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The role of correlations in a sequence of quantum observations on empirical measures

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Pith's one-line read This paper derives closed-form covariance formulas for histograms of finite outcome strings from generic quantum instruments and introduces a relative-entropy measure that quantifies how much Markov-model compression loses.

desk verdict Sound extension of the ED1 formalism to length-L sequences, but the spectral-gap condition and the covariance-only KL divergence need fixing. read the letter →

arxiv 2411.08214 v1 pith:RYX5R3UW submitted 2024-11-12 quant-ph cond-mat.stat-mech

Joseph A. Smiga , Gabriel T. Landi This is my paper · ORCID
classification quant-phcond-mat.stat-mech MSC 81P1581P4562B10 PACS 03.65.Ta02.50.Cw05.40.-a
keywords quantuminstrumentsempiricaldistributionsmeasurementcorrelationsjumpscovariancematrixrelativeentropyFisherinformationfullcountingstatistics
topics The Measurement Problem
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

This paper establishes that for a stationary sequence of outcomes produced by repeatedly measuring a quantum system with any quantum instrument, the covariance matrix of the empirical distribution of length-L substrings has a closed-form expression in the large-data limit, Eq. (11), with the correlation matrix Ψ given explicitly in terms of the instrument superoperator and its Drazin inverse, Eq. (32). This reduces the complicated many-body problem of correlated quantum measurement records to linear algebra, making the Fisher information and other statistical quantities directly computable from the instrument alone. The paper further introduces a relative-entropy measure I_m^L that quantifies how much information is lost when the true string statistics are compared with an order-m Markov-model approximation, thereby giving a quantitative signature of non-Markovian correlations. A sympathetic reader would care because these formulas turn full counting statistics and quantum metrology for a wide class of measurement schemes into a single tractable calculation.

What carries the argument

The load-bearing object is the instrument superoperator M = \sum_x M_x together with its Drazin inverse (1-M)^+, the inverse defined on the subspace orthogonal to the steady-state eigenspace. The correlation matrix Ψ constructed from these objects, Eq. (32), compresses the entire infinite tail of measurement correlations into a single finite matrix entry that appears in the covariance formula, so that quantum backaction and memory effects are fully captured by linear algebra. For L > 1, the extra overlap terms p_{y←x}(\ell)-p_y, \ell=1,...,L-1, handle the fact that a substring of length L is counted multiple times when it overlaps its predecessor, which is a purely combinatorial correction independent of quantum dynamics.

What would settle it

Simulate the spin-conserving periodic spin chain of Sec. V C (κ=0) with random projective measurements and compute the empirical distribution of q_+ for N=$10^{5}$: the multimodal structure visible in Fig. 8 for rapid measurements would persist rather than become Gaussian, directly violating the Gaussian assumption that underlies Eqs. (11) and (38).

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Extended reading notes

Core claim

The central claim is that for a stationary quantum-instrument process with unique steady state π, the covariance of the empirical distribution of length-L substrings is, for large N, Σ_{xy} = 1/(N−L+1)[p_x(δ_{xy}−p_y)+Ψ_{yx}p_x+Ψ_{xy}p_y], where Ψ_{yx} = \sum_{\ell=1}^{L-1}[p_{y←x}(\ell)-p_y] + (1/p_x)\langle\langle 1|M_y(1-M)^+M_x|\pi\rangle\rangle. Here M = \sum_x M_x is the completely positive instrument superoperator, (1-M)^+ its Drazin inverse, and the overlap terms p_{y←x}(\ell)-p_y account for the counting of overlapping substrings. The paper proves this in Appendix A by eigendecomposing M and summing the geometric series of transient eigenvalues, and it specializes the result to quantum jumps, where M_x = -J_x $L_0^{{-1}}$ is built from the jump and no-jump superoperators. It then defines the empirical-distribution mutual information I_m^L = D_{KL}(Σ\|$Σ^{{(m,L)}}$), the KL divergence between the true covariance and that of the process marginalized to an order-m Markov model, and shows this quantity is zero for Markov processes and independent of N when the means coincide.

Load-bearing premise

The whole approach presumes that the empirical distribution becomes Gaussian in the large-N limit, which in turn requires that all transient eigenvalues of the instrument superoperator M have modulus strictly less than one so that correlations decay fast enough; if some eigenvalue sits on the unit circle, as in the conserved-spin chain of Sec. V C, the covariance formula and the information measure no longer apply.

