Pith. sign in

REVIEW 2 major objections 5 minor 1 cited by

Timescales, Squeezing and Heisenberg Scalings in Many-Body Continuous Sensing

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read The paper proposes a finite-time metric for continuous quantum sensing, proves its N²T² bound, and gives two dissipative sensors that meet it.

desk verdict Useful new finite-time metric for continuous metrology, with an exact N^2T^2 bound and two Heisenberg-limited sensors; the spin-squeezer leg has a real gap in the SM that needs fixing, but the core idea is sound. read the letter →

arxiv 2505.04591 v1 pith:BGZIG23T submitted 2025-05-07 quant-ph

Gideon Lee , Ron Belyansky , Liang Jiang , Aashish A. Clerk This is my paper · ORCID
classification quant-ph
keywords continuousquantummetrologyFisherinformationHeisenbergscalingopensystemsspinsqueezingsuperradiancephotodetection
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 argues that the usual asymptotic 'sensitivity' metric for continuous quantum sensors is misleading, since it can be inflated arbitrarily by making a sensor slow, and proposes instead the optimized finite-time environmental quantum Fisher information, $I_E^{\mathrm{opt}}(T)=\max_{\{\Gamma_j\}} I_E(\{\Gamma_i\};T)$, which counts both integration time and system size as finite resources. For a restricted class of dissipative sensors the paper derives an exact formula for this quantity from the stationary noise spectrum, and proves the bound $I_E^{\mathrm{opt}}(T)\le N^2T^2$. It then exhibits two $N$-qubit sensors—high-temperature superradiance and dissipative spin squeezing—that reach $N^2T^2$ Heisenberg scaling for collective magnetic fields, with the spin-squeezed sensor having the additional property that its quantum limit is reached by plain photodetection of the cavity output. If the claims hold, experimenters gain a defensible definition of Heisenberg scaling and two concrete setups that achieve it with simple measurements.

What carries the argument

The load-bearing object is the two-sided (pseudo-density) master equation of Eq. (3), whose trace yields the global QFI and whose modulus yields the environmental QFI. For the restricted class, Eq. (10) converts these QFIs into integrals of the stationary symmetrized autocorrelation function $C_{ZZ}(\tau)$, with the environment contribution being the global contribution minus a correction built from the two-time correlation $C_{ZZ}(\tau_1+\tau_2)$. The optimized metric $I_E^{\mathrm{opt}}(T)$ then captures a tradeoff: very weak waveguide coupling emits no signal, while very strong coupling dephases the sensor before the parameter is imprinted; the optimum sits near $\Gamma\sim 1/T$. For the spin squeezer, the second load-bearing object is its pure dark state—a state annihilated by all jump operators—which makes direct photodetection of the output field the QFI-saturating measurement.

What would settle it

Measure the estimation error of the dissipative spin squeezer with even $N$, large squeezing $r$, and $\Gamma\simeq 1.89/T$ over a fixed time $T$; the claim predicts an error scaling as the inverse of $0.191\,J(J+1)\,T^2/2$ with $J=N/2$, so a visibly weaker-than-quadratic $N$ scaling, or a much larger constant, would falsify it.

Watch

Extended reading notes

Core claim

On its own terms, the central discovery is that continuous sensing has a well-defined Heisenberg limit once time is treated as a finite resource. The proposed figure of merit is the optimized finite-time environmental QFI, $I_E^{\mathrm{opt}}(T)\equiv\max_{\{\Gamma_j\}}I_E(\{\Gamma_i\};T)$, which is bounded by $N^2T^2$ for any collective generator $\hat Z=\hat J_{\vec r}$. For sensors with $\hat H_0=0$ whose steady state is either maximally mixed with Hermitian jump operators or a pure dark state with purely dissipative dynamics, the paper derives the exact identity $I_E(T)=I_G(T)-4\int_0^T\!d\tau_1\int_0^{\tau_1}\!d\tau_2\,C_{ZZ}(\tau_1+\tau_2)$. Using this formula, the high-temperature superradiant sensor gives $I_E^{\mathrm{opt}}\simeq 0.1912\,J(J+1)\,T^2/3$ for all three field directions, and the dissipative spin squeezer gives $I_E^{\mathrm{opt}}[\hat J_x]\simeq 0.191\,J(J+1)\,T^2/2$, both of order $N^2T^2$; for the spin squeezer the optimal measurement is direct photodetection.

