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Optimal asymptotic precision bounds for nonlinear quantum metrology under collective dephasing

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Under collective dephasing, a squeezed one-axis-twisted state with an echo readout reaches the same N^-3/2 quadratic frequency-estimation precision as a maximally entangled GHZ state.

desk verdict The CSS scaling and the QFI machinery are solid, but the N^{-3/2} optimality claim is an acknowledged conjecture sitting at the edge of the HP approximation, so the abstract oversells it. read the letter →

arxiv 2501.00189 v1 pith:LN7E7PIU submitted 2024-12-30 quant-ph

Francisco Riberi , Lorenza Viola This is my paper · ORCID
classification quant-ph
keywords quantummetrologynonlinearcollectivedephasingone-axis-twistedstatesFisherinformationHolstein-PrimakoffmappingfrequencyestimationZenoregime
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 asks how much precision a quadratic (nonlinear) frequency sensor can retain when all N qubit probes suffer identical classical dephasing, with arbitrary temporal correlations in the noise. It establishes that under Markovian dephasing every state considered — product, squeezed, or maximally entangled — is capped at the linear Heisenberg scaling $N^{-1}$, so entanglement provides no advantage. With temporally correlated noise in the short-time Zeno regime, a product coherent spin state already reaches $N^{-5}$/4, while a properly squeezed one-axis-twisted state followed by an anti-squeezing echo readout reaches $N^{-3}$/2, matching a generalized GHZ state. The paper proves this last bound is achievable and argues, through a conjecture deferred to the conclusion, that $N^{-3}$/2 is asymptotically optimal. It also constructs a two-observable ratio estimator that removes noise-induced bias at twice the measurement cost without sacrificing the quantum Cramér-Rao precision.

What carries the argument

The load-bearing object is the family $\mathcal K_{\hat z}$ of Gaussian (properly squeezed) one-axis-twisted states $|{\rm OATS}\rangle=e^{-i\beta J_z}e^{-i\mu J_x^2}|{\rm CSS}\rangle$ in the Holstein-Primakoff low-excitation regime, where $J_+\simeq\sqrt{2J}\,a$, $J_z=J-a^\dagger a$, and the noisy Hamiltonian becomes a driven oscillator with quadratic position coupling $H\simeq b(\sin\theta\,\hat x+J\cos\theta)^2+(\sin\theta\,\hat x+J\cos\theta)\xi(t)$. In this representation the Wigner function remains Gaussian, the symmetric logarithmic derivative is exactly solvable as a polynomial at most quadratic in $\hat x,\hat p$, and the quantum Fisher information splits as $F_Q=F_Q^{(A)}+F_Q^{(B)}$ with a vanishing cross term. The leading contribution is controlled by the effective squeezing parameter $\delta=\cos^2\beta+(1+4J^2\mu^2)\sin^2\beta+2J\mu\sin(2\beta)$, and the associated optimal POVM is the anti-squeezed (echo) observable $e^{i\eta J_x^2/2}J_y e^{-i\eta J_x^2/2}$, which avoids single-particle resolution.

What would settle it

Simulate the full N-qubit spin dynamics without the Holstein-Primakoff approximation for the perfect-echo state $\mu=(2J)^{-1/2}$, $\beta=-\pi/2$ under collective dephasing in the Zeno regime, optimizing over the encoding time near $\tau_{\rm opt}=1/(2\kappa_0\omega_c)|\csc\theta_{\rm opt}|(N\delta/2)^{-1/2}$ with $\theta_{\rm opt}=\arccos\sqrt{2/3}$; if the time-optimized variance scales worse than $N^{-3/2}$ (for instance $N^{-5/4}$) as $N$ grows, the central claim collapses. A cold-atom implementation measuring $J_y$ after the anti-squeezing echo would provide the same check experimentally.

