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Enhancement of non-Gaussianity and nonclassicality of pair coherent states with postselected von Neumann measurement

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that postselected von Neumann measurements with anomalous weak values enhance both non-Gaussianity and nonclassicality of pair coherent states, producing deeper squeezing, stronger entanglement, and more sub-Poissonian…

desk verdict A legitimate extension of the authors' weak-measurement state-engineering framework to pair coherent states, but the numerical evidence rests on unverified appendix formulas and a misdefined g^(2). read the letter →

arxiv 2412.12824 v2 pith:XGFKKJHT submitted 2024-12-17 quant-ph

Yi-Fang Ren , Janarbek Yuanbek , Yusuf Turek This is my paper · ORCID
classification quant-ph
keywords paircoherentstatespostselectedvonNeumannmeasurementweakvalueamplificationnon-GaussianitynonclassicalityquadratureandsumsqueezingHillery-ZubairyentanglementEPRcorrelation
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

Pair coherent states (PCS) are two-mode non-Gaussian states whose nonclassicality can be improved by photon addition or subtraction, but those operations are inefficient. This paper claims that a postselected von Neumann measurement on one mode achieves a comparable enhancement without adding or subtracting photons: with an anomalous weak value and moderate coupling, the output state shows deeper quadrature squeezing, newly appearing sum squeezing, more sub-Poissonian statistics, and stronger HZ and EPR entanglement. The authors also show that the measurement-enhanced state, used as a teleportation channel, keeps the average fidelity above the classical threshold for anomalous weak values, though not above the initial PCS fidelity. The practical interest is a state-optimization scheme based only on weak coupling and postselection, which they argue is easier to implement and has a non-negligible success probability.

What carries the argument

The load-bearing object is the displaced-superposition output state $|\Psi\rangle$ of Eq. (10): a PCS displaced by $+\Gamma/2$ and by $-\Gamma/2$, with weights set by the weak value $\langle\sigma_x\rangle_w=\tan(\alpha/2)$. This object carries the argument because every reported figure is an expectation value evaluated on it using the infinite sums in Appendix A (Eqs. A1–A13); in the weak-coupling limit those sums reduce the state to a PCS superposed with its single-photon-added and single-photon-subtracted versions, which is why the anomalous weak value can amplify squeezing, correlations, and Wigner negativity. The computation is closed by standard criteria: $Q_i$ for quadrature squeezing, $S_{ab}$ for sum squeezing, $g^{(2)}_{ab}$ and $g^{(2)}(0)$ for photon statistics, the HZ inequality, the EPR variance, and the VBK teleportation fidelity.

What would settle it

Truncate the PCS expansion at increasing $N$ and numerically evaluate the displaced-superposition state for small $\gamma$ and $\Gamma$; a persistent mismatch with Eqs. (A1)–(A13) would indicate an analytic error and would invalidate the figures. In addition, recompute $g_a^{(2)}(0)$ and $g_b^{(2)}(0)$ with the standard normalization $\langle a^{\dagger 2}a^2\rangle/\langle a^\dagger a\rangle^2$ and check whether the sub-Poissonian curves in Fig. 6 survive.

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Extended reading notes

Core claim

The central claim is that applying the postselected von Neumann measurement $H_{\rm int}=g\sigma_x\otimes P_x$ to the $a$-mode of a PCS, and postselecting the polarization pointer on $|H\rangle$, produces a normalized superposition $$|\Psi\rangle=\frac{\$\lambda$}{2}\left[(1+\langle\sigma_x\rangle_w)D\!\left(\frac{\Gamma}{2}\right)+(1-\langle\sigma_x\rangle_w)D^\dagger\!\left(\frac{\Gamma}{2}\right)\right]|\gamma,\delta\rangle,$$ whose nonclassicality exceeds that of the initial state. For weak coupling $\Gamma\ll1$ the output reduces to a superposition of the PCS, a single-photon-added PCS, and a single-photon-subtracted PCS, so the anomalous weak value acts as an effective photon operation. The paper reports that with parameters such as $\Gamma=0.3$, weak value $\langle\sigma_x\rangle_w\simeq5.761$, and suitable $\gamma$, the final state has deeper squeezing along the $F_2$ quadrature (about 19 percent more squeezing), nonzero sum squeezing, stronger sub-Poissonian statistics, and larger HZ and EPR correlations than the initial PCS, while the scaled joint Wigner function shows more negativity and interference structure. It also reports that teleportation through the enhanced channel remains successful for anomalous weak values in the weak-measurement regime, although the average fidelity does not beat the initial PCS channel.

Load-bearing premise

The numerical evidence for enhancement comes from analytic expectation values listed in Appendix A (Eqs. A1–A13) that are stated without derivation, and from photon-statistics formulas whose denominator appears to be missing a factor; if those expressions contain errors, the reported enhancement curves do not follow.

