Postselected amplification applied to Mach-Zehnder-interferometer for phase shift measurement of optical coherent states

J.L. Li; L.P. Qin; X.Q. Li; Y.Z. Niu

REVIEW 4 major objections 6 minor 1 cited by

Postselected amplification applied to Mach-Zehnder-interferometer for phase shift measurement of optical coherent states

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Postselected amplification of a small phase shift in a Mach-Zehnder interferometer is carried by a weak value while the dark-port output remains a single coherent state, buying technical robustness rather than a precision-scaling advantage.

desk verdict The central claim survives—postselected phase amplification in a coherent-state MZI yields a single coherent state with AAV weak value—but the quantitative quadrature and sensitivity formulas have factor errors that must be fixed before the paper's performance conclusions can be trusted. read the letter →

arxiv 2411.16334 v1 pith:E6UQSN7N submitted 2024-11-25 quant-ph

J.L. Li , Y.Z. Niu , L.P. Qin , X.Q. Li This is my paper · ORCID

classification quant-ph PACS 03.65.Ta42.50.-p

keywords postselectedamplificationweak-valueMach-Zehnderinterferometercoherentstatephaseshiftestimationhomodynequadratureshot-noiselimitdetectorsaturation

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 proposes a postselected amplification scheme for measuring a small phase shift imprinted on one arm of a Mach-Zehnder interferometer through which a coherent-state laser passes. It argues that, although the two arms of the interferometer do not form a quantum superposition entangled with the optical field, the normally dark output port still carries an amplified phase shift $\tilde{\chi}_A = A_w \chi$ in the Aharonov-Albert-Vaidman limit, with $A_w = c_1/(c_1+c_2)$ playing the role of a weak value. Beyond that limit the paper supplies an exact expression, Eq. (10), for the amplified phase. The consequence that matters is that the postselected output remains a single coherent state, so the photon-number scaling of the precision stays at the shot-noise limit $1/\sqrt{N}$ rather than improving to $1/N$. A reader should care because the scheme cleanly separates postselection-based amplification from entangled-meter metrology and identifies a practical regime in which postselection improves robustness to detector saturation and local-oscillator phase error without sacrificing shot-noise-limited sensitivity.

What carries the argument

The load-bearing object is the product-state transformation of the coherent field through the interferometer, Eq. (3), together with the recombination formula for the dark-port amplitude $\alpha_f$. Defining $c_1 = \cos\theta_2/\sqrt{2}$ and $c_2 = -e^{i\gamma}\sin\theta_2/\sqrt{2}$ lets the paper rewrite $\alpha_f = (c_1 e^{i\chi}+c_2)\alpha$ and identify $A_w = c_1/(c_1+c_2) = \langle f|\hat{A}|i\rangle/\langle f|i\rangle$ with $\hat{A} = |1\rangle\langle 1|$; this is the Aharonov-Albert-Vaidman weak value, a named quantity that can be large when $c_1+c_2$ is small. The phase is read out by homodyne field-quadrature measurement, $X_\xi = \langle \alpha_f | \hat{X}_\xi | \alpha_f\rangle = \sqrt{I_f}\sin\tilde{\chi}$, with the local-oscillator phase chosen as $\xi = \pi/2 + \lambda$ for maximal sensitivity. The single-coherent-state form of $|\alpha_f\rangle$ is what lets the amplified phase appear as a rotation of the field amplitude, rather than as an interference effect between two coherent-state components.

What would settle it

Send a coherent state through the interferometer with known $\chi \sim 10^{-2}$, set $\gamma=0$, and scan $\theta_2$ while performing homodyne tomography on port 3. If the reconstructed Wigner function shows negativity or interference fringes, or if the measured quadrature variance departs from the coherent-state value $1/4$, the product-state assumption fails. Conversely, the claim that precision stays at $1/\sqrt{N}$ can be tested by checking whether the estimation error improves with photon number as $1/\sqrt{N}$ rather than $1/N$.