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Extended reading notes

Core claim

By mapping the spin dynamics to a bosonic mode through the Holstein-Primakoff transformation and solving the symmetric-logarithmic-derivative equation in phase space, the paper obtains an exact asymptotic quantum Fisher information for properly squeezed one-axis-twisted states under collective dephasing. For the optimal signal direction $\theta_{\rm opt}=\arccos\sqrt{2/3}$ and Zeno-regime noise $\kappa(t)\simeq\kappa_0^2(\omega_c t)^2$, the time-optimized precision is $\Delta\hat b_{\rm opt}^{\rm OATS}\propto\delta^{-1/4}N^{-5/4}$, where $\delta$ is the effective quadrature-squeezing parameter. At the perfect-echo parameters $\mu=(2J)^{-1/2}$, $\beta=-\pi/2$, one has $\delta=2J$, which pushes the scaling to $\Delta\hat b_{\rm opt}^{\rm PE}\propto N^{-3/2}$, the same exponent as the maximally entangled $|\Phi\rangle$ state. The saturating readout is the interaction-based echo observable $O=e^{i\eta J_x^2/2}J_y e^{-i\eta J_x^2/2}$, corresponding to the leading part of the SLD; the cross term with the quadratic part vanishes exactly for these states. The claimed asymptotic optimality of the $N^{-3/2}$ bound is stated in Sec. V as a conjecture, with the matching upper bound expected to follow from techniques used in the linear metrology setting.

Load-bearing premise

The load-bearing premise is that the Holstein-Primakoff low-excitation mapping stays quantitatively faithful for the perfect-echo one-axis-twisted parameters, even though Eq. (D15) shows the mean excitation number is already of order N and the paper itself places that state at the fringe of the mapping's validity.

Editorial extensions

If this is right

  • If the N^-3/2 bound is asymptotically optimal, quadratic encoding under collective dephasing offers a super-Heisenberg advantage over the linear N^-1 limit that is accessible with experimentally available squeezed states rather than fragile GHZ states.
  • Under Markovian collective dephasing, none of the considered input states beats N^-1; any precision enhancement in that regime would have to come from reducing noise correlations, not from input entanglement or squeezing.
  • The state-independent optimal angle $\theta_{\rm opt}=\arccos\sqrt{2/3}$ gives a concrete tuning prescription for generalized Ramsey protocols with quadratic encoding.
  • The ratio estimator yields an asymptotically unbiased frequency estimate under dephasing at the cost of doubling the measurement resources, with no asymptotic loss in precision relative to the quantum Cramér-Rao bound.
  • Together the achievability result and the optimality conjecture imply the generalized no-go $N^{-(k-p/2)}$ for k-th order nonlinearities under p-body collective dephasing, so interactions can beat the linear Heisenberg limit but cannot beat the generalized classical bound.

Reading between the lines

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

  • If the deferred optimality proof is completed, N^-3/2 becomes a universal ceiling for quadratic encoding under one-body collective dephasing; a natural next test is whether two-axis countertwisting or higher-order squeezing can enter a different excitation regime and change the exponent.
  • Because the perfect-echo state sits at the fringe of the Holstein-Primakoff regime, exact finite-N spin simulations could determine whether the N^-3/2 exponent survives outside the Gaussian approximation or merely the prefactor changes.
  • The ratio-estimator construction should extend to entangled one-axis-twisted inputs through a second-order cumulant expansion, making bias-free estimation available for the same states that achieve the N^-3/2 scaling.
  • The results suggest a resource hierarchy for nonlinear metrology: temporal noise correlations are what unlocks sub-Heisenberg scaling, while squeezing substitutes for entanglement as the state resource.
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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

3 major / 6 minor

Summary. The paper analyzes frequency estimation with quadratic signal encoding under classical collective dephasing with arbitrary temporal correlations, for N qubits described by H(t)=b J_z^2 + J_z ξ(t). It derives quantum Fisher information bounds for product coherent spin states and for a family of properly squeezed one-axis-twisted states (OATS), using a Holstein-Primakoff mapping and phase-space methods. The main scaling results are: Markovian collective dephasing limits all considered Gaussian states to N^{-1}; in the non-Markovian Zeno regime, CSS inputs reach N^{-5/4} and a perfect-echo OATS with an interaction-based readout reaches N^{-3/2}, the same scaling as a maximally entangled generalized GHZ (Phi) state. The paper also constructs a two-observable ratio estimator that is asymptotically unbiased under dephasing and claims that it reaches the same precision as the standard estimator. The abstract states that the N^{-3/2} scaling is 'proved asymptotically optimal.'

