The reviewed record of science sign in
Pith

arxiv: 2607.06431 · v1 · pith:JCVJ63YK · submitted 2026-07-07 · quant-ph

Unbiased Estimation of Conditional Covariance for Quantum Optomechanics

Katsuta Sakai , Nobuyuki Matsumoto This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-07-08 05:54 UTCglm-5.2pith:JCVJ63YKrecord.jsonopen to challenge →

classification quant-ph PACS 42.50.Wk03.65.Ta07.10.Cm
keywords covarianceconditionalconventionalestimatorlinear-gaussianmacroscopicquantumretrodictive
0
0 comments X

The pith

Three trajectories fix quantum covariance bias in mg-scale mirror

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

This paper proves and demonstrates that a third trajectory — the smoothed estimate, which uses the complete measurement record — can algebraically cancel the unknown backward covariance that biases the conventional two-trajectory method. The key identity, Eq. (3), combines three pairwise trajectory-difference variances in a way that exactly recovers the forward conditional covariance V without assuming V^(E) ≈ V. The authors validate this on a 7.71-mg suspended mirror: the three-trajectory reconstruction agrees with an independently parameterized Riccati prediction (chi-squared p=0.263 for finite-band closure), while the conventional estimator shows a resolved systematic bias (chi-squared p=0.024 against the forward-covariance target). The method retains off-diagonal covariance elements that the conventional approach cancels, and works under feedback cooling where the symmetry assumption fails most badly. The authors argue this opens a practical route to entanglement certification for macroscopic oscillators, including kilogram-scale gravitational-wave test masses.

Core claim

The central discovery is the exact algebraic identity V = [Var(x_fwd - x_bwd) + Var(x_smooth - x_fwd) - Var(x_smooth - x_bwd)] / 2, derived from the orthogonality property of the smoothing error in linear-Gaussian estimation. This identity eliminates the unknown future-likelihood covariance V^(E) that biases the conventional two-trajectory estimator, and it recovers the full covariance matrix including off-diagonal elements. Experimental validation on a milligram-scale mirror shows statistical consistency with a fixed-parameter Riccati prediction, while the conventional estimator shows a resolved systematic mismatch.

What carries the argument

Three record-derived trajectories: the causal (forward Kalman) estimate conditioned on past measurements, the future-likelihood (retrodictive) estimate conditioned on future measurements, and the two-filter smoothed estimate conditioned on the complete record. Their pairwise difference variances, combined via the identity in Eq. (3), yield the forward conditional covariance without assuming forward-backward symmetry.

If this is right

  • The estimator can be applied to bipartite systems for Gaussian entanglement certification via the PPT criterion, where the conventional estimator would falsely classify entangled states as separable (the paper shows a representative point where the bias flips the conclusion).
  • The method scales to kilogram-scale gravitational-wave test masses, where feedback cooling is essential and forward-backward symmetry is broken, enabling measurement-based entanglement tests in proposed interferometer experiments.
  • The frequency-resolved spectral closure test (Fig. 4) provides a falsifiability mechanism: the spectral identity S_V(omega) = [S_FB + S_SF - S_SB]/2 must hold at every frequency, not just in integrated form, making accidental agreement harder.
  • The information-weighted fallback (Eq. S170) extends the method to cases where the future likelihood is singular or ill-conditioned, which is relevant when correlated force and readout noise cancel in the measured record.

Load-bearing premise

The load-bearing premise is that the experimental system is accurately described by the linear-Gaussian state-space model with noise covariances fixed by independent calibrations. In particular, the residual stabilized laser-frequency noise was not measured out-of-loop and was not included as a fitted state-space noise source; if the true residual frequency noise deviates significantly from the nominal model, it could introduce an unmodeled observability limitation affecting.

What would settle it

An independent out-of-loop measurement of the stabilized laser-frequency noise showing it is substantially larger than the nominal model would introduce an unmodeled covariance contribution that could break the closure agreement. Alternatively, if the system exhibits significant non-Gaussianity or nonlinearity outside the linear-Gaussian regime, the orthogonality property underlying Eq. (3) would fail and the estimator would become biased.