Editorial extensions

If this is right

  • For any instrument-based measurement sequence, the Fisher information for estimating a parameter from the empirical distribution is F = N(\partial_\theta p)^T(P+ΨP+PΨ^T)^{-1}\partial_\theta p, so metrological error bars can be computed from the instrument superoperators alone, without Monte Carlo simulation of the record.
  • The measure I_m^L vanishes exactly when the underlying process is Markov of order at most m (for L ≥ m+1), so it functions as a quantitative non-Markovianity witness for sequential quantum measurements.
  • Even for instruments whose individual outcomes are independent, the ED_L covariance contains overlap contributions from the second line of Eq. (32), so higher-order empirical distributions carry information about sequence structure that ED1 cannot see.
  • In the quantum-jump case, substituting M_x=-J_x L_0^{-1} into Eq. (32) gives a closed expression for Ψ in terms of the Lindblad generator, making the covariance and the information measure computable directly from the master equation and jump operators.

Reading between the lines

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

  • The same Ψ-based formulas should, if the spectral-gap assumption holds, apply to any rapidly measured open system, including weak continuous measurements treated as a high-rate instrument; a direct Monte Carlo test of Eq. (11) for a weakly measured cavity would cleanly separate the validity of the Gaussian assumption from the spectral-gap condition.
  • Because I_m^L is independent of N when the first moments match, one could use the measured histograms from an experimental record to fit the smallest Markov order m that makes I_m^L statistically indistinguishable from zero, turning the measure into a data-driven model-selection tool that requires no knowledge of the underlying instrument.
  • The same matrices that appear in the covariance also determine the Fisher information, suggesting that the formalism could be used to quantify how much estimation error is avoided by keeping correlations in the data, i.e., the gap between the true estimator variance and the iid-optimal bound; the paper touches on this via Eq. (30) but does not fully exploit the comparison.
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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 / 4 minor

Summary. The paper studies the empirical distributions of length-L outcome blocks generated by a stationary sequence of quantum-instrument measurements. Its central technical results are the large-N covariance formula Eq. (11) with the correlation matrix Psi given by Eq. (31) for L=1 and Eq. (32) for general L, both expressed through the Drazin inverse (1-M)^+ of the unconditional instrument superoperator M. The paper then defines, in Eq. (38), a quantity I_m^L based on the KL divergence between Gaussian covariances, meant to quantify how correlations in the measurement string affect reconstruction under an order-m Markov model. The formalism is illustrated on an amplitude-damped qubit, a boundary-driven XX spin chain, and a periodically measured spin chain, and it is connected to Fisher information for parameter estimation.

Significance. If the large-N Psi formulas are valid, they provide a broadly applicable full-counting-statistics tool for sequential quantum measurements, extending the earlier ED1 treatment of Ref. [34] to general EDL and to quantum jump processes. The derivations in Appendices A-C are mostly coherent, the analytic bit-flip case provides a useful closed-form check, and no quantities are fitted to data: the information measure is compared against an explicit Markov benchmark. The main gap is that the spectral assumptions stated in Appendix A do not justify the geometric-series resummation for all instruments covered by the paper's claims; this is fixable by strengthening the assumption to a spectral gap and restricting the scope accordingly.