Watch

Extended reading notes

Core claim

The central discovery is that postselected amplification of a phase shift in a Mach-Zehnder interferometer works for optical coherent states even when the which-path degrees of freedom are not entangled with the optical field. The beam-splitter action is taken as $U_{BS_1}|\alpha;0\rangle = |\alpha\cos\theta_1; -i\alpha\sin\theta_1\rangle$, a product of two coherent states, and the dark-port output is a single coherent state $|\alpha_f\rangle$ with $\alpha_f = (c_1 e^{i\chi} + c_2)\alpha$. For small $\chi$, this gives the amplified phase $\tilde{\chi}_A = A_w \chi$ with $A_w = c_1/(c_1+c_2)$, the standard weak-value result, and beyond the small-$\chi$ limit the exact amplified phase is $\tilde{\chi}_B = \arctan\big((\sin\chi\cos\theta_2 - \sin\gamma\sin\theta_2)/(\cos\chi\cos\theta_2 - \cos\gamma\sin\theta_2)\big)$. Because the output is a coherent state rather than a superposition of two coherent states, the precision follows the $1/\sqrt{N}$ shot-noise scaling, and the practical benefit of postselection is technical: the amplified phase is easier to read out in the presence of local-oscillator phase error and photodetector saturation.

Load-bearing premise

The derivation assumes that after the first beam splitter the two interferometer arms are in a product state of two coherent states, so the dark-port output is one coherent state; if the input field were non-classical or the beam splitter entangled the path modes, the output would be a superposition and the phase-extraction and scaling analysis would break down.

Editorial extensions

If this is right

A small phase shift can be extracted from the dark-port homodyne signal as $\tilde{\chi}_A = A_w \chi$, making the readout less sensitive to local-oscillator phase error than a direct phase-shift measurement.
Because the output is a single coherent state, postselection cannot push the precision beyond the shot-noise limit; the scheme is a technical-robustness tool, not a way to beat standard quantum scaling.
The exact beyond-AAV formula Eq. (10) lets an experimenter calibrate the true phase $\chi$ from the measured amplified phase, extending the scheme to larger phase shifts.
Averaging over $M$ repeated pulses reduces the estimation uncertainty by $1/\sqrt{M}$, compensating for the photon loss caused by postselection and making the scheme practical for static parameter estimation.
The same interferometric setup can be used for measuring relativistic-gravity-induced phase shifts with the added benefit of postselection-based robustness.

Reading between the lines

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

This suggests a general rule: in any linear-optics interferometer whose beamsplitter leaves a product of coherent states, postselection cannot change the photon-number scaling; the entire benefit is improved tolerance to technical imperfections.
The paper does not compute Fisher information, but the single-coherent-state output implies the available information about $\chi$ at the optimal quadrature is proportional to the photon number $N$, so postselection appears to redistribute measurement resources over repeated runs rather than create new information.
A direct experiment could verify the saturation claim: with a bright input ($N$ well above $N_{\rm sat}$) and strong postselection, the estimation error should remain small even where the un-postselected homodyne currents would be saturated.
If the forthcoming nonlinear-coupling extension is made, the product-state output would still be a single coherent state, so any precision enhancement would have to originate in the nonlinear interaction itself rather than in postselection.

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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

Colleague,

The key claim of this paper is that postselection in an MZI with a coherent input leaves the dark-port output as a single coherent state—not a cat state—and that the AAV weak value emerges in the linearized limit. That claim survives reading. The exact beyond-AAV expression for the amplified phase (their Eq. 10) and the explicit demonstration that the photon-number scaling stays 1/√N are genuinely useful. This is a modest but legitimate contribution, not a breakthrough.

The derivation of α_f from the beam-splitter unitaries is correct. The identification of A_w = c1/(c1+c2) as the AAV weak value checks out. The numerical examples in Fig. 2 show the expected amplification. The technical-advantage sections on LO phase error and detector saturation are sensible; the saturation analysis with the exponential model is a nice addition, though the qualitative conclusion that postselection helps when detectors saturate is not new (see Refs. [21, 22, 26, 27]).