Figures

Figures reproduced from arXiv: 2607.06431 by Katsuta Sakai, Nobuyuki Matsumoto.

Figure 1
Figure 1. Figure 1: shows how strongly the conventional two￾trajectory estimator is biased when the symmetry as￾sumption V (E) ≃ V fails. We quantify the deterministic covariance mismatch by the covariance-space distance dM(Vconv, V ) = " 1 2 X i (log λi) 2 #1/2 . (8) FIG. 1. Bias of the conventional retrodictive esti￾mator. The color scale shows the covariance-space distance dM between the conventional two-trajectory covaria… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

Continuous measurements can prepare macroscopic mechanical oscillators in conditional quantum states, but their covariance is difficult to verify. The conventional retrodictive estimator assumes a forward--backward covariance symmetry and can be biased, because physical dynamics such as feedback damping reduces the observability of the state from future records. Here, we derive an exact linear-Gaussian estimator from causal, retrodictive, and smoothed trajectories. For a milligram-scale mirror, it agrees with a Riccati prediction based on parameters fixed independently, while the conventional estimate exhibits a large bias in the covariance-space metric, $d_M \sim 5$. Our method paves the way toward unbiased testing of macroscopic entanglement within a calibrated linear-Gaussian model, which will be applicable to tabletop mirrors as well as gravitational-wave kg-scale test masses.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 5 minor

Summary. This manuscript derives and experimentally validates a three-trajectory estimator for the forward conditional covariance V in linear-Gaussian quantum optomechanics. The key identity, Eq. (3), combines the causal (forward), retrodictive (backward), and smoothed trajectories to eliminate the unknown future-likelihood covariance V^(E) that biases the conventional two-trajectory estimator Var(→x−←x)/2. The derivation rests on the orthogonality principle of linear-Gaussian smoothing [Eqs. (S28)–(S32)] and is exact under standard linear-Gaussian assumptions. The experimental validation uses a 7.71-mg suspended mirror with all filter parameters fixed by independent calibrations (cavity scans, optical-spring curve, ring-down, intensity-noise measurement). The three-trajectory reconstruction agrees with the finite-band Riccati prediction (χ²_V = 3.99, p = 0.263, ν = 3), while the conventional estimator shows a resolved systematic bias (χ²_V = 9.40, p = 0.0244 against the forward-covariance target). A frequency-resolved closure test (Fig. 4) provides an additional, harder-to-satisfy check across all four spectral components.

Significance. The paper makes a clean methodological contribution: the estimator identity in Eq. (3)/Eq. (S33) is exact, parameter-free, and derived from first principles of linear estimation theory. The experimental validation is carefully designed — parameters are fixed independently before covariance evaluation, and the finite-band closure test is a genuine falsifiable prediction. The frequency-resolved spectral test (Fig. 4) is a stronger check than integrated moments alone and adds substantial credibility. The motivation is well-grounded: the forward–backward symmetry assumption V^(E) ≃ V fails precisely in regimes with feedback damping and detuning, which are the regimes of interest for macroscopic quantum-state certification. The application to entanglement testing (Fig. 1, star point) illustrates the practical stakes. The transparency about the residual laser-frequency-noise limitation (§S2.2.5, §S3.3.1) is commendable and strengthens the paper's credibility.