major comments (2)
  1. [Appendix A, Eqs. (A5)-(A7) and (A14)-(A15)] The eigendecomposition assumes a unique steady state and |Re lambda_j| < 1, but the geometric-series resummation leading to (1-M)^+ requires |lambda_j| < 1. The weaker condition admits instruments with non-trivial peripheral spectrum. A concrete example is M = D o U, where U is the cyclic shift on a d=4 computational basis, U|k> = |k+1 mod 4>, and D is dephasing in that basis. This instrument is trace-preserving, has unique steady state I/4, and its eigenvalues are 1, i, -1, -i, so |Re lambda_j| < 1 holds for all j >= 1. However, p_{y<-x}(ell) = delta_{y, x+ell mod 4} is periodic, and the sum defining Psi_N in Eq. (7) has no unique large-N limit: along N = 4k the value depends on y-x mod 4, while along other subsequences it differs. Hence Eq. (31) is not an approximation to Psi for this stationary instrument, and the covariance formula Eq. (11) as stated does not apply. The assumption should be strengthened to a spectral gap, |lambda_j| < 1 for all j >= 1, and this restriction should be stated in Sec. III and Appendix A; the discussion in Sec. V C of conserved total spin acknowledges a different failure and does not cover the periodic case.
  2. [Sec. IV B, Eq. (38)] The quantity I_m^L is defined as DKL(Sigma || Sigma^(m,L)), the KL divergence between two Gaussian distributions with the same mean, not as the KL divergence between the process distributions P_L and Q_m^L. Writing DKL(P_L || Q_m^L) is therefore an overloading that is justified only if the empirical distributions are (at least asymptotically) Gaussian and only when m+1 >= L so the means coincide. Equation (35) is likewise a covariance-only divergence. This distinction is not cosmetic because Sec. V C exhibits a stationary process for which the empirical distribution is multimodal; for such processes I_m^L is not a relative entropy of the empirical distributions. Please rename the quantity (for example, 'Gaussian covariance divergence') or verify the Gaussian approximation for the models in which it is interpreted as a relative entropy.
minor comments (4)
  1. [Sec. IV B, after Eq. (38)] 'Meteorological' should be 'metrological'.
  2. [Secs. II A-II C] The symbol M is used both for the alphabet of outcomes and for the unconditional instrument superoperator in Eq. (14); using |M| for alphabet cardinality or another symbol for the superoperator would remove ambiguity.
  3. [Appendix C] The constraint argument drops the boundary error epsilon_N after noting it is strictly bounded; please clarify explicitly that the support constraints are exact only up to these boundary corrections and that the O(N^{-1}) width claim follows from the bound rather than from a distributional calculation.
  4. [Fig. 8 caption] The quantities q_+ and P(q_+) are used in the caption but not defined there; please define them explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new Psi formulas and the relative-entropy measure are derived from quantum-instrument dynamics and compared to an explicit Markov benchmark, not fitted to the target quantities.

full rationale

The central derivation chain is self-contained. The paper starts from the quantum-instrument expression for the conditional probability p_{y<-x}(ell), Eq. (A4)/(A12), and derives the large-N forms of the covariance kernel, Eqs. (A7) and (A15), by an eigendecomposition and geometric-series summation in Appendix A. These results are not obtained by fitting the covariance or the information measure, and the information measure I_m^L in Eq. (38) is compared against an explicit order-m Markov benchmark whose covariance is independently constructed as Sigma^{(m,L)} in Eq. (37). The worked examples, such as the amplitude-damped qubit leading to Eq. (44), are closed-form consequences of the formalism rather than inputs. The main self-citations, chiefly Ref. [34] for the large-N Gaussian covariance formula Eq. (8), are prior work by the same group, but they are parameter-free and externally checkable; they serve as inputs to the new calculation rather than as conclusions whose derivation is presupposed by the paper's central claim. The paper also explicitly acknowledges a related limitation in Sec. V C regarding stationary processes with conserved total spin. One rigor concern is that Appendix A states the eigendecomposition assumption as |Re lambda_j| < 1 in Eqs. (A5) and (A14), while the geometric-series step requires |lambda_j| < 1. This is a validity gap for instruments with nontrivial peripheral spectrum, but it is a correctness issue, not a circular reduction: no target quantity is defined in terms of itself, and no fitted parameter is renamed as a prediction. Therefore the paper does not exhibit self-definitional, fitted-input, or self-citation-load-bearing circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numbers are fitted to data. The central formulas are derived for a fixed instrument and steady state; the only inputs are the instrument superoperators. The remaining assumptions are the steady-state, spectral-gap, Gaussianity, and no-dark-state conditions listed.

assumptions (5)
  • domain assumption The measurement process is stationary with initial state equal to the steady state pi of the instrument map M.
    Stated in Sec II C; all substring marginals are translationally invariant, which is used throughout and in Eq (10).
  • domain assumption The superoperator M has a unique steady state and all non-unit eigenvalues lie inside the unit disk; the text states |Re lambda_j| < 1 for j >= 1.
    Appendix A, Eqs (A5) and (A14); required for the geometric series in Psi to converge and for (1-M)^+ to be the Drazin inverse.
  • domain assumption The empirical distribution is asymptotically multivariate Gaussian for large N.
    Assumed following Ref [34] in Sec II B; the paper shows in Sec V C a stationary process where this fails, so it is not universal.
  • domain assumption For quantum jumps, there are no dark states, so a jump is eventually observed and the integral of the no-jump propagator is -L0^{-1}.
    Sec II D, Eq (24); needed to define the jump instrument M_x = -J_x L0^{-1}.
  • domain assumption The nullspace of the ED covariance Sigma and Sigma_P is one-dimensional, spanned by the all-ones vector.
    Sec IV A and Appendix B; used to define pseudo-determinant and Drazin-inverse KL divergence for degenerate Gaussians.