Load-bearing premise

The load-bearing premise is the derivation of Eq. (10), which is proven only for a maximally mixed steady state with a self-adjoint dissipator or for a pure dark state with purely dissipative dynamics, and for the spin squeezer it also rests on a leading-order cumulant expansion and a large-$r$ approximation.

Editorial extensions

If this is right

  • Heisenberg scaling in continuous sensing becomes precise: for any collective generator $\hat Z=\hat J_{\vec r}$ and fixed $T$, $I_E^{\mathrm{opt}}(T)\le N^2T^2$, so an $N^2$ scaling cannot be an artifact of a slow sensor.
  • Optimizing the qubit–waveguide coupling of the high-temperature superradiant sensor at $\Gamma_\alpha\simeq 1.89/T$ yields $I_E^{\mathrm{opt}}\simeq 0.1912\,J(J+1)\,T^2/3$ for fields along $x$, $y$, or $z$.
  • The dissipative spin squeezer, at large squeezing $r$ and $\Gamma\simeq 1.89/T$, yields $I_E^{\mathrm{opt}}[\hat J_x]\simeq 0.191\,J(J+1)\,T^2/2$, a prefactor $3/2$ larger than the thermal sensor.
  • Any sensor hosting a pure dark state saturates its environmental QFI with direct photodetection of the emitted field; no coherent-absorber decoder is required.
  • The conventional sensitivity $S_Z$ is not bounded by $N$ and can display spurious super-Heisenberg scaling, so it is not a reliable certification of Heisenberg scaling.

Reading between the lines

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

  • A testable extension is to run the same fixed-$T$ optimization on the odd-$N$ spin squeezer, where the paper's numerics suggest $N^2$ scaling without spin squeezing; confirming that would show that environment–system entanglement alone can produce Heisenberg scaling.
  • If Eq. (10) holds beyond the two proven cases, the environmental QFI of a candidate sensor could be predicted directly from its measured noise spectrum, turning the metric into a practical screening tool.
  • The same resource accounting could be exported to waveform estimation or finite-bandwidth sensing, where the filter-function form of $I_E$ connects the optimal integration time to the bandwidth of the signal.
  • A curious open question is whether the exact identity is the leading term of a general relation between environmental QFI and noise spectra, with the maximally-mixed and pure-dark cases as endpoints.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. This paper addresses the definition of Heisenberg scaling in continuous quantum sensing. It proposes the optimized finite-time environmental QFI, I_E^opt(T) = max_{Γ_i} I_E({Γ_i};T), as a resource-correct figure of merit, argues that the asymptotic sensitivity S_Z is inadequate because it can diverge with N through slow sensor timescales, and proves a general bound I_E^opt(T) ≤ N^2 T^2 for collective generators, with a tightened filter-function bound 0.262 N^2 T^2 for the restricted class studied. Two sensors are analyzed: a high-temperature superradiant model, for which the environmental QFI is computed exactly and optimized, and a dissipative spin squeezer, for which a leading-order cumulant calculation yields the same N^2 scaling. The paper further claims that for the spin-squeezed sensor, direct photodetection of the cavity output saturates the QFI, avoiding the coherent-absorber decoder needed for the high-temperature sensor.