The soft spots are real but concentrated. The quadrature formulas (11) and (12) are missing a factor of √2: from Eq. (5), |α_f| = √(N/2)√(1−sin(2θ2)cosχ), not (√N/2) times that. And in Eq. (15), the factor [1−sin(2θ2)cosχ] should be its square root. These are not merely prefactor typos; they change the claimed sensitivity and the comparison with conventional homodyne detection. The phase-extraction ambiguity for large amplification (branching of arctan/arcsin) is not discussed, which could matter in practice. The abstract also has a confusing notation where path states and coherent states are both called |1⟩, |2⟩.

None of this undermines the central claim. The paper's quantitative conclusions cannot be trusted until the algebra is fixed, but the qualitative message—that postselection amplifies the phase while preserving coherent-state structure and shot-noise scaling—is sound.

I'd send it to a standard quantum-metrology referee. It deserves a careful review, and the authors should be asked to correct Eqs. (11), (12), and (15) and to discuss the phase-unwrapping issue. For my own work, I probably wouldn't cite it in the next year, but I would bring it to a reading group to illustrate the MZI versus WVA distinction.

Referee Report

4 major / 6 minor

Summary. The manuscript considers a Mach-Zehnder interferometer with an input coherent state and proposes a postselected phase-shift measurement at the dark port, where the output field remains a single coherent state whose phase is amplified. In the small-phase (AAV) limit the amplification coefficient is identified with the Aharonov-Albert-Vaidman weak value, and an exact expression for the amplified phase is given. The paper then analyzes homodyne quadrature detection of the dark-port field, including signal-to-noise ratios, averaging over repeated measurements, and technical advantages with respect to local-oscillator phase errors and photodetector saturation. The central conclusion is that the amplified phase can be extracted from a quadrature measurement with shot-noise-limited precision 1/√N, and that postselection offers technical, not fundamental, advantages in this coherent-state setting.

Significance. The core derivation (Eqs. (3)-(10)) is self-contained, and the observation that the dark-port output is a single coherent state rather than a cat-like superposition is a useful conceptual clarification of the difference between interferometric postselection and qubit-coupled weak-value amplification. The paper correctly avoids claiming any quantum-enhanced precision scaling and explicitly notes the distinction from entangled qubit-oscillator models. Its potential value lies in the technical-advantage analysis (tolerance to LO phase errors and detector saturation). However, the quantitative formulas for the quadrature signal and sensitivity contain algebraic errors that must be corrected before the performance claims can be accepted.

major comments (4)

[Eq. (11)] The quadrature amplitude in Eq. (11) is too small by a factor of 1/√2. From Eq. (5) with γ=0, |α_f| = sqrt(N/2)(cosθ2 − sinθ2) in the AAV-limit amplitude, so with ξ=π/2+λ the homodyne signal is X_ξ = sqrt(N/2)(cosθ2 − sinθ2) sin~χ_A. Equation (11) instead has √N/2, which is smaller by √2. This is not a convention issue: it is inconsistent with the authors' own Eq. (5) and with Eq. (14), which appears to have been derived from the correct amplitude.
[Eq. (12)] The same factor-of-1/√2 error appears in the beyond-AAV signal. From Eq. (5) with γ=0, |α_f| = sqrt(N/2) sqrt(1 − sin(2θ2) cosχ), so Eq. (12) should read X_ξ = sqrt(N/2) sqrt(1 − sin(2θ2) cosχ) sin~χ_B, not (√N/2) times the bracket. The printed form underestimates the signal amplitude by √2.
[Eq. (15)] Equation (15) is missing the square root on the interference factor. Error propagation from X = |α_f| sin~χ with δX = 1/2 gives δ~χ = 1/(2|α_f||cos~χ|), hence ~R_S/N = √(2N) sqrt(1 − sin(2θ2) cosχ) |cos~χ_B| ~χ_B. The printed [1 − sin(2θ2) cosχ] without the square root overestimates the sensitivity when the bracket is small; it is also inconsistent with Eq. (14), since in the AAV limit 1 − sin(2θ2) cosχ reduces to (cosθ2 − sinθ2)^2 and the square root is needed to recover Eq. (14).
[Eqs. (14)-(15)] The sensitivity is defined as ~χ/δ~χ, i.e., for the amplified phase. For estimating the physical phase χ, the relevant quantity is χ/δχ with δχ = δ~χ/|d~χ_B/dχ|. In the AAV limit these coincide, but for the beyond-AAV examples (e.g., Fig. 2 with χ=10^{-2}) the nonlinearity of ~χ_B(χ) means the reported ~R_S/N is not the estimation sensitivity for χ. The authors should either compute the derivative with respect to χ or explicitly state that they report the SNR of the amplified phase rather than of the true phase.