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Pith. "Pith review of The role of correlations in a sequence of quantum observations on empirical measures." pith.science (2026). https://pith.science/paper/RYX5R3UW

@misc{pith2026241108214,
  author       = {Pith},
  title        = {Pith review of: The role of correlations in a sequence of quantum observations on empirical measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RYX5R3UW}},
  note         = {Machine review of arXiv:2411.08214}
}
read the original abstract

The outcome of continuously measuring a quantum system is a string of data whose intricate correlation properties reflect the underlying quantum dynamics. In this paper we study the role of these correlation in reconstructing the probabilities of finite sequences of outcomes, the so-called empirical distributions. Our approach is cast in terms of generic quantum instruments, and therefore encompass all types of sequential and continuous quantum measurements. We also show how this specializes to important cases, such as quantum jumps. To quantify the precise role of correlations, we introduce a relative-entropy based measure that quantifies the range of correlations in the string, and the influence that these correlations have in reconstructing finite sequences.

Figures

Figures reproduced from arXiv: 2411.08214 by the authors.

Figure 1
Figure 1. FIG. 1. Reconstructing the empirical distributions from a [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Visualization of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Illustrations of examples explored in this paper. (a) An amplitude-damped described in Sec. V A. (b) A boundary [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Information contained in correlations between mea [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The rate of Fisher information per observation in the [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The KL-divergence of [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Correlational information encoded in the boundary [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Distribution of empirical distributions for an [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The KL-divergence of [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Graph of observing sequence that emphasizes going “into” and “out of” sequences of data from a [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 10
Figure 10. Figure 10: For notational simplicity, denote ⃗xL−1 = ⃗x while y denotes the complete L-sequence. There is also a normalization constraint, as before, X z qz = 1 . (C2) However, this with the constraints from Eq. (C1) have a redundancy. To see this, write I⃗x = X z qz⃗x O⃗x = X z…

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Works this paper leans on

48 extracted references · 42 canonical work pages

  1. [34]

    J. A. Smiga, M. Radaelli, F. C. Binder, and G. T. Landi, Phys. Rev. Res. 5, 033150 (2023), publisher: American Physical Society

    work page 2023

  2. [1]

    M. B. Plenio and P. L. Knight, Rev. Mod. Phys. 70, 101 (1998), publisher: American Physical Society

    work page 1998

  3. [2]

    B¨ uttiker, Physical Review B46, 12485 (1992)

    M. B¨ uttiker, Physical Review B46, 12485 (1992)

    work page 1992

  4. [3]

    H. M. Wiseman and G. J. Milburn, Quantum Measure- ment and Control (Cambridge University Press, New York, 2009)

    work page 2009

  5. [4]

    G. T. Landi, M. J. Kewming, M. T. Mitchison, and P. P. Potts, PRX Quantum 5, 020201 (2024), arXiv:2303.04270 [cond-mat, physics:quant-ph]

    arXiv 2024

  6. [5]

    Piilo, S

    J. Piilo, S. Maniscalco, K. H¨ ark¨ onen, and K.-A. Suomi- nen, Phys. Rev. Lett. 100, 180402 (2008), publisher: American Physical Society

    work page 2008

  7. [6]

    Piilo, K

    J. Piilo, K. H¨ ark¨ onen, S. Maniscalco, and K.-A. Suomi- nen, Phys. Rev. A 79, 062112 (2009), publisher: Ameri- can Physical Society

    work page 2009

  8. [7]

    Patterns in the jump-channel statistics of open quantum systems,

    G. T. Landi, “Patterns in the jump-channel statistics of open quantum systems,” (2023), arXiv:2305.07957 [cond-mat, physics:quant-ph]

    arXiv 2023

Show all 48 references
  1. [8]