Significance. The proposed metric directly addresses a known ambiguity in continuous metrology, and the explicit pathological examples in the SM (Kac-factor scaling, uncorrelated-qubit series) make the case concretely. The high-temperature superradiant sensor is solved exactly, and the optimized result I_E^opt ≃ 0.1912 J(J+1)/3 T^2 is a useful benchmark. The bound I_E ≤ 0.262 N^2 T^2 is a clean, parameter-free statement. The dark-state photodetection result is practically significant: if valid, it removes the need for complex decoder networks in a broad class of dissipative sensors. The main unresolved issue is that the spin-squeezer half of the central claim rests on an incomplete derivation and an uncontrolled approximation, so the headline 'striking advantage' is not yet fully established. No computational code or data is provided for the numerical checks.

major comments (2)
  1. [Supplemental Material, Eq. (S59)] The identity that reduces the double integral ⟨Z e^{i H_eff† τ1} e^{-i H_eff τ2} Z⟩ to the symmetrized autocorrelator C_ZZ(τ1+τ2) (or an equivalent form) is left blank in Eq. (S59). This identity is the final step in deriving Eq. (S52), i.e. Eq. (10) of the main text, for the dissipative spin squeezer. Without it, the application of Eq. (10) to the spin squeezer, and hence Eq. (18) and the associated photodetection-optimality claim, are not established. Please supply the missing derivation.
  2. [Main text, Eqs. (17)-(18); SM Eq. (S66)] The exponential form C_JxJx(τ) ≈ 2⟨ΔJ_x^2⟩ e^{-2Γτ} is obtained from a leading-order cumulant expansion combined with a large-r stationary approximation, with no estimate of the neglected O(e^{-4r}) terms or of higher-order cumulants. The closed-form result Eq. (18) and the claimed N^2 scaling for the spin-squeezed sensor inherit this approximation. The numerical agreement cited in Fig. 1(c) is not independently verifiable because no code or data are included. Please provide either a rigorous error bound on the approximation, a derivation of the correlator beyond leading order, or the numerical data and code used for Fig. 1(c).
minor comments (5)
  1. [Main text, footnote [32]] The footnote contains the unresolved placeholder 'Eq. XXX'; it should refer to Eq. (2).
  2. [SM, 'Photodetection is optimal...' section] The cross-reference '(see Sec. )' is left blank; please fill in the correct section number.
  3. [Main text, paragraph after Eq. (17)] The comparison with full numerical simulation is attributed to 'Fig. 1b', but the corresponding simulation data appear in Fig. 1(c); Fig. 1(b) shows the high-temperature I_E(Γ) curve. Please correct the figure reference.
  4. [Fig. 2(a)] The label 'Inf.-temp. superradiance model' is inconsistent with the 'high-temperature' terminology used throughout the text.
  5. [Main text, Eq. (10)] The phrase 'completely pure' could be misread as any pure steady state; the SM correctly specifies that the derivation requires a dark state of a dissipation-only Liouvillian. Please make this condition explicit in the main text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the QFI formulas are derived from the two-sided master equation, the optimized coupling maximizes a closed-form expression rather than fitting data, and the central scaling claims do not reduce to their inputs by construction.

full rationale

The derivation chain is self-contained rather than circular. Equations (9), (10), (13), (14), and (18) are obtained from the two-sided master equation via a perturbative Dyson expansion and the quantum regression theorem, not assumed. The optimized coupling Gamma is the maximizer of the derived analytic IE(Gamma;T), not a fitted parameter renamed as a prediction, and the bound I_opt_E <= N^2 T^2 follows from IE <= IG and CZZ(0) <= N^2/2 rather than from the target scaling. Self-citations supply background results (hidden time-reversal symmetry, dissipative spin-squeezing steady states, large-r variance), but these are parameter-free published results and are not used to assume the paper's main QFI expressions; the explicit decoder construction in SM Eq. (S47) and the SM photodetection proof provide independent support. This pass did find a genuine rigor gap in the spin-squeezer leg: SM Eq. (S59), intended to reduce the no-jump correlator to CZZ(tau1+tau2), is blank, and Eq. (17) rests on a leading-order cumulant and large-r stationary approximation. That is an omitted derivation step and an approximation, not a circular reduction, so it should be weighed as a correctness risk rather than as circularity. The high-temperature superradiant sensor is exact and is not affected by that gap.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No physical entities are invented; the new object is a defined figure of merit, the optimized finite-time environmental QFI, which is a functional of the output state rather than a new force or particle. No data-fitting free parameters are used: the coupling Γ is the analytic maximizer of a derived expression, and the squeezing parameter r is a control parameter taken in a well-defined limit.