minor comments (6)

[Abstract and Introduction] The phrase 'optical coherent states (|1⟩ and |2⟩)' should read '(|α1⟩ and |α2⟩)'; the current notation is confusing and appears to be a typo.
[Eq. (7)] The expression for |f⟩ has an unbalanced parenthesis: '|f ⟩ = cosθ2|1⟩ +i sinθ2e−iγ |2⟩)/√2' should omit the extra closing parenthesis before the division.
[Below Eq. (7)] The statement that 'the phase shift χ is amplified as ~χ_A = A_w χ' holds only when c1+c2 is real (e.g., γ=0); for complex c1+c2 the phase of the output is not simply A_w χ. This condition should be stated explicitly.
[Fig. 3 caption] The word 'squeezing uncertainty' is misleading: a coherent state has no squeezing, and the reduction described is simply the usual averaging over repeated measurements.
[Eq. (12)] The subscript 'Byd' appears to be a typo for 'Beyond' or similar and should be standardized.
[Eq. (22)] The notation for the inverse sine is awkward; the argument should be written as X_ξ / [|α_f| (kmax/Nsat)] with clearer bracketing.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the amplified-phase relation is derived from explicit coherent-state propagation through the interferometer, with no fitted parameter or self-citation carrying the central claim.

full rationale

The paper's central claim, Eq. (8), is obtained by direct calculation, not by importing a result. Starting from the beam-splitter transformation Eq. (3), the authors apply the sequence of unitaries U_BS2 U_2 U_1 U_BS1 to the coherent state |α;0⟩ and obtain the output amplitudes α_f and α_fbar in Eq. (5). Equation (6) then expands e^{iχ} ≈ 1+iχ, defines c1 and c2 from the beam-splitter coefficients, and groups the result as (c1+c2)e^{iA_w χ}α with A_w = c1/(c1+c2). This is a mathematical rearrangement of the exact small-χ expansion, not an input assumed to equal the desired prediction. Equation (7) then identifies A_w with the Aharonov-Albert-Vaidman weak value for the projector |1⟩⟨1| and the explicitly given pre- and postselected path states; this identification is exact and does not smuggle in the result. Beyond the AAV limit, Eq. (10) provides the full nonlinear expression for the amplified phase, again by direct trigonometry from α_f, so the central claim is self-contained. The homodyne quadrature results in Eqs. (11)-(15) follow from the coherent-state expectation X_ξ = |α_f| cos(arg α_f - ξ) with the stated local-oscillator phase; possible algebraic prefactor issues are correctness concerns, not circularity. The self-citations to Refs. [16,17,22] are used for comparison with the qubit-coupled coherent-state model and for the general technical-advantage discussion, but the MZI-specific derivation does not depend on those papers' theorems or fitted values. No parameter is fitted to data, no uniqueness theorem is imported from the authors' prior work, and no empirical pattern is relabeled as a derivation. The paper is therefore not circular.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The theory introduces no new entities or fitted parameters. Its physical content is a standard linear-optics calculation, so the axiomatic load is low, but the quantitative SNR claims contain algebraic inconsistencies.

assumptions (3)

standard math Coherent states transform under a beam splitter as product states, with the unused port in the vacuum state.
Used in Eq (3) to write the state after BS1 as a product, which is the basis for the single-coherent-state output.
domain assumption The interferometer is lossless and single-mode, described by the scattering matrix in Eq (2).
Assumed in the unitary evolution of Eqs (1)-(5).
domain assumption Homodyne detection measures the field quadrature with shot noise governed by the coherent state statistics.
Used in Eqs (11)-(15) for the signal-to-noise and sensitivity analysis.

