Optical phase estimation via homodyne measurement in the presence of saturation effect of photodetectors

Jialin Li; Lupei Qin; Xin-Qi Li; Yazhi Niu

REVIEW 3 major objections 5 minor 20 references

Optical phase estimation via homodyne measurement in the presence of saturation effect of photodetectors

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read Optical phase can still be estimated from saturated photodetectors by inverting each detector's saturation curve before taking the current difference.

desk verdict Clean bias-correction idea, but the precision formula drops the shot-noise variance and is unsupportable. read the letter →

arxiv 2501.06768 v1 pith:KPWBQFSR submitted 2025-01-12 quant-ph

Jialin Li , Yazhi Niu , Lupei Qin , Xin-Qi Li This is my paper · ORCID

classification quant-ph

keywords opticalphaseestimationhomodynemeasurementphotodetectorsaturationnonlineardetectorresponsecoherentstatesstandardquantumlimitinversefunctionpostselection

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 extends optical phase estimation by homodyne measurement from the usual linear detector response to the nonlinear, saturated regime. For coherent light it derives the average detector current $I_j=I_{\max}(1-e^{-N_j/N_{\rm sat}})$ and shows that applying the inverse function $F$ to the two detector currents gives $F(I_2)-F(I_1)=2|\alpha\beta|\sin(\chi-\phi)$, so the unknown phase $\chi$ can still be extracted when the detectors are saturated. The accompanying precision formula scales as $1/\sqrt{MN}$, i.e., the standard quantum limit. This matters because proposed high-intensity laser interferometers for relativistic-gravity effects push photon numbers into the saturation territory of photodetectors, where the standard linear protocol would give a wrong phase.

What carries the argument

The load-bearing object is the saturating exponential response $\mu_j(n)=k_{\max}(1-e^{-n/N_{\rm sat}})$ for the average number of photoelectrons produced by $n$ incident photons. Its key property is that averaging over the Poisson photon-number distribution of a coherent field preserves the exponential form: $\langle\mu_j(n)\rangle=\mu_j(\langle n\rangle)$, yielding the current $I_j=I_{\max}(1-e^{-N_j/\tilde N_{\rm sat}})$. The inverse function $F(I_j)$ then converts measured currents back into mean photon numbers, reducing the saturated homodyne problem to the same difference identity $N_2-N_1=2|\alpha\beta|\sin(\chi-\phi)$ that is used in the linear regime.

What would settle it

Take a calibrated photodetector, send coherent pulses of known mean photon number $N$ spanning from well below to well above $N_{\rm sat}$, and record the average current; the paper's Eq. (5) predicts $I=I_{\max}(1-e^{-N/\tilde N_{\rm sat}})$, so any systematic deviation from that curve at fixed $N$ would falsify the saturation model that Eq. (6) relies on.

Watch

Extended reading notes

Core claim

The central discovery is that the saturation nonlinearity can be removed by a deterministic inversion. Starting from the model $\mu_j(n)=k_{\max}^{(j)}(1-e^{-n/N_{\rm sat}^{(j)}})$ for the average photoelectron number and averaging it over the Poisson photon statistics of a coherent state, the paper obtains the same exponential form for the current, $I_j=I_{\max}^{(j)}(1-e^{-N_j/\tilde N_{\rm sat}^{(j)}})$, with $\tilde N_{\rm sat}^{(j)}\simeq N_{\rm sat}^{(j)}$. Defining $F$ as the inverse of this current--photon-number relation, the phase is read from $F(I_2)-F(I_1)=2|\alpha\beta|\sin(\chi-\phi)$. The precision of this readout, including shot-to-shot detector fluctuations and $M$ repeated measurements, is $\delta\chi\sim 1/\sqrt{MN}$---the same standard quantum limit as the linear-regime measurement---so the generalization does not cost precision, provided the saturation curve is known and the detector is not oversaturated.

Load-bearing premise

The argument stands or falls on the assumption that a photodetector's average photoelectron count follows exactly the saturating exponential $\mu_j(n)=k_{\max}(1-e^{-n/N_{\rm sat}})$ and that $N_{\rm sat}$ and $k_{\max}$ are known precisely when the measured current is inverted.

