spectral region, Journal of Mathematical Physics 25, 3194 (1984), https://pubs.aip.org/aip/jmp/article- pdf/25/11/3194/19309030/3194 1 online.pdf

    1984

  37. [45]

    Ciulli, F

    S. Ciulli, F. Geniet, G. Mennessier, and T. D. Spearman, Resonance determination by stabilized analytic continu- ation of theoretical data, and comparison with the mo- ments method, Phys. Rev. D36, 3494 (1987)

    1987

  38. [46]

    Gorman and T

    N. Gorman and T. D. Spearman, Resonance pole deter- mination in a quantum-mechanical model, Il Nuovo Ci- mento A (1965-1970)99, 741 (1988)

    1965

  39. [47]

    Hastie, R

    T. Hastie, R. Tibshirani, and J. H. Friedman,The El- ements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd ed. (Springer, 2009)

    2009

  40. [48]

    Mohammadipour and X

    P. Mohammadipour and X. Li, Direct analysis of zero- noise extrapolation: Polynomial methods, error bounds, and simultaneous physical-algorithmic error mitigation (2025), arXiv:2502.20673 [quant-ph]

    2025

  41. [49]

    We remark here that theL 2 (pseudo-)norm defined in (4) is not the only choice one can make. In fact, there are four differentL 2 type norms one can write down where the integrand is expressed as the square of the real part of the function, the imaginary part of the function, ...

  42. [50]

    Ciulli and T

    S. Ciulli and T. Spearman, Analytic continuation from data points with unequal errors, Journal of Mathematical Physics23, 1752 (1982)

    1982

  43. [51]

    Ciulli and T

    S. Ciulli and T. Spearman, Search for physical structures on the boundary by optimal analytic continuation from a finite set of interior data points, Physical Review D27, 1580 (1983)

    1983

  44. [52]

    A. Rath, V. Vitale, S. Murciano, M. Votto, J. Dubail, R. Kueng, C. Branciard, P. Calabrese, and B. Vermer- sch, Entanglement barrier and its symmetry resolution: theory and experiment, PRX Quantum4, 010318 (2023), arXiv:2209.04393 [cond-mat, physics:quant-ph]

    2023 arXiv

  45. [53]

    Porras and J

    D. Porras and J. I. Cirac, Effective quantum spin systems with trapped ions, Phys. Rev. Lett.92, 207901 (2004)

    2004

  46. [54]

    The mean and covariance matrix here are jackknife- corrected

  47. [55]

    E. H. Lieb, Convex trace functions and the wigner- yanase-dyson conjecture, Les rencontres physiciens- 7 math´ ematiciens de Strasbourg-RCP2519, 0 (1973)

    1973

  48. [56]

    Uhlmann, Relative entropy and the wigner-yanase- dyson-lieb concavity in an interpolation theory, Commu- nications in Mathematical Physics54, 21 (1977)

    A. Uhlmann, Relative entropy and the wigner-yanase- dyson-lieb concavity in an interpolation theory, Commu- nications in Mathematical Physics54, 21 (1977)

    1977

  49. [57]

    Petz, Quasi-entropies for finite quantum systems, Re- ports on mathematical physics23, 57 (1986)

    D. Petz, Quasi-entropies for finite quantum systems, Re- ports on mathematical physics23, 57 (1986)

    1986

  50. [58]

    M¨ uller-Lennert, F

    M. M¨ uller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, On quantum R´ enyi entropies: A new generalization and some properties, Journal of Math- ematical Physics54, 122203 (2013), arXiv:1306.3142 [quant-ph]

    2013 arXiv

  51. [59]

    M. M. Wilde, A. Winter, and D. Yang, Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched R´ enyi Relative Entropy, Communications in Mathematical Physics331, 593 (2014), arXiv:1306.1586 [quant-ph]

    2014 arXiv

  52. [60]

    K. M. R. Audenaert and N. Datta,α-z-R´ enyi rela- tive entropies, Journal of Mathematical Physics56, 022202 (2015), https://pubs.aip.org/aip/jmp/article- pdf/doi/10.1063/1.4906367/15997344/022202 1 online.pdf

    2015 doi

  53. [61]

    Kudler-Flam, L

    J. Kudler-Flam, L. Nie, and A. Vijay, R´ enyi mutual information in quantum field theory, tensor networks, and gravity, Journal of High Energy Physics2024, 195 (2024)

    2024

  54. [62]

    Elben and B

    A. Elben and B. Vermersch, Randommeas.jl: A julia package for randomized measurements in quantum de- vices (2025), arXiv:2509.12749 [quant-ph]. Stabilized Analytic Continuation In this appendix, we review the key features of the SAC method outlined in references [38–42]. Addit...

    2025 arXiv

  55. [63]

    Due to errors in the data points, it now no longer makes sense to choose the subtraction point to correspond to one of the data points

    These points constitute an ellipsoid in data space whose principal axes correspond toe I . Due to errors in the data points, it now no longer makes sense to choose the subtraction point to correspond to one of the data points. To incorporate errors, the subtraction point can i...

  56. [64]

    Due to convexity ofδ 2 min, the minimum value is attained on the boundary of this domain which satisfiesχ 2 =χ 2

  57. [65]

    G[y, y0, λ] =δ2 min[y, y0] +λ(χ 2[y]−χ 2

    Thus, we can recast the problem of finding the data point satisfyingχ 2 =χ 2 0 which minimizesδ 2 min as a Lagrange multiplier problem. G[y, y0, λ] =δ2 min[y, y0] +λ(χ 2[y]−χ 2

  58. [66]

    (20) We then have the following two equations: ∂G ∂y0 = ∂G ∂yI = 0. These equations simplify to NX J=1 B−1 IJ (yJ −y 01J ) + λ ϵ2 I (yI −d I ) = 0 =⇒(y I −y 01I ) +λ NX J=1 BIJ (yJ −d J ) ϵ2 J = 0 (21) NX I,J=1 1I B−1 IJ (yJ −y 01J ) = 0 =⇒ NX I=1 1I (yI −d I ) ϵ2 I = 0 (22) I...

  59. [67]

    Onceλhas been determined, we can compute the minimal norm asδ 2 min =λ 2 PN r=1 σrp2 r

    This is the only step that needs to be performed numerically. Onceλhas been determined, we can compute the minimal norm asδ 2 min =λ 2 PN r=1 σrp2 r. Now, just as in the noiseless case,δ 2 min is really a function ofαsince the data valuesd i ≡d ′ i;α are functions forα. Thus, ...

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