assumptions (6)
  • domain assumption The sensor and waveguides form a Markovian open system governed by the GKSL master equation, Eq. (2), with ideal waveguides whose full emitted state is measurable.
    This underlies the environmental QFI definition and the two-sided master equation in Eq. (3); non-Markovian or lossy detection would change the accessible information.
  • domain assumption The initial state is the unique dissipative steady state of the θ=0 Lindbladian, and the protocol is a sudden quench at t=0.
    All QFI expressions use this initial condition, and the resource accounting assumes no cost for the state preparation time.
  • domain assumption The generator Z is a permutation-symmetric collective spin operator, Z = J_r, with zero mean in the θ=0 steady state, and the dynamics are restricted to the maximum-J subspace.
    The N^2T^2 bound and the Heisenberg scaling statements depend on this collective-spin resource constraint.
  • ad hoc to paper Eq. (10) is derived only for two special classes: maximally mixed steady state with self-adjoint Lindbladian and Hermitian jump operators, or pure dark state with dissipation-only dynamics.
    The central formula for the environmental QFI in terms of CZZ is proven in SM Eqs. (S42) and (S58) under these restrictions, not for general continuous sensors.
  • ad hoc to paper For the dissipative spin squeezer, the autocorrelation CJxJx(τ) is evaluated with a leading-order cumulant expansion and a large-r stationary approximation, giving exponential decay with rate 2Γ.
    The analytic Heisenberg-scaling expression for the spin-squeezer, Eq. (18), depends on this approximation; the paper cites numerical agreement but does not provide an exact analytic solution.
  • domain assumption For the photodetection optimality proof, the sensor is initialized in a pure dark state, the parameter enters only through a Hamiltonian θZ, and the detection record is coarse-grained into no-photon versus some-photon outcomes.
    The saturation proof in the SM relies on these properties and on the binary distribution having only a quadratic θ-dependence to the relevant order.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Timescales, Squeezing and Heisenberg Scalings in Many-Body Continuous Sensing." pith.science (2026). https://pith.science/paper/BGZIG23T

@misc{pith2026250504591,
  author       = {Pith},
  title        = {Pith review of: Timescales, Squeezing and Heisenberg Scalings in Many-Body Continuous Sensing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BGZIG23T}},
  note         = {Machine review of arXiv:2505.04591}
}
read the original abstract

The continuous monitoring of driven-dissipative systems offers new avenues for quantum advantage in metrology. This approach mixes temporal and spatial correlations in a manner distinct from traditional metrology, leading to ambiguities in how one identifies Heisenberg scalings (e.g.~standard asymptotic metrics like the sensitivity are not bounded by system size). Here, we propose a new metric for continuous sensing, the optimized finite-time environmental quantum Fisher information (QFI), that remedies the above issues by simultaneously treating time and system size as finite resources. In addition to having direct experimental relevance, this quantity is rigorously bounded by both system size and integration time, allowing for a precise formulation of Heisenberg scaling. We also introduce two many-body continuous sensors: the high-temperature superradiant sensor, and the dissipative spin squeezer. Both exhibit Heisenberg scaling of a collective magnetic field for multiple directions. The spin squeezed sensor has a striking advantage over previously studied many-body continuous sensors: the optimal measurement achieving the full QFI does not require the construction of a complex decoder system, but can be achieved using direct photodetection of the cavity output field.