Significance. If the central claims are fully established, this is a valuable contribution to nonlinear quantum metrology under realistic dephasing. The analytic SLD and QFI derivation for Gaussian OATS in Sec. III B and Appendix D is detailed and self-contained, with the noiseless and exact Phi-state limits recovered consistently, and the numerical checks in Figs. 2 and 3 support the CSS scalings. The ratio estimator is a practically useful construction that addresses a real bias problem. However, the headline optimality result rests on two load-bearing points that are not yet established: the controlled validity of the Holstein-Primakoff approximation at the perfect-echo working point, and the matching upper bound that would justify the word 'optimal.'

major comments (3)
  1. [Sec. III B 1 and Appendix D, Eq. (D15)] The achievability of the N^{-3/2} scaling for the perfect-echo OATS relies on the Holstein-Primakoff Hamiltonian in Eq. (8), but the paper's own Eq. (D15) gives <a-dagger a> ≈ J/2 at t=0 for the PE parameters μ=(2J)^{-1/2}, β=-π/2, i.e. the excitation number is of order N rather than ≪J. The manuscript itself states in Sec. III B that this state 'lies at the fringe of the HP regime of applicability.' The replacement sqrt(1-a-dagger a/(2J))→1 drops corrections of order unity in J_z, so the effective bosonic Hamiltonian is not a controlled approximation at the exact working point used for the headline exponent. The illustrative phase-space plot in Fig. 1 uses N=10 and cannot validate the asymptotic scaling. I ask for a quantitative assessment, for example exact finite-J simulation of the spin dynamics for the PE protocol up to moderate N, or a rigorous bound on the error induced by the HP truncation in the QFI; without this, the N^{-3/2} achievability claim is not fully supported.
  2. [Abstract and Sec. V] The abstract claims that the N^{-3/2} scaling 'we prove is asymptotically optimal,' but Sec. V explicitly states 'we do not pursue a formal proof here' and frames the matching no-go as a conjecture based on ideas from Ref. [28], which is itself 'in preparation.' What the paper actually proves is that the PE OATS attains the same exponent as the Phi state. That is an equality with a known state, not an upper bound over all strategies. Please either supply the missing optimality proof or revise the abstract and Sec. V to state that N^{-3/2} is attained by the PE OATS and is conjectured, not proved, to be asymptotically optimal.
  3. [Sec. III B 2, Eq. (20)] The QFI optimization is performed using only the term F_Q^(A) in Eq. (20), and the text in Appendix D states that F_Q^(B) can be neglected for α<1/2, while for α=1/2 both terms scale equally. The perfect-echo OATS is precisely the boundary case α=1/2, so the claim that dropping F_Q^(B) 'does not otherwise change the main conclusions' needs an explicit demonstration at the PE parameters. This does not affect the claimed exponent, but it affects the constants in Table I and the statement that the readout in Eq. (22) saturates the QCRB.
minor comments (6)
  1. [Eq. (1)] The notation 'Ju,u ∈ {u,y,z}' contains a typo; it should read J_u with u ∈ {x,y,z}.
  2. [Sec. II A] The sentence 'In the limit of temporally uncorrelated, Markovian noise is spectrum is flat' should read 'the noise spectrum is flat.'
  3. [Fig. 1 caption] 'inducediffusion along the Jy-direction' is missing a space and should read 'induces diffusion.'
  4. [Sec. III B 1] The set K_hat{z} of properly squeezed states is used before being formally defined; please give an explicit definition in the main text rather than only in Appendix D.
  5. [Sec. IV B] The extension of the ratio estimator to OATS states is described only as 'conceptually straightforward' and is not derived. Since the abstract and Sec. I state that the ratio estimator works 'without detriment to the achievable precision,' the scope of that claim should be clarified: a derivation or a caveat is needed for the entangled OATS case.
  6. [Ref. [28]] The optimality conjecture for the no-go bound relies on Ref. [28], which is listed as 'in preparation.' For a journal submission, the status of this reference should be made explicit, or the conjecture should be stated without depending on an unpublished work.

Circularity Check

1 steps flagged · score 4.0 of 10

Achievability derivation is self-contained, but the advertised 'asymptotically optimal' proof is deferred to the authors' own in-preparation no-go conjecture, leaving the optimality half of the headline reliant on self-citation.