Editorial extensions

If this is right

For mean photon numbers up to a few times $N_{\rm sat}$, the inverse-response protocol $F(I_2)-F(I_1)$ recovers the phase $\chi$ that the standard linear protocol would systematically misestimate.
The precision of the saturated-regime estimate remains $\sim 1/\sqrt{MN}$, the standard quantum limit, rather than degrading with saturation, as long as the response curve and its parameters are known.
Once both detectors are fully saturated, the currents carry no phase information and no inversion can recover $\chi$; the paper identifies this oversaturation regime as an in-principle limitation.
To operate high-intensity lasers without entering oversaturation, the paper points to postselection as a way to keep photodetectors below the saturation threshold.
The derivation relies on coherent-state Poisson statistics, so the simple inversion works for the coherent signal states used throughout the paper.

Reading between the lines

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

Beyond the paper, the same inversion strategy should apply to any monotone detector nonlinearity, not only the exponential saturation model; the essential requirement is that the current remains an invertible function of the mean photon number.
The paper does not quantify how errors in $N_{\rm sat}$ or $k_{\max}$ propagate into the phase estimate; a natural extension is to compute the sensitivity $\partial\chi/\partial N_{\rm sat}$ and design a calibration procedure.
A testable prediction is that, for a fixed phase $\chi$, the phase extracted with $F$ should stay flat as the input intensity is swept through saturation, whereas the linear protocol shows a growing bias; this could be checked with an independently calibrated detector.
For non-coherent states the simplification $\langle\mu_j(n)\rangle=\mu_j(\langle n\rangle)$ generally fails, so extending the method to squeezed or thermal light would require a different averaging; the present result is specific to coherent states.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know. First, the paper's average-current formula for saturating detectors, Eq. (5), is correct: Poisson-averaging the saturation model from Harris et al. gives I = Imax(1 - exp(-N/Nsat)), and inverting that to recover the phase bias is valid if the model and its parameters are exact. Second, the precision result, Eq. (8), is wrong. The paper claims the photoelectron variance is just sigma_j^2, but the law of total variance gives Var(k_j) = sigma_j^2 + Var_n[mu_j(n)], and the second term is the shot-noise contribution from Poisson photon statistics. Dropping it makes Eq. (8) predict zero phase uncertainty for an ideal detector, which contradicts balanced homodyne detection's 1/sqrt(N) limit. This isn't a minor correction; the whole point of Eq. (8) is to quantify the precision after saturation. So the paper's central new claim fails.

What is actually good: the derivation of Eq. (5) is clean and the inversion idea in Eq. (6) is a legitimate way to correct bias when the response is known. The numerical example in Table I illustrates the saturation effect on the average current clearly. The paper is readable and the self-reported lack of comparison with real detector response curves is acknowledged implicitly by the model assumption.

Soft spots in proportion: the variance error is the load-bearing flaw, and it is fatal for Eq. (8). The secondary issue is that the whole inversion strategy assumes exact knowledge of N_sat and I_max; the paper doesn't quantify sensitivity or compare with measured curves. That would matter even if the variance were fixed. The false variance equality is not a matter of interpretation; it's an elementary probability error.

Who is this for? Someone working on high-intensity homodyne phase estimation who wants to know whether saturation breaks the standard protocol. The bias-correction part could be useful if the noise were handled correctly, but as written the paper doesn't deliver a consistent precision statement. I wouldn't cite it. For peer review: I would desk reject it in this form; the error is decisive and the remaining content is not substantial enough to warrant referee time. If the authors correct Eq. (8) by including the shot-noise term, the bias-correction part could be a short note.

Referee Report

3 major / 5 minor

Summary. The paper studies optical phase estimation via homodyne detection when photodetectors exhibit saturation. It models the mean photoelectron count with μ_j(n)=k_max(1-e^{-n/N_sat}), derives the average current I_j=I_max(1-e^{-N_j/\tilde N_sat}) in Eq. (5), and introduces the inverse function F so that F(I_2)-F(I_1)=N_2-N_1, enabling phase extraction in the nonlinear regime via Eq. (6). The authors then use error propagation to derive a phase-uncertainty formula Eq. (8) and claim it scales as 1/√(MN) with photon number and measurement repetitions. They illustrate the bias of the standard linear protocol when the detector saturates and argue that the inverse-function correction improves the phase estimate.