Figures

Figures reproduced from arXiv: 2505.04591 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quantum Synchronization

    quant-ph 2026-07 conditional novelty 2.0 of 10

    Quantum synchronization — rhythm-locking in open quantum systems — has matured into an experimentally demonstrated field spanning few-body oscillators, many-body phases, and time crystals, as documented in this compre...

Reference graph

Works this paper leans on

62 extracted references · 38 canonical work pages · cited by 1 Pith paper

  1. [1]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 96, 010401 (2006)

    2006

  2. [2]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330–1336 (2004)

    work page 2004

  3. [3]

    S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994)

    1994

  4. [4]

    M. G. A. PARIS, International Journal of Quantum Information 07, 125 (2009), https://doi.org/10.1142/S0219749909004839

    doi:10.1142/s0219749909004839 2009

  5. [5]

    C. W. Helstrom, Journal of Statistical Physics 1, 231–252 (1969)

    work page 1969

  6. [6]

    A. H. Kiilerich and K. Mølmer, Estimation of atomic interaction parameters by photon counting (2014), arXiv:1403.1192 [quant-ph]

    work page Pith review arXiv 2014

  7. [7]

    Catana, L

    C. Catana, L. Bouten, and M. Gut ¸˘ a, Journal of Physics A: Mathematical and Theoretical 48, 365301 (2015)

    work page 2015

  8. [8]

    Ilias, D

    T. Ilias, D. Yang, S. F. Huelga, and M. B. Plenio, PRX Quantum 3, 10.1103/prxquantum.3.010354 (2022)

    doi:10.1103/prxquantum.3.010354 2022

Show all 62 references
  1. [9]

    D. Yang, S. F. Huelga, and M. B. Plenio, Phys. Rev. X 13, 031012 (2023)

    2023

  2. [10]

    Godley and M

    A. Godley and M. Guta, Quantum 7, 973 (2023)

    2023

  3. [11]

    A. H. Kiilerich and K. Mølmer, Physical Review A 94, 10.1103/physreva.94.032103 (2016)

    2016 doi

  4. [12]

    J. A. Smiga and G. T. Landi, The role of correlations in a sequence of quantum observations on empirical measures (2024), arXiv:2411.08214 [quant-ph]

    2024 arXiv

  5. [13]

    J. A. Smiga, M. Radaelli, F. C. Binder, and G. T. Landi, Physical Review Research 5, 10.1103/physrevre- search.5.033150 (2023)

    2023 doi

  6. [14]

    Radaelli, J

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

    2024

  7. [15]

    Radaelli, G

    M. Radaelli, G. T. Landi, K. Modi, and F. C. Binder, New Journal of Physics 25, 053037 (2023)

    2023

  8. [16]

    Macieszczak, M

    K. Macieszczak, M. Gut ¸˘ a, I. Lesanovsky, and J. P. Garra- han, Physical Review A 93, 10.1103/physreva.93.022103 (2016)

    2016 doi

  9. [17]

    Fern´ andez-Lorenzo and D

    S. Fern´ andez-Lorenzo and D. Porras, Physical Review A 96, 10.1103/physreva.96.013817 (2017)

    2017 doi

  10. [19]

    Garbe, M

    L. Garbe, M. Bina, A. Keller, M. G. Paris, and S. Felicetti, Physical Review Letters 124, 10.1103/phys- revlett.124.120504 (2020)

    2020 doi

  11. [20]

    Di Candia, F

    R. Di Candia, F. Minganti, K. V. Petrovnin, G. S. Paraoanu, and S. Felicetti, npj Quantum Information 9, 10.1038/s41534-023-00690-z (2023)

    2023 doi

  12. [21]

    D. Yang, M. Ketkar, K. Audenaert, S. F. Huelga, and M. B. Plenio, Quantum cramer-rao precision limit of noisy continuous sensing (2025), arXiv:2504.12400 [quant-ph]

    2025 arXiv

  13. [22]

    Cabot, F

    A. Cabot, F. Carollo, and I. Lesanovsky, Quantum en- hanced parameter estimation with monitored quantum nonequilibrium systems using inefficient photo detection (2025), arXiv:2503.21753 [quant-ph]