    Hatridge, S

    M. Hatridge, S. Shankar, M. Mirrahimi, F. Schackert, K. Geerlings, T. Brecht, K. M. Sliwa, B. Abdo, L. Frun- zio, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret, Science 339, 178 (2013), publisher: American Associa- tion for the Advancement of Science

    2013

  2. [9]

    M. S. Blok, C. Bonato, M. L. Markham, D. J. Twitchen, V. V. Dobrovitski, and R. Hanson, Nature Phys 10, 189 (2014), publisher: Nature Publishing Group

    2014

  3. [10]

    Aharonov, A

    D. Aharonov, A. Kitaev, and J. Preskill, Phys. Rev. Lett. 96, 050504 (2006), publisher: American Physical Society

    2006

  4. [11]

    J. Zou, S. Bosco, and D. Loss, npj Quantum Inf 10, 1 (2024), publisher: Nature Publishing Group

    2024

  5. [12]

    Effect of Correlated Er- rors on Quantum Memory,

    S. Bagewadi and A. Chatterjee, “Effect of Correlated Er- rors on Quantum Memory,” (2024), arXiv:2408.08786 [quant-ph]

    2024 arXiv

  6. [13]

    Schmid, Annals of Physics 173, 103 (1987)

    A. Schmid, Annals of Physics 173, 103 (1987)

    1987

  7. [14]

    Dittrich and R

    T. Dittrich and R. Graham, Phys. Rev. A 42, 4647 (1990), publisher: American Physical Society

    1990

  8. [15]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 96, 010401 (2006), publisher: American Physical Society

    2006

  9. [16]

    Yang, Phys

    Y. Yang, Phys. Rev. Lett. 123, 110501 (2019), publisher: American Physical Society

    2019

  10. [17]

    Altherr and Y

    A. Altherr and Y. Yang, Phys. Rev. Lett. 127, 060501 (2021), publisher: American Physical Society

    2021

  11. [18]

    C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sand- berg, and M. Zimmermann, Rev. Mod. Phys. 52, 341 (1980), publisher: American Physical Society

    1980

  12. [19]

    Radaelli, G

    M. Radaelli, G. T. Landi, K. Modi, and F. C. Binder, New J. Phys. 25, 053037 (2023), arXiv:2206.00463 [cond- mat, physics:quant-ph, stat]

    2023 arXiv

  13. [20]

    Tilloy, Phys

    A. Tilloy, Phys. Rev. A 98, 010104 (2018)

    2018

  14. [21]

    Guilmin, P

    P. Guilmin, P. Rouchon, and A. Tilloy, IF AC- PapersOnLine 22nd IF AC World Congress, 56, 5164 (2023)

    2023

  15. [22]

    Parameters estimation by fitting correlation functions of continu- ous quantum measurement,

    P. Guilmin, P. Rouchon, and A. Tilloy, “Parameters estimation by fitting correlation functions of continu- ous quantum measurement,” (2024), arXiv:2410.11955 [quant-ph]

    2024 arXiv

  16. [23]

    V. T. Vo, T. V. Vu, and Y. Hasegawa, J. Phys. A: Math. Theor. 55, 405004 (2022), publisher: IOP Publishing

    2022

  17. [24]

    Hasegawa, Nat Commun 14, 2828 (2023), number: 1 Publisher: Nature Publishing Group

    Y. Hasegawa, Nat Commun 14, 2828 (2023), number: 1 Publisher: Nature Publishing Group

    2023

  18. [25]

    Hasegawa and T

    Y. Hasegawa and T. V. Vu, Physical Review Letters 123, 110602 (2019), arXiv:1902.06376

    2019 arXiv

  19. [26]

    Guarnieri, G

    G. Guarnieri, G. T. Landi, S. R. Clark, and J. Goold, Physical Review Research 1, 033021 (2019), arXiv:1901.10428v2

    2019 arXiv

  20. [27]

    Hasegawa, Physical Review Letters 125, 050601 (2020), arXiv:1911.11982

    Y. Hasegawa, Physical Review Letters 125, 050601 (2020), arXiv:1911.11982

    2020 arXiv

  21. [28]

    Hasegawa, Physical Review Letters 126, 010602 (2021), arXiv:2003.08557

    Y. Hasegawa, Physical Review Letters 126, 010602 (2021), arXiv:2003.08557

    2021 arXiv

  22. [29]