Significance. The mean-current calculation leading to Eq. (5) is straightforward and correct, and the idea of undoing the saturation nonlinearity by applying the inverse of the assumed response to the measured currents is sensible. However, the paper's precision analysis is invalid: the variance of k_j is not σ_j², as claimed, because the photon-number fluctuations contribute through the law of total variance. This missing term is the usual shot noise, and its omission produces an unphysical formula that would give zero phase uncertainty for an ideal detector. The claimed 1/√(MN) scaling is therefore not established. The conceptual contribution of the bias correction is limited to the exact validity of the saturation model, and the quantitative results need substantial revision.

major comments (3)

[Eq. (8) and the variance derivation following Eq. (7)] The variance calculation leading to Eq. (8) is incorrect. From the mixture distribution Eq. (4), the law of total variance gives Var(k_j) = E_n[Var(k_j|n)] + Var_n(E[k_j|n]) = σ_j² + Var_n[μ_j(n)]. In the linear regime N_j ≪ N_sat, this second term equals (k_max/N_sat)^2 N_j, which is precisely the photon-number shot noise responsible for the standard quantum limit. The text's claim that δ²k_j = σ_j² omits this term; consequently Eq. (8) predicts zero phase uncertainty for a noiseless detector (σ=0) and does not reproduce the SQL. Eq. (8) and the subsequent scaling claim δχ ∼ 1/√(MN) are therefore mathematically unsupported.
[Eq. (6) and the saturation model of Eq. (3)] The inversion formula Eq. (6) is definitional: F is constructed as the inverse of the assumed mean-current relation Eq. (5), so F(I_2)-F(I_1)=N_2-N_1 holds by construction. The method's practical utility depends on the saturation model Eq. (3) being exact and on knowing N_sat and k_max precisely. The manuscript does not analyze how parameter uncertainty or model misspecification propagates into the phase estimate, nor does it compare the model with measured detector response data. Without such an analysis, the claimed 'improved estimation' in the saturation regime is not quantitatively supported beyond the idealized model.
[Final paragraph and the scaling claim after Eq. (8)] Even accepting the derivation of Eq. (8), the claimed SQL scaling is not obtained: in the linear response regime N_j ≪ N_sat, Eq. (8) simplifies to δχ ∝ σ/(|αβ|), which depends on the local-oscillator amplitude |β| and can be made arbitrarily small by increasing the LO power while keeping σ fixed. This is not the standard quantum limit, which for balanced homodyne detection should be independent of the LO amplitude once shot noise is included. The omitted term Var(μ_j(n)) is required to cancel this |β| dependence and recover the genuine 1/√N scaling.

minor comments (5)

[After Eq. (5)] The equality ⟨μ_j(n)⟩ = μ_j(⟨n⟩) is stated without qualification; it is only approximate, because ⟨μ_j(n)⟩ = k_max(1 - e^{-N_j/\tilde N_sat}) while μ_j(⟨n⟩) = k_max(1 - e^{-N_j/N_sat}), and the two expressions coincide only in the limit N_sat → ∞. Please state this explicitly.
[Equation (5) and the following text] The notation for the saturation parameter is confusing: the paper uses N_sat, \tilde N_sat, and the textual abbreviation ~N_sat at different points. Since \tilde N_sat = (1-e^{-1/N_sat})^{-1} differs from N_sat, the text should clearly indicate whenever the approximation \tilde N_sat ≈ N_sat is being used.
[References] Reference [4] contains a typo: 'adn' should be 'and'.
[Figure 2] In Fig. 2, the estimator for χ~ is not defined; the authors should specify how the estimated phase is obtained from the measured currents in both the linear and nonlinear protocols.
[After Eq. (5)] The phrase 'This relation is not obvious at all in priori' should read 'a priori'.