major comments (2)
  1. [§S3.3.1, Eqs. (S153)–(S156)] The forward filter uses a stationary (steady-state) Kalman gain K, computed from the discrete algebraic Riccati equation (S156), rather than a time-varying gain. Similarly, the future-likelihood information matrix J is stationary [Eq. (S163)]. This is a modeling choice that is not explicitly justified in the main text. For the 10-s experimental record with a 280-Hz optical-spring-shifted resonance and feedback damping at Q_eff ≈ 250, the transient time of the filter is of order Q_eff/(π f_eff) ≈ 0.3 s, which is short relative to 10 s but not negligible for the finite-record statistics. The paper should state that the stationary-gain approximation is adequate and briefly justify it, or note the fraction of the record affected by transients.
  2. [Fig. 1 and main text discussion of d_M] The covariance-space distance d_M [Eq. (8)] is computed between V_conv = (V + V^(E))/2 and V. The map in Fig. 1 assumes the ideal normalized technical-noise level N_x = 0, with the experimental operating point at d_M ≈ 4.3 (ideal) or d_M ≈ 4.9 (with N_x = 2.61). The main text later reports d_M ~ 5 for the experimental data. It would help the reader to clarify the relationship between the Fig. 1 theoretical map (which uses the model V and V^(E) at each point in parameter space) and the experimentally measured d_M from the actual trajectory-difference data. Specifically, is the experimental d_M ~ 5 the same quantity as the d_M ≈ 4.9 from the model map, or are these computed differently? A sentence connecting these would avoid confusion.
minor comments (5)
  1. Abstract states 'd_M ~ 5'; the main text gives d_M ≈ 4.9 (with N_x = 2.61) and the experimental χ²_V = 9.40. Consider stating the experimental d_M value explicitly in the abstract for precision.
  2. Eq. (9): the notation 'χ²_V = δv^T C^{-1}_{δv} δv' has a subscript placement that could be misread. Consider writing χ²_V = δv^T C^{-1}_{δv} δv more clearly, or defining C_{δv}^{-1} explicitly before use.
  3. The OU approximation for thermal noise [Eq. (S127), Table S2] uses three poles. The paper could briefly state the fractional accuracy of this approximation over the 130 Hz–2 kHz band to reassure readers that the approximation error is subdominant.
  4. The paper could briefly mention whether the 12% quadrature calibration uncertainty (σ_α/α = 12%) is the dominant contribution to the propagated covariance C_{δv}, or whether finite-record statistics dominate. This would help readers assess the relative importance of calibration versus statistics.
  5. Reference [23] is a self-citation to a prior preprint by some of the authors, noting a correction to a previous analysis. This is appropriately transparent but the footnote could briefly state what was corrected for readers unfamiliar with that work.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for a careful and constructive report. Both major comments identify legitimate points of clarification. We will (1) add an explicit justification for the stationary-gain approximation, including the transient timescale and the fraction of the record affected, and (2) add a sentence clarifying the relationship between the model-based d_M values in Fig. 1 and the experimentally reported d_M ~ 5. Both revisions are minor and do not affect any results or conclusions.

read point-by-point responses

  1. Referee: The forward filter uses a stationary (steady-state) Kalman gain K, computed from the discrete algebraic Riccati equation (S156), rather than a time-varying gain. Similarly, the future-likelihood information matrix J is stationary [Eq. (S163)]. This is a modeling choice that is not explicitly justified in the main text. For the 10-s experimental record with a 280-Hz optical-spring-shifted resonance and feedback damping at Q_eff ≈ 250, the transient time of the filter is of order Q_eff/(π f_eff) ≈ 0.3 s, which is short relative to 10 s but not negligible for the finite-record statistics. The paper should state that the stationary-gain approximation is adequate and briefly justify it, or note the fraction of the record affected by transients.

    Authors: The referee is correct that the stationary-gain approximation is a modeling choice that should be explicitly justified. We will add a brief discussion to Section S3.3.1. The key points are as follows. The filter transient timescale is set by the convergence rate of the forward Riccati equation and the backward information-matrix recursion. For the forward filter, the relevant timescale is of order Q_eff/(π f_eff) ≈ 250/(π × 280 Hz) ≈ 0.28 s, as the referee notes. For the backward information filter, the convergence timescale is comparable, since the same mechanical susceptibility governs the rate at which future measurements accumulate information about the present state. For a 10-s record, the transient fraction is approximately 2 × 0.3 s / 10 s ≈ 6% (accounting for transients at both ends of the record). Moreover, the spectral covariance-closure test (Fig. 4) integrates over 130 Hz–2 kHz, so the low-frequency spectral bins most affected by any residual transient leakage are outside the primary integration band. The finite-band closure test (χ²_V = 3.99, p = 0.263) already confirms that the stationary-gain reconstruction is statistically consistent with the fixed-parameter model over the verification band. We will state these points explicitly in the revised Supplemental Material. revision: yes