  1. self citation load bearing [Abstract; Sec. V (Conclusion), paragraph on no-go conjecture; Ref. [28]]
    "The use of nonclassical spin-squeezed states and a nonlinear readout further allows for an N−3/2 precision scaling, which we prove is asymptotically optimal. ... While we do not pursue a formal proof here, we expect it may be obtained using ideas and techniques similar to those employed for the linear case28."

    The abstract's optimality claim is not derived in the paper: Sec. V explicitly disclaims a formal proof and delegates the matching upper bound to a conjecture based on 'ideas and techniques similar to those employed for the linear case [28]', where Ref. [28] is the same authors' in-preparation work. Thus the 'asymptotically optimal' part of the headline rests on an unverified self-citation rather than on a derivation contained in this manuscript. The achievability half (PE OATS reaching N^{-3/2}) is independently computed from the QFI in the HP approximation, which is why the circularity score is not higher.

full rationale

The main derivations are not circular: the QFI for the Gaussian OATS is computed analytically from the noisy Wigner function and SLD equation, and the N^{-3/2} scaling is benchmarked against the exactly solvable Phi state, with no parameters fitted to data. The choice of PE OATS parameters (mu, beta, theta) is an optimization over state and readout variables, not a fit to the target result. The HP-approximation caveat is a correctness risk rather than circularity: Eq. (D15) gives <a-dagger a> ~ J/2 for the PE parameters, and the paper itself says the state 'lies at the fringe of the HP regime of applicability.' The one load-bearing circularity-type issue is the optimality claim: the abstract says the N^{-3/2} scaling 'we prove is asymptotically optimal,' but Sec. V says 'we do not pursue a formal proof here' and delegates the no-go to the authors' own in-preparation linear-metrology work [28]. Because the achievability derivation has independent content and only the optimality half reduces to self-citation, a score of 4 is appropriate.

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

No parameters are fitted to data; the noise constants (gamma, kappa0, omega_c) are environmental inputs, and state parameters (mu, beta, theta) are optimization variables. The only new mathematical object is the SLD solution in Eqs. (18)-(21), which is derived from the dynamics. No new physical entities are postulated. The main caveat is that the claimed optimality of N^(-3/2) rests on a conjectured no-go, not on a derived upper bound.

assumptions (4)
  • domain assumption The dephasing environment is classical, Gaussian, zero-mean, and stationary, with known two-time correlation C(t) and spectrum S(omega).
    Sec. IIA models noise as a real stochastic process xi(t) in Eq. (1); the cumulant expansion truncates at second order, giving the dephasing factor kappa(t) in Eq. (2).
  • domain assumption In the Zeno regime the noise spectrum is cutoff above omega_c so that kappa(t) approximately kappa0^2 (omega_c t)^2 for omega_c t much less than 1.
    Introduced in Sec. IIA and used throughout Sec. III to derive the N^(-5/4) and N^(-3/2) scalings; depends on S(omega) vanishing for |omega| greater than about omega_c.
  • domain assumption The Holstein-Primakoff low-excitation approximation a-dagger a/(2J) much less than 1 remains valid for the optimized OATS parameters and encoding times.
    Sec. IIIB1 and Eq. (D15) restrict initial states to K_z; the PE OATS achieving N^(-3/2) has <a-dagger a> about J/2, at the boundary of validity, as acknowledged in Sec. IIIB.
  • standard math The symmetric logarithmic derivative for the Gaussian states of interest is a polynomial at most quadratic in quadratures x and p.
    Appendix D4 solves the SLD equation by posing this ansatz; it is standard for Gaussian states and is checked term-by-term against the quadratic Wigner function.

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Pith. "Pith review of Optimal asymptotic precision bounds for nonlinear quantum metrology under collective dephasing." pith.science (2026). https://pith.science/paper/LN7E7PIU

@misc{pith2026250100189,
  author       = {Pith},
  title        = {Pith review of: Optimal asymptotic precision bounds for nonlinear quantum metrology under collective dephasing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LN7E7PIU}},
  note         = {Machine review of arXiv:2501.00189}
}
abstract