Circularity Check

1 steps flagged · score 6.0 of 10

The phase-extraction formula is the inverse of the assumed saturation response by construction, so the 'improved estimation' is not an independent prediction; the precision formula also has a separate mathematical error.

self definitional [Equation (6), following Eq. (5)]
"Based on Eq. (5), we can introduce an inverse function F and reexpress Eq. (5) as N_j = F (I_j ). Then we have F (I_2) − F (I_1) = N_2 − N_1 = 2|αβ| sin(χ − ϕ) ."

F is introduced as the inverse of the assumed saturation curve I_j(N_j)=I_max(1-exp(-N_j/N_sat)), so Eq. (6) is the identity F(I(N))=N combined with the beam-splitter relation N2-N1=2|αβ|sin(χ-φ). The phase-extraction claim is therefore guaranteed by the definition of F once the saturation model of Eq. (3) is accepted; it is not a new physical prediction and contains no independent validation of the saturation curve. The paper presents this transformation as the central generalization, but the result reduces by construction to the input response model.

full rationale

The main derivation is self-contained algebra from an externally cited saturation model, not a self-citation chain. However, the central result Eq. (6) is definitional: F is defined as the inverse of Eq. (5), so F(I2)-F(I1)=N2-N1 holds identically and cannot be tested independently of the assumed μ_j(n)=k_max(1-e^{-n/N_sat}). The 'improvement' over linear response is a restatement of invertibility, not an empirical prediction. I also note a separate, non-circular correctness issue: the paper claims δ²k_j=σ²_j from the mixture ~P_j(k), but by the law of total variance Var(k_j)=σ²_j+Var_n[μ_j(n)], so Eq. (8) drops the Poissonian shot-noise contribution; for ideal detectors it would predict zero phase uncertainty, contradicting the SQL. This error does not itself make the derivation circular. The only self-citation, Ref. [13], is used in the summary to recommend postselection for oversaturation and is not load-bearing for Eqs. (5)-(8).

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central derivation rests on an assumed exponential saturation response (Eq. 3) and a Gaussian conditional noise model (Eq. 2), both imported from Refs. [8,9]. The recovery of the phase is the inverse of this assumed response, so the improvement is contingent on the model being exact and the parameters being known. No new physical entity is introduced.

free parameters (4)

N_sat = 10^17 photons (chosen for the numerical example)
Saturation threshold in Eq. (3); the inverse F(I) and all illustrative results depend on this value.
k_max/N_sat = 0.1 (chosen)
Opto-electric conversion ratio used in Table I and Fig. 2.
τ_w = 0.1 ms (chosen)
Detector response window that sets the current scale I=e k_max/τ_w.
σ_j = not specified
Variance of the conditional photoelectron response in Eq. (2); it controls the claimed precision in Eq. (8) but is never assigned or measured.

assumptions (6)

domain assumption Gaussian conditional response P_j(k|n) with variance σ_j²
Eq. (2), following Refs. [8,9]; the precision formula depends on this noise model.
domain assumption Exponential saturation response μ_j(n)=k_max(1-e^{-n/N_sat})
Eq. (3), imported from Refs. [8,9]; all subsequent results are consequences of this assumed detector nonlinearity.
standard math Poissonian photon statistics for coherent states
Used in Eq. (4) to average μ_j(n) over the incident photon number distribution.
standard math Beam-splitter transformation for homodyne mixing
Used before Eq. (1) to relate detector fields to signal and local oscillator amplitudes.
domain assumption Identical detectors with equal Imax, σ, and Ñsat
Assumed before Eq. (8) to simplify the precision formula; real detectors would have parameter asymmetry.
standard math First-order error propagation
Used in Eq. (7) to convert current fluctuations into phase uncertainty; valid only for small fluctuations.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Optical phase estimation via homodyne measurement in the presence of saturation effect of photodetectors." pith.science (2026). https://pith.science/paper/KPWBQFSR

@misc{pith2026250106768,
  author       = {Pith},
  title        = {Pith review of: Optical phase estimation via homodyne measurement in the presence of saturation effect of photodetectors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KPWBQFSR}},
  note         = {Machine review of arXiv:2501.06768}
}

read the original abstract

For optical phase estimation via homodyne measurement, we generalize the theory from detector's linear to nonlinear response regime, which accounts for the presence of saturation effect. For optical coherent light, we carry out analytic expressions for detector's current and estimate precision. Using specific device parameters, we illustrate the improved estimation after accounting for the saturation effect.