  2. Referee: The covariance-space distance d_M [Eq. (8)] is computed between V_conv = (V + V^(E))/2 and V. The map in Fig. 1 assumes the ideal normalized technical-noise level N_x = 0, with the experimental operating point at d_M ≈ 4.3 (ideal) or d_M ≈ 4.9 (with N_x = 2.61). The main text later reports d_M ~ 5 for the experimental data. It would help the reader to clarify the relationship between the Fig. 1 theoretical map (which uses the model V and V^(E) at each point in parameter space) and the experimentally measured d_M from the actual trajectory-difference data. Specifically, is the experimental d_M ~ 5 the same quantity as the d_M ≈ 4.9 from the model map, or are these computed differently? A sentence connecting these would avoid confusion.

    Authors: We agree that the relationship between the model-based d_M values in Fig. 1 and the experimentally reported d_M ~ 5 should be clarified. These are computed differently and are not the same quantity. In Fig. 1, d_M is computed purely from the model: at each point in parameter space, the Riccati equation gives V, the future-likelihood recursion gives V^(E), and d_M is evaluated between V_conv = (V + V^(E))/2 and V using Eq. (8). The values d_M ≈ 4.3 (N_x = 0) and d_M ≈ 4.9 (N_x = 2.61) are model predictions of the bias that would affect the conventional estimator at the experimental operating point. The experimentally reported d_M ~ 5 in the main text is a different quantity: it is the covariance-space distance between the conventional two-trajectory estimate reconstructed from the actual experimental trajectory-difference data and the three-trajectory unbiased estimate (or equivalently the finite-band Riccati prediction), both evaluated over the 130 Hz–2 kHz integration band. The approximate agreement between the model-predicted bias (d_M ≈ 4.9) and the experimentally observed bias (d_M ~ 5) is consistent with the expectation that the experimental conventional estimator is biased by approximately the amount predicted by the model. We will add a sentence to the main text making this distinction explicit. revision: yes

Circularity Check

0 steps flagged

No circularity found. The estimator identity is derived from standard external estimation-theory results, and the experimental validation uses independently fixed parameters.

full rationale

The central estimator identity (Eq. 3 / Eq. S33) is derived algebraically from the orthogonality principle of linear-Gaussian smoothing (Eqs. S28–S32), which is cited to Rauch-Tung-Striebel [13] and Mayne [14] — external foundational references from 1965–1966. No self-citation is load-bearing for the derivation. The experimental comparison fixes all filter parameters from independent auxiliary calibrations (cavity scans, optical-spring curve, ring-down, mass measurements), with f_eff and Q_eff fitted only to the ordinary displacement spectrum and fixed before any covariance evaluation. The detuning is selected by zero-crossing analysis, not by matching covariance. The Riccati prediction and the three-trajectory reconstruction share model parameters but compute V through independent pathways (deterministic model equations vs. empirical trajectory-difference statistics), making the agreement a falsifiable test rather than a tautology. The paper explicitly states: 'Since the covariance is obtained from pairwise trajectory-difference statistics rather than assigned by the Riccati equation, the agreement provides a falsifiable covariance-closure test.' Self-citations (Refs [9, 20, 22, 26, 32]) involve author Matsumoto but are cited for the experimental apparatus, entanglement proposals, or calibration formulas — none are load-bearing for the estimator identity or the covariance-closure test. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The paper does not invent new physical entities. It introduces a new estimator (the three-trajectory reconstruction) and an augmented state-space model, but these are mathematical constructions within the standard linear-Gaussian framework, not new physical postulates.