    2025

  14. [23]

    A. Khan, F. Albarelli, and A. Datta, A tensor network approach to sensing quantum light-matter interactions (2025), arXiv:2504.12399 [quant-ph]

    2025

  15. [24]

    Gammelmark and K

    S. Gammelmark and K. Mølmer, Phys. Rev. A 87, 032115 (2013)

    2013

  16. [25]

    Tsang, Phys

    M. Tsang, Phys. Rev. Lett. 108, 170502 (2012)

    2012

  17. [26]

    Tsang, H

    M. Tsang, H. M. Wiseman, and C. M. Caves, Phys. Rev. Lett. 106, 090401 (2011)

    2011

  18. [27]

    J. W. Gardner, T. Gefen, S. A. Haine, J. J. Hope, and Y. Chen, Phys. Rev. Lett. 132, 130801 (2024)

    2024

  19. [28]

    J. W. Gardner, T. Gefen, S. A. Haine, J. J. Hope, J. Preskill, Y. Chen, and L. McCuller, Stochastic wave- form estimation at the fundamental quantum limit (2024), arXiv:2404.13867 [quant-ph]

    2024 arXiv

  20. [29]

    Gammelmark and K

    S. Gammelmark and K. Mølmer, Phys. Rev. Lett. 112, 170401 (2014)

    2014

  21. [30]

    [29] considered a more general situation where the Hamiltonian and jump operator may also be time-dependent

    Note however, that Ref. [29] considered a more general situation where the Hamiltonian and jump operator may also be time-dependent. 7

  22. [31]

    See Supplemental Material for the complete derivation of analytical results in the main text

  23. [32]

    We make the natural assumption that this is the steady state of Eq

    The initial condition corresponds to the initial state of the sensor qubits. We make the natural assumption that this is the steady state of Eq. XXX, i.e. before ˆH(θ) is turned on, the sensor is in the dissipative steady state determined by ˆH0 and the coupling to the waveguides

  24. [33]

    Cabot, F

    A. Cabot, F. Carollo, and I. Lesanovsky, Phys. Rev. Lett. 132, 050801 (2024)

    2024

  25. [34]

    For example, the sensor proposed in [33] has a steady state which hasN 2 variance in ˆJy,z, but does not exhibit a Heisenberg scaling along those axes

  26. [35]

    Stannigel, P

    K. Stannigel, P. Rabl, and P. Zoller, New Journal of Physics 14, 063014 (2012)

    2012

  27. [36]

    Roberts, A

    D. Roberts, A. Lingenfelter, and A. Clerk, PRX Quan- tum 2, 020336 (2021)

    2021

  28. [37]

    Iemini, R

    F. Iemini, R. Fazio, and A. Sanpera, Physical Review A 109, 10.1103/physreva.109.l050203 (2024)

    2024 doi

  29. [38]

    Iemini, A

    F. Iemini, A. Russomanno, J. Keeling, M. Schir` o, M. Dal- monte, and R. Fazio, Phys. Rev. Lett. 121, 035301 (2018)

    2018

  30. [39]

    Tsang, Quantum reversal: a general theory of coher- ent quantum absorbers (2024), arXiv:2402.02502 [quant- ph]

    M. Tsang, Quantum reversal: a general theory of coher- ent quantum absorbers (2024), arXiv:2402.02502 [quant- ph]

    2024 arXiv

  31. [40]

    Roberts and A

    D. Roberts and A. A. Clerk, Phys. Rev. X 10, 021022 (2020)

    2020

  32. [41]

    Roberts and A

    D. Roberts and A. A. Clerk, Phys. Rev. Lett.130, 063601 (2023)

    2023

  33. [42]

    M. Yao, A. Lingenfelter, R. Belyansky, D. Roberts, and A. A. Clerk, Phys. Rev. Lett. 134, 130404 (2025)

    2025

  34. [43]