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Cite this review

Pith. "Pith review of Postselected amplification applied to Mach-Zehnder-interferometer for phase shift measurement of optical coherent states." pith.science (2026). https://pith.science/paper/E6UQSN7N

@misc{pith2026241116334,
  author       = {Pith},
  title        = {Pith review of: Postselected amplification applied to Mach-Zehnder-interferometer for phase shift measurement of optical coherent states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E6UQSN7N}},
  note         = {Machine review of arXiv:2411.16334}
}

abstract

We propose a postselected amplification (PSA) scheme for phase shift measurement of optical coherent states when passing through the Mach-Zehnder-interferometer (MZI). Different from the usual weak-value-amplification (WVA) formulation, the which-path states of the MZI ($\left| 1 \right\rangle $ and $\left| 2 \right\rangle $) cannot be described as sub-system states entangled with the optical coherent states ($\left| 1 \right\rangle $ and $\left| 2 \right\rangle $) separated by the beam-splitter. However, we obtain the same result of the usual WVA in the Aharonov-Albert-Vaidman (AAV) limit, but beyond this limit, the result is different, e.g., the photon-number scaling can be different. We explicitly carry out the amplified phase shift, which is extracted out from the field-quadrature measurement in the dark port of the MZI. We also evaluate the performance quality of the proposed scheme, and analyze the technical advantages by considering possible errors in the quadrature measurement.

Figures

Figures reproduced from arXiv: 2411.16334 by the authors.

**Figure 2.** FIG. 2: Phase shift amplification in the postselected photon [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** FIG. 4: Estimate error ratio, [PITH_FULL_IMAGE:figures/full_fig_p005_4.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

Optical phase estimation via homodyne measurement in the presence of saturation effect of photodetectors
quant-ph 2025-01 reject novelty 4.0 of 10

For coherent light, a saturating photodetector's average current is I=I_max(1-exp(-N/Ñ_sat)); inverting this relation recovers the optical phase in the nonlinear response regime.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Hosten and P

O. Hosten and P . G. Kwiat, Observation of the Spin Hall Effect of Light via weak measurements , Science 319, 787 (2008)

work page 2008

[2] [2]

Further, one can modulate the transmission- reﬂection coefﬁcient of BS2

If setting γ = 0 , i.e., not introducing the element of phase shifter in the MZI, then making θ2 =π/ 4 would result in port 3 being a completely dark port (no light is going out from this port). Further, one can modulate the transmission- reﬂection coefﬁcient of BS2. Making θ2 slightly deviate fromπ/ 4, one can realize the ampliﬁcation, since c1 +c2 is sm...

work page

[3] [3]

D. J. Starling, P . B. Dixon, A. N. Jordan, and J. C. Howell, Optimizing the signal-to-noise ratio of a beam-deﬂection measurement with interferometric weak values , Phys. Rev. A 80, 041803 (2009)

work page 2009

[4] [4]

D. J. Starling, P . B. Dixon, N. S.Williams, A. N. Jordan, and J. C. Howell, Continuous phase ampliﬁcation with a sagnac interferometer, Phys. Rev. A 82, 011802 (2010)

work page 2010

[5] [5]

P . B. Dixon, D. J. Starling, A. N. Jordan, and J. C. How- ell, Ultrasensitive beam deﬂection measurement via inter- ferometric weak value ampliﬁcation , Phys. Rev. Lett. 102, 173601 (2009)

work page 2009

[6] [6]

is valid in the so-called AA V limit. Beyond this limit, one can reexpress the result of α f as α f = |α f |ei(~χ B +λ ), (9) where |α f | = √ N 2 √ 1 − sin(2θ2) cos(χ −γ), ~χ B = arctan ( sinχ cosθ2 − sinγ sinθ2 cosχ cosθ2 − cosγ sinθ2 ) . (10) Just like ~χ A, ~χ B is the ampliﬁed phase shift. Once ~χ B is measured out, the true phase shift χ can be extr...

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