free parameters (4)
  • f_eff = 283.5 ± 6.7 Hz
    Segment-specific effective optical-spring resonance frequency, fitted to the ordinary displacement spectrum before covariance evaluation.
  • Q_eff = 250 ± 13
    Segment-specific effective quality factor, fitted to the ordinary displacement spectrum before covariance evaluation.
  • Nx = 2.61 (+2.61/-1.31)
    Effective intensity-quadrature occupation, fixed by independent relative-intensity-noise measurement.
  • OU approximation coefficients (a_j, C_j) = See Table S2
    Coefficients for the Ornstein-Uhlenbeck approximation of the suspension thermal-force noise, chosen to fit the 1/f spectrum over 130 Hz - 2 kHz.
axioms (4)
  • domain assumption The symmetrically ordered first and second moments of the quantum system obey the same linear-estimation algebra as a classical Gaussian process.
    Invoked in the main text and Section S1 to justify applying classical Kalman-Bucy and smoothing theory to quantum conditional states.
  • standard math The smoothing error is orthogonal to any square-integrable function of the complete measurement record.
    The core mathematical identity (Eq. 3) relies on this orthogonality property from linear estimation theory.
  • domain assumption The experimental system is accurately described by the augmented linear-Gaussian state-space model with the specified noise covariances.
    The entire reconstruction and Riccati comparison depend on the state-space model (Eqs. S132-S140) being an accurate representation of the physical system over the verification band.
  • ad hoc to paper The residual stabilized laser-frequency noise is subdominant and can be treated as a systematic sensitivity rather than a fitted state-space noise source.
    Section S3.3.1 acknowledges no out-of-loop measurement was available; the nominal model is used instead, which is a necessary simplification for the current analysis.

pith-pipeline@v1.1.0-glm · 30472 in / 2621 out tokens · 444981 ms · 2026-07-08T05:54:17.755291+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · 1 internal anchor

  1. [1]

    Wieczorek, S

    W. Wieczorek, S. G. Hofer, J. Hoelscher-Obermaier, R. Riedinger, K. Hammerer, and M. Aspelmeyer, Phys. Rev. Lett. 114, 223601 (2015)

    work page 2015

  2. [2]

    D. J. Wilson, V. Sudhir, N. Piro, R. Schilling, A. H. Ghadimi, and T. J. Kippenberg, Nature524, 325 (2015)

    work page 2015

  3. [3]

    Rossi, D

    M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, Nature563, 53 (2018)

    work page 2018

  4. [4]

    Rossi, D

    M. Rossi, D. Mason, J. Chen, and A. Schliesser, Phys. Rev. Lett.123, 163601 (2019)

    work page 2019

  5. [5]

    R. E. Kalman and R. S. Bucy, Journal of Basic Engineering83, 95 (1961)

    work page 1961

  6. [6]

    C. Meng, G. A. Brawley, S. Khademi, E. M. Bridge, J. S. Bennett, and W. P. Bowen, Sci. Adv.8, eabm7585 (2022)

    work page 2022

  7. [7]

    Huang, A

    G. Huang, A. Beccari, N. J. Engelsen, and T. J. Kippenberg, Nature626, 512 (2024)

    work page 2024

  8. [8]

    Müller-Ebhardt, H

    H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, and Y. Chen, Phys. Rev. Lett.100, 013601 (2008)

    work page 2008

  9. [9]

    D. Miki, N. Matsumoto, A. Matsumura, T. Shichijo, Y. Sugiyama, K. Yamamoto, and N. Yamamoto, Phys. Rev. A107, 032410 (2023)

    work page 2023

  10. [10]

    Gammelmark, B

    S. Gammelmark, B. Julsgaard, and K. Mølmer, Phys. Rev. Lett.111, 160401 (2013)

    work page 2013

  11. [11]

    Zhang and K

    J. Zhang and K. Mølmer, Phys. Rev. A96, 062131 (2017)

    work page 2017

  12. [12]

    Hatakeyama, R

    K. Hatakeyama, R. Fukuzumi, A. Matsumura, D. Miki, and K. Yamamoto, arXiv preprint arXiv:2603.16821 10.48550/arXiv.2603.16821 (2026), arXiv v2 (2026), arXiv:2603.16821 [quant-ph]

    work page doi:10.48550/arxiv.2603.16821 2026

  13. [13]

    H. E. Rauch, F. Tung, and C. T. Striebel, AIAA J.3, 1445 (1965)

    work page 1965

  14. [14]

    D. Q. Mayne, Automatica4, 73 (1966)

    work page 1966

  15. [15]