    A. M. Timpanaro, G. Guarnieri, J. Goold, and G. T. Landi, Physical Review Letters 123, 090604 (2019), arXiv:1904.07574

    2019 arXiv

  23. [30]

    Prech, P

    K. Prech, P. Johansson, E. Nyholm, G. T. Landi, C. Ver- dozzi, P. Samuelsson, and P. P. Potts, Physical Review Research 5, 023155 (2023)

    2023

  24. [31]

    Milz and K

    S. Milz and K. Modi, PRX Quantum 2, 030201 (2021), publisher: American Physical Society

    2021

  25. [32]

    Parameter estimation for quantum jump unraveling,

    M. Radaelli, J. A. Smiga, G. T. Landi, and F. C. Binder, “Parameter estimation for quantum jump unraveling,” (2024), arXiv:2402.06556 [quant-ph]

    2024

  26. [33]

    Bertini, A

    L. Bertini, A. Faggionato, and D. Gabrielli, Ann. Inst. H. Poincar´ e Probab. Statist.51, 867 (2015)

    2015

  27. [35]

    Consistency of empirical distributions of sequences of graph statistics in networks with dependent edges,

    J. R. Stewart, “Consistency of empirical distributions of sequences of graph statistics in networks with dependent edges,” (2024), arXiv:2404.11438 [math, stat]

    2024 arXiv

  28. [36]

    E. B. Davies and J. T. Lewis, Commun. Math. Phys. 17, 239 (1970)

    1970

  29. [37]

    Lindblad, Commun

    G. Lindblad, Commun. Math. Phys. 65, 281 (1979)

    1979

  30. [38]

    Kullback and R

    S. Kullback and R. A. Leibler, Ann. Math. Stat. 22, 79 (1951), publisher: Institute of Mathematical Statistics

    1951

  31. [39]

    Yuan, Phys

    X. Yuan, Phys. Rev. A 99, 032317 (2019), publisher: American Physical Society

    2019

  32. [40]

    C. R. Rao, in Breakthroughs in Statistics: Foundations and Basic Theory, Springer Series in Statistics, edited by S. Kotz and N. L. Johnson (Springer, New York, NY,

  33. [41]

    S. M. Kay, in Fundamentals of statistical signal process- ing: Estimation theory, Vol. 1 (Prentice Hall PTR, Upper Saddle River, NJ, 2013) 20th ed., pp. 27–81. 13

    2013

  34. [42]

    G. T. Landi, E. Novais, M. J. de Oliveira, and D. Karevski, Phys. Rev. E 90, 042142 (2014), publisher: American Physical Society

    2014

  35. [43]

    Schuab, E

    L. Schuab, E. Pereira, and G. T. Landi, Phys. Rev. E 94, 042122 (2016), publisher: American Physical Society

    2016

  36. [44]

    S. H. S. Silva, G. T. Landi, and E. Pereira, Phys. Rev. E 107, 054123 (2023), publisher: American Physical So- ciety

    2023

  37. [45]

    Coppola, E

    M. Coppola, E. Tirrito, D. Karevski, and M. Collura, Phys. Rev. B 105, 094303 (2022), publisher: American Physical Society

    2022

  38. [46]

    Optimal time es- timation and the clock uncertainty relation for stochastic processes,

    K. Prech, G. T. Landi, F. Meier, N. Nurgalieva, P. P. Potts, R. Silva, and M. T. Mitchison, “Optimal time es- timation and the clock uncertainty relation for stochastic processes,” (2024), arXiv:2406.19450 [cond-mat]. Appendix A: Ψ matrix for quantum jumps In this appendix, we...

    2024

  39. [47]

    1 − 1 N + N −1X ℓ=2 1 − ℓ N λℓ−1 j # ⟨ ⟨1|MyPjMx|π⟩ ⟩ = 1 px X j≥1 1 1 − λj

    Single measurement empirical distribution First, we consider the special case of empirical distributions with L = 1 measurements. This is generalized to longer, finite-sequence case in the next section. Recall that [Eq. (7)] Ψyx = N −1X ℓ=1 1 − ℓ N [py←x(ℓ) − py] , (A2) for py...

  40. [48]

    into” and “out of

    Sequence of measurements As described in Ref. [34], the calculations for the Fisher information contained in the empirical distribution of indi- vidual measurements in a stochastic process extends naturally to the distribution of finite sequences of measurements. Namely, the F...

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