Interactions among sensors can provide, in addition to entanglement, an important resource for boosting the precision in quantum estimation protocols. Dephasing noise, however, remains a leading source of decoherence in state-of-the-art quantum sensing platforms. We analyze the impact of classical {\em collective dephasing with arbitrary temporal correlations} on the performance of generalized Ramsey interferometry protocols with \emph{quadratic} encoding of a target frequency parameter. The optimal asymptotic precision bounds are derived for both product coherent spin states and for a class of experimentally relevant entangled spin-squeezed states of $N$ qubit sensors. While, as in linear metrology, entanglement offers no advantage if the noise is Markovian, a precision scaling of $N^{-1}$ is reachable with classical input states in the quadratic setting, which is improved to $N^{-5/4}$ when temporal correlations are present and the Zeno regime is accessible. The use of nonclassical spin-squeezed states and a nonlinear readout further allows for an $N^{-3/2}$ precision scaling, which we prove is asymptotically optimal. We also show how to counter {\em noise-induced bias} by introducing a simple ratio estimator which relies on detecting two suitable system observables, and show that it remains asymptotically unbiased in the presence of dephasing, without detriment to the achievable precision.

Figures

Figures reproduced from arXiv: 2501.00189 by the authors.

Figure 1
Figure 1. Importantly, however, restrictions must be imposed to ensure that the low-excitation condition is obeyed throughout the time regime of interest. As we explicitly show by express￾ing the mean excitation number ⟨a †a⟩ in phase-space repre￾sentation [see in particular Eq. (D15)], the subset of states Kzˆ ⊂ Gzˆ that are adequately described by Eq. (8) must have [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

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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. Higher-order noise statistics restore Heisenberg scaling under collective dephasing

    quant-ph 2026-07 accept novelty 7.5 of 10

    At fixed single-atom T2, finite-rate compound-Poisson collective dephasing saturates the GHZ decoherence rate and restores Heisenberg scaling, while Gaussian diffusion is the exact worst case.

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Works this paper leans on

15 extracted references · 14 canonical work pages · cited by 1 Pith paper

  1. [1]

    Holstein-Primakoff description of the dynamics While important on theoretical grounds, computing the asymptotic performance of the Φ state is of little practical value, as the generation of N-partite entangled GHZ states for large N remains very challenging. However, the non- classical OATSs, which include CSS as a particular limit, are accessible in curr...

    work page

  2. [2]

    one-body

    Quantum Fisher information and optimal precision scaling Crucially, in addition to providing a useful visualization, the phase-space representation enables for an exact solution of the SLD operator that determines the QFI. While again we present the full details of the calculation elsewhere (Appendix D 4), the salient aspects may be summarized as follows....

    work page

  3. [3]

    The bosonic Josephson junction BECs have been heavily studied as a system for implementing precision interferometry2,38,43. At sufficiently low temperature, the condensate is well described by the single-particle wavefunction ψ(r,t), which obeys the Gross-Pitaevskii equation, i ∂ ∂t ψ(r,t) = − 1 2m ∇2 +Vext(r) +4πas m |ψ(r,t)|2 ψ(r,t), where Vext(r,t) is ...

    work page

  4. [4]

    A recent proposal 47 makes use of two bosonic modes, a probe P and an ancilla A, to achieve this

    Towards nonlinear metrology with conditional operations As stressed in the main text, it would be desirable to generate a quadratic sensing Hamiltonian which allows estimation of an external field parameter, as opposed to some intrinsic property of the sensors themselves. A recent proposal 47 makes use of two bosonic modes, a probe P and an ancilla A, to ...

    work page

  5. [5]

    (1) in the main text with k = 2, we consider the set C ˆn(θ , φ ) of all CSS

    CSS evolution under quadratic dynamics Given the Hamiltonian in Eq. (1) in the main text with k = 2, we consider the set C ˆn(θ , φ ) of all CSS. While our first aim is to re-derive known precision bounds for the noiseless setting, for later use we show how to obtain expectation values of relevant 12 observables in the case where dephasing noise is also p...

    work page

  6. [6]

    We can compute the QFI of any CSS |θ , φ ⟩ as FQ[ρCSS(τ)] =4ν(⟨J4 z (τ)⟩ − ⟨J2 z (τ)⟩2) =4ν∆J2 z (τ)