    Tsang, Phys

    M. Tsang, Phys. Rev. A105, 042213 (2022)

    work page 2022

  16. [16]

    Guevara and H

    I. Guevara and H. M. Wiseman, Phys. Rev. Lett.115, 180407 (2015)

    work page 2015

  17. [17]

    K. T. Laverick, P. Warszawski, A. Chantasri, and H. M. Wiseman, PRX Quantum4, 040340 (2023)

    work page 2023

  18. [18]

    L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys. Rev. Lett.84, 2722 (2000)

    work page 2000

  19. [19]

    Simon, Phys

    R. Simon, Phys. Rev. Lett.84, 2726 (2000)

    work page 2000

  20. [20]

    Matsumoto, M

    N. Matsumoto, M. Sugawara, S. Suzuki, N. Abe, K. Komori, Y. Michimura, Y. Aso, S. B. Cataño-Lopez, and K. Edamatsu, Phys. Rev. Lett.122, 071101 (2019)

    work page 2019

  21. [21]

    G. I. González and P. R. Saulson, J. Acoust. Soc. Am.96, 207 (1994)

    work page 1994

  22. [23]

    Note that in v5 of arXiv:2008.10848, the difference between the prediction and retrodiction power spectra was used instead of the power spectrum of the difference between the two estimates, thereby omitting the required cross-spectrum

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  23. [24]

    H. Miao, S. Danilishin, H. Müller-Ebhardt, H. Rehbein, K. Somiya, and Y. Chen, Phys. Rev. A81, 012114 (2010)

    work page 2010

  24. [25]

    R. A. Thomas, M. Parniak, C. Østfeldt, C. B. Møller, C. Bærentsen, Y. Tsaturyan, A. Schliesser, J. Appel, E. Zeuthen, and E. S. Polzik, Nat. Phys.17, 228 (2021)

    work page 2021

  25. [26]

    Matsumoto, K

    N. Matsumoto, K. Sakai, K. Hatakeyama, K. Izumi, D. Miki, S. Iso, A. Matsumura, and K. Yamamoto, arXiv preprint arXiv:2507.12899 10.48550/arXiv.2507.12899 (2025), arXiv:2507.12899 [quant-ph]

    work page doi:10.48550/arxiv.2507.12899 2025

  26. [27]

    Mayne, Automatica4, 73 (1966)

    D. Mayne, Automatica4, 73 (1966). 32

    work page 1966

  27. [28]

    Fraser and J

    D. Fraser and J. Potter, IEEE Transactions on Automatic Control14, 387 (1969)

    work page 1969

  28. [29]

    R. K. Mehra,On Optimal and Suboptimal Linear Smoothing, Contractor Report NASA-CR-97072 (National Aeronautics and Space Administration (NASA), 1968)

    work page 1968

  29. [30]

    J. E. W. Jr, A. S. Willsky, and N. R. S. Jr, Stochastics5, 1 (1981), https://doi.org/10.1080/17442508108833172

    work page doi:10.1080/17442508108833172 1981

  30. [31]

    Khademi, J

    S. Khademi, J. J. Slim, K. T. Laverick, J. Chang, J. Guo, S. Gröblacher, H. M. Wiseman, and W. P. Bowen, Post-processed estimation of quantum state trajectories (2025), arXiv:2510.16754 [quant-ph]

    work page arXiv 2025

  31. [32]

    Matsumoto, K

    N. Matsumoto, K. Komori, S. Ito, Y. Michimura, and Y. Aso, Phys. Rev. A94, 033822 (2016)

    work page 2016

  32. [33]

    E. H. Moore, Bulletin of the American Mathematical Society26, 394 (1920)

    work page 1920

  33. [34]

    Penrose, Proceedings of the Cambridge Philosophical Society51, 406 (1955)

    R. Penrose, Proceedings of the Cambridge Philosophical Society51, 406 (1955)

    work page 1955

  34. [35]

    Sugiyama, T

    Y. Sugiyama, T. Shichijo, N. Matsumoto, A. Matsumura, D. Miki, and K. Yamamoto, Phys. Rev. A107, 033515 (2023)

    work page 2023