Figures

Figures reproduced from arXiv: 2501.06768 by the authors.

**Figure 2.** FIG. 2: Error ratio [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

20 extracted references · 19 canonical work pages

[1]

Helstrom, Quantum Detection and Estimation Theory (Academic, New Y ork, 1976)

C.W. Helstrom, Quantum Detection and Estimation Theory (Academic, New Y ork, 1976)

work page 1976
[2]

The electric currents in D1 and D2 are proportional to the incident photon numbers, which can be denoted as I1 = rN1 and I2 = rN2, with r the opto-electric conversion constant

The intensities of the two opti- cal pulses are characterized by the mean photon numbers N1 = |~α 1|2 and N2 = |~α 2|2. The electric currents in D1 and D2 are proportional to the incident photon numbers, which can be denoted as I1 = rN1 and I2 = rN2, with r the opto-electric conversion constant. From the differenc e of I2 and I1, we can obtain δ ˜I = I2 −...

work page
[3]

1, and |β |2 = 10 15

01, Nsat = 10 17, kmax/N sat = 0. 1, and |β |2 = 10 15. approximated by N (j) sat . Then, comparing Eqs. (3) and (5), we have ⟨µ j(n)⟩ = µ j(⟨n⟩). Here ⟨·⟩means an average over the distribution function Pj (n). This relation is not obvious at all in priori. Based on Eq. (5), we can introduce an inverse function F and reexpress Eq. (5) as Nj = F (Ij ). The...

work page
[4]

01 0 . 01 0 . 16 0 . 01 0 . 16

work page
[5]

1 0 . 1 1 . 60 0 . 1 1 . 53 1 1 16 . 02 0 . 63 10 . 13 2 2 32 . 04 0 . 87 13 . 85 3 3 48 . 07 0 . 95 15 . 22 Below, using speciﬁc parameters consistent with the es- timation in Refs. [5–7], we illustrate the effect of satura- tion and the improved estimation of χ in the nonlinear re- sponse regime, by applying the above results of Eqs. (5) and (6). Let us...

work page
[6]

Quantum Phase and Phase Dependent Measurements , edited by W. P . Schleich and S.M. Barnett [Phys. Scr. T48, 13 (1993)]

work page 1993
[7]

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

2009
[8]

Giovannetti, S

V . Giovannetti, S. Lloyd, adn L. Maccone, Quantum- enhanced measurements: Beating the standard quantum 4 limit, Science 306, 1330 (2004)

work page 2004

Show all 20 references

[9]

S. P . Kish and T. C. Ralph, Quantum-limited measure- ment of space-time curvature with scaling beyond the con- ventional Heisenberg limit , Phys. Rev. A 96, 041801(R) (2017)

2017
[10]

Sabin, Quantum detection of wormholes, Sci.Rep

C. Sabin, Quantum detection of wormholes, Sci.Rep. 7, 716 (2017)

2017
[11]

Sanchidrian-V aca and C

C. Sanchidrian-V aca and C. Sabin, Parameter esti- mation of wormholes beyond the Heisenberg limit , arXiv:1811.02395

arXiv
[12]

Harris, R

J. Harris, R. W. Boyd,and J. S. Lundeen, W eak value am- pliﬁcation can outperform conventional measurement in the presence of detector saturation , Phys. Rev. Lett. 118, 070802 (2017)

2017
[13]

L. Xu, Z. Liu, A. Datta, G. C. Knee, J. S. Lundeen, Y . Lu, and L. Zhang, Approaching quantum-limited metrology with imperfect detectors by using weak-value ampliﬁcation , Phys. Rev. Lett. 125, 080501 (2020)

2020
[14]