    Noiseless CSS asymptotic precision Let us now show assume that ξ (t) ≡ 0 and show that the generalized SQL scaling for quadratic encoding, ∆ˆb ∝ J−3/2, can be saturated by a member of the set C ˆn(θ , φ ). We can compute the QFI of any CSS |θ , φ ⟩ as FQ[ρCSS(τ)] =4ν(⟨J4 z (τ)⟩ − ⟨J2 z (τ)⟩2) =4ν∆J2 z (τ). The mean values ⟨J2 z (τ)⟩, ⟨J4 z (τ)⟩ in the lim...

    work page

  7. [7]

    (B13) This coincides with Eq

    Noiseless asymptotic precision of Φ states If we consider the generalized GHZ state Φ defined in the main text, the corresponding QFI may be easily obtained in terms of the variance of J2 z , FQ[ρΦ(τ)] =4ν ∆(J2 z )2(τ) =4ν ⟨J4 z (τ)⟩ − ⟨J2 z (τ)⟩2 = νJ4, and leads, via the QCRB, to the following bound to the precision: ∆ˆb(τ) ≥ 1√ T τ J2 . (B13) This coin...

    work page

  8. [8]

    Evaluating uncertainty Eq

    Noisy CSS asymptotic precision The performance of the noiseless optimal strategy using CSS states may be also exactly assessed in the presence of collective dephasing, κ(τ) ̸= 0. Evaluating uncertainty Eq. (B12) for θ = π/4 and again working at b0 = 0, we find: ∆ˆb(τ) =2J sinh(2κ(τ)) +e2κ(τ) + cosh(2κ(τ)) T τ J3 . (C1) 14 Importantly, the presence of deca...

    work page

Show all 15 references
  1. [9]

    Noisy asymptotic precision of Φ states Despite the fact that the generalized GHZ state |Φ⟩ is entangled, its time-evolved density operator remains effectively one of a two-level system in the presence of collective dephasing; explicitly, ¯ρΦ(t) =1 2 |J,J⟩ˆz ˆz⟨J,J| + eiJ2bte−J...

  2. [10]

    Here, for convenience, we take said axis to be along the ˆx direction, ˜H(t) =bJ 2 x + Jxξ (t) ≡ ˜Hˆx(t)

    Equivalence between fixed Hamiltonian and fixed initial state settings In our original statement of the problem, we are interested in quantifying the sensing performance of a certain class of spin squeezed states, to be specified below, evolving under the action of a noisy qua...

  3. [11]

    Here we present a brief summary of its most salient features, as useful in our context

    Quantum mechanics in phase space The phase-space formulation provides an equivalent description of quantum mechanics, alternative to the usual one relying on operators acting on a Hilbert space. Here we present a brief summary of its most salient features, as useful in our con...

  4. [12]

    Computation of the Wigner function As a starting point, let us write the Wigner function of the CSS in the HP approximation. Since, as noted in the main text, |CSS⟩ˆz is the ground state of H = a†a in the low excitation limit, we can write state |0⟩ in the position representat...

  5. [13]

    We focus on sensing small phase space displacements, ϕ = δbτ ≪ 1, at the operating point b0 = 0

    Ultimate precision bounds We are now in a position to compute the SLD, and consequently derive the precision limits achievable by members of the set of properly squeezed Gaussian states in the HP limit. We focus on sensing small phase space displacements, ϕ = δbτ ≪ 1, at the o...

  6. [14]

    (B12), with the relevant expectation values, Eqs

    QCRB performance Our starting point is the uncertainty in Eq. (B12), with the relevant expectation values, Eqs. (26)-(27) and Eqs. (B3)-(B4), evaluated for φ = 0 as well as b0 = 0: ∆ˆbCSS(τ)2 = e2κ(τ) + 2J − sin(θ )2 sinh(2κ(τ)) 8J3T τ cos(θ )2 sin(θ )2 . (E1) The goal is to o...

  7. [15]

    Quantum sensing,

    Ratio estimator performance Replacing the relevant mean values, Eqs. (26)-(27), and Eqs. (B3)-(B4), into the squared uncertainty associated to the ratio estimator, Eq. (28), by taking b0 = 0, φ = 0, we find ∆ˆbCSS R (τ)2 = e2κ(τ) cos(θ )2 + cosh(2κ(τ))sin(θ )2 + 2J sin(θ )2 si...

    2017 arXiv

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