    Roberts and A

    D. Roberts and A. A. Clerk, Phys. Rev. Lett.131, 190403 (2023)

    2023

  35. [44]

    Lingenfelter, M

    A. Lingenfelter, M. Yao, A. Pocklington, Y.-X. Wang, A. Irfan, W. Pfaff, and A. A. Clerk, Phys. Rev. X 14, 021028 (2024)

    2024

  36. [45]

    Groszkowski, M

    P. Groszkowski, M. Koppenh¨ ofer, H.-K. Lau, and A. A. Clerk, Phys. Rev. X 12, 011015 (2022)

    2022

  37. [46]

    G. S. Agarwal and R. R. Puri, Phys. Rev. A 41, 3782 (1990)

    1990

  38. [47]

    E. G. Dalla Torre, J. Otterbach, E. Demler, V. Vuletic, and M. D. Lukin, Physical Review Letters 110, 10.1103/physrevlett.110.120402 (2013)

    2013 doi

  39. [48]

    In contrast, N odd does not yield a pure steady state [45]. Nevertheless, numerical results suggest that N odd also yields a finite-time environmental QFI that goes like N 2, although this is far less useful than the N even case, see [31] for further comments

  40. [49]

    Finally, we note that in the large r limit, we may obtain similar results if one in- stead tried to sense a z magnetic field, i.e

    holds for any system that hosts a pure dark state, and hence holds for N even. Finally, we note that in the large r limit, we may obtain similar results if one in- stead tried to sense a z magnetic field, i.e. for the choice ˆZ = ˆJz: one has Heisenberg-limited scaling, and th...

    2020

  41. [50]

    This is implicit in the decoder construction of [9]; we provide a complementary proof in [31]

  42. [51]

    A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010)

    2010

  43. [52]

    Midha and S

    S. Midha and S. Gopalakrishnan, Metrology of open quantum systems from emitted radiation (2025), arXiv:2504.13815 [quant-ph]

    2025 arXiv

  44. [53]

    J. L. Daletskii and S. Krein, AMS Translations 2, (47)1 (1965)

    1965

  45. [54]

    Carlsson, Perturbation theory for the matrix square root and matrix modulus (2018), arXiv:1810.01464 [math.FA]

    M. Carlsson, Perturbation theory for the matrix square root and matrix modulus (2018), arXiv:1810.01464 [math.FA]

    2018 arXiv

  46. [55]

    J. J. Sakurai and J. Napolitano, Modern Quantum Me- chanics (Cambridge University Press, 2021)

    2021

  47. [56]

    C. W. Gardiner, Phys. Rev. Lett. 70, 2269 (1993)

    1993

  48. [57]

    H. J. Carmichael, Phys. Rev. Lett. 70, 2273 (1993)

    1993

  49. [58]

    R. Kubo, J. Phys. Soc. Jpn. 17, 1100 (1962)

    1962

  50. [59]

    Timescales and Heisenberg Scalings in Many-Body Continuous Sensing

    A. Pocklington and A. A. Clerk, Phys. Rev. Lett. 134, 050603 (2025). 1 Supplemental Material: “Timescales and Heisenberg Scalings in Many-Body Continuous Sensing” METROLOGY AS THE DUAL OF NOISE Continuous sensing and qubit dephasing In this appendix, we perform the mapping of ...

    2025

  51. [60]

    The steady state is the maximally mixed state

  52. [61]

    completely dephasing

    The Lindbladian is self-adjoint. Beyond the high-temperature model, highly mixed steady states are highly common; at any rate for any continuous sensing protocol, one may put in any steady state, including the maximally mixed state. Using just assumption (1), we first arrive a...

  53. [62]

    The steady state is a pure dark state

  54. [63]

    The system is dissipation only. The first fact is interesting for two reasons: First, there are large classes of Lindbladians with pure dark states, and second, for any such sensor, the optimal measurement is simply photodetection on the output field (see Sec. ). Hence, we wil...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.