Aharonov, D

Y . Aharonov, D. Z. Albert, and L. V aidman,How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100 , Phys. Rev. Lett. 60, 1351 (1988)

1988
[15]

A. N. Jordan, J. Martinez-Rincon, and J. C. Howell, Tech- nical advantages for weak-value ampliﬁcation: when less is more, Phys. Rev. X 4, 011031 (2014)

2014
[16]

J. Ren, L. Qin, W. Feng, and X. Q. Li, W eak- value-ampliﬁcation analysis beyond the Aharonov-Albert- V aidman limit, Phys. Rev. A 102, 042601 (2020)

2020
[17]

J. Li, Y . Niu, L. Qin, and X. Q. Li, Postselected am- pliﬁcation applied to Mach-Zehnder-interferometer for phase shift measurement of optical coherent states , arXiv:2411.16334v1

arXiv
[18]

D. R. M. Arvidsson-Shukur, N. Y . Halpern, H. Lepage, A. A. Lasek, C. H. W. Barnes, and S. Lloyd, Quantum ad- vantage in postselected metrology, Nat. Commun. 11, 3775 (2020)

2020
[19]

Lupu-Gladstein, Y

N. Lupu-Gladstein, Y . B. Yilmaz, D. R. M. Arvidsson- Shukur, A. Brodutch, A. O. T. Pang, A. M. Steinberg, and N. Y . Halpern, Negative Quasiprobabilities Enhance Phase Estimation in Quantum-Optics Experiment , Phys. Rev. Lett. 128, 220504 (2022)

2022
[20]

Yang, Theory of Compression Channels for Post-selected Quantum Metrology, arXiv:2311.06679v2

J. Yang, Theory of Compression Channels for Post-selected Quantum Metrology, arXiv:2311.06679v2. ..PRL 2024

2024 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Helstrom, Quantum Detection and Estimation Theory (Academic, New Y ork, 1976)

C.W. Helstrom, Quantum Detection and Estimation Theory (Academic, New Y ork, 1976)

work page 1976

[2] [2]

The electric currents in D1 and D2 are proportional to the incident photon numbers, which can be denoted as I1 = rN1 and I2 = rN2, with r the opto-electric conversion constant

The intensities of the two opti- cal pulses are characterized by the mean photon numbers N1 = |~α 1|2 and N2 = |~α 2|2. The electric currents in D1 and D2 are proportional to the incident photon numbers, which can be denoted as I1 = rN1 and I2 = rN2, with r the opto-electric conversion constant. From the differenc e of I2 and I1, we can obtain δ ˜I = I2 −...

work page

[3] [3]

1, and |β |2 = 10 15

01, Nsat = 10 17, kmax/N sat = 0. 1, and |β |2 = 10 15. approximated by N (j) sat . Then, comparing Eqs. (3) and (5), we have ⟨µ j(n)⟩ = µ j(⟨n⟩). Here ⟨·⟩means an average over the distribution function Pj (n). This relation is not obvious at all in priori. Based on Eq. (5), we can introduce an inverse function F and reexpress Eq. (5) as Nj = F (Ij ). The...

work page

[4] [4]

01 0 . 01 0 . 16 0 . 01 0 . 16

work page

[5] [5]

1 0 . 1 1 . 60 0 . 1 1 . 53 1 1 16 . 02 0 . 63 10 . 13 2 2 32 . 04 0 . 87 13 . 85 3 3 48 . 07 0 . 95 15 . 22 Below, using speciﬁc parameters consistent with the es- timation in Refs. [5–7], we illustrate the effect of satura- tion and the improved estimation of χ in the nonlinear re- sponse regime, by applying the above results of Eqs. (5) and (6). Let us...

work page

[6] [6]

Quantum Phase and Phase Dependent Measurements , edited by W. P . Schleich and S.M. Barnett [Phys. Scr. T48, 13 (1993)]

work page 1993

[7] [7]

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

2009

[8] [8]

Giovannetti, S

V . Giovannetti, S. Lloyd, adn L. Maccone, Quantum- enhanced measurements: Beating the standard quantum 4 limit, Science 306, 1330 (2004)

work page 2004