Editorial extensions

If this is right

  • For $\delta=0$ and $\Gamma=0.3$, the $F_2$ quadrature squeezing reaches about $-0.172$ near $\gamma=10$, a 19 percent improvement over the initial PCS's $-0.125$.
  • Sum squeezing, which is identically zero for the initial PCS, appears in a range of $\gamma$ for anomalous weak values and nonzero coupling, with the minimum near $\gamma\simeq0.5$ for larger $\Gamma$.
  • For PND $\delta\ge2$, both HZ correlation and EPR correlation become smaller (stronger entanglement) than the initial PCS in parameter regions set by $\gamma$ and $\alpha$.
  • In the weak-coupling regime, the final state approximates a superposition of PCS, photon-added PCS, and photon-subtracted PCS, so the scheme can prepare displaced Fock states and cat-like superpositions by postselecting on the $b$-mode.
  • With $\delta=0$ and $\gamma>0.9$, the VBK teleportation average fidelity through the enhanced PCS channel stays above 0.5 for all $\Gamma\le1$ and anomalous weak values, though it does not exceed the fidelity of the initial PCS channel.

Reading between the lines

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

  • If the Appendix sums check out, the same displaced-superposition mechanism should apply to other two-mode states such as two-mode squeezed vacuum; a conditional weak measurement on both modes could generate entangled superpositions of coherent states.
  • The reported gains are for a postselected subensemble; a full resource analysis that includes the postselection probability would show whether the enhancement is worthwhile for protocols, since large anomalous weak values are accompanied by lower success rates.
  • The weak-coupling reduction to a PCS plus photon-added and photon-subtracted terms suggests a concrete experimental test: compare the output state produced by the measurement with a heralded photon-added or photon-subtracted PCS to extract the effective gain.
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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 / 4 minor

Summary. The manuscript proposes a scheme to enhance the non-Gaussianity and nonclassicality of pair coherent states (PCS) by applying a postselected von Neumann measurement to one of the two modes. After deriving the exact output state in Eq. (10) and its normalization, the authors compute squeezing parameters, cross- and auto-correlation functions, Hillery-Zubairy and EPR correlations, the joint Wigner function, and teleportation fidelity, using the analytic expectation values listed in Appendix A. The central claim is that, for anomalous weak values and suitable coupling strengths, the output state possesses deeper squeezing, sum squeezing, more sub-Poissonian statistics, and stronger entanglement than the initial PCS. The paper also discusses the potential of the scheme for quantum state engineering without photon addition or subtraction.

Significance. If the numerical results are correct, the paper offers a new approach to enhancing the nonclassical properties of PCS that avoids photon addition/subtraction operations and may be relevant for quantum information tasks. The derivation of the exact output state (Eq. 10) is a clear strength, and the treatment of the measurement model is careful. However, the quantitative evidence for the enhancement claims is entirely based on the Appendix A expectation values, which are not derived and contain apparent summation-limit errors. The definitional error in the second-order correlation function further undermines the credibility of the reported photon statistics. These issues prevent independent verification of the central claim.

major comments (3)
  1. [§IV-B, Eqs. (29)-(30)] The second-order correlation function is defined with the denominator ⟨a†a⟩ instead of ⟨a†a⟩². For a coherent-like state with ⟨a†a⟩∼|γ|² at large γ, the printed definition would diverge, while Fig. 6 reports g^(2)→1. This error invalidates the sub-Poissonian statistics claim as presented; the definition should be corrected to g^(2)_a = ⟨a†²a²⟩/⟨a†a⟩² and similarly for mode b.
  2. [Appendix A, Eqs. (A1)-(A12)] Many of the infinite sums in this appendix contain terms with negative factorials at n=0, e.g., (n−1)! in P12, P22, P52, and (n−2)! in P42, K21, K22, without specifying that the sums start at n≥1 or n≥2. As written, these expressions are undefined. Since every quantitative result in Figures 3–8 and 12 is computed from these formulas, the central claim of enhancement cannot be verified. The authors must either derive these expressions with correct summation limits or provide a reproducible numerical implementation (e.g., code or a symbolic derivation in an appendix).
  3. [§VI, page 10] The statement that the initial PCS with δ=0 is Gaussian is incorrect; PCS are non-Gaussian two-mode states for all δ, including δ=0. This mischaracterizes the baseline state in the Wigner function comparison (Figs. 9 and 10) and the discussion of non-Gaussianity enhancement. The claim should be removed or corrected.
minor comments (4)
  1. [§III-A, Eq. (17)] The definition of F2 in Eq. (17) appears to have a sign error: both terms should have opposite signs (e−iϵ(a+b) − eiϵ(a†+b†)) to satisfy the commutation relation [F1,F2]=i/2. The subsequent formulas in Eqs. (20)–(21) are consistent with the correct definition, so this is likely a typo, but it should be fixed.
  2. [§IV-A, Fig. 5] The caption reports the weak value ⟨σx⟩w=5.761, while the text for the same figure states ⟨σx⟩w=5.671 (α=8π/9, tan(4π/9)=5.671). Please make the value consistent.
  3. [References] References [11] and [55] are duplicates (A. Gábris and G. S. Agarwal, same title and journal). Please remove the duplicate.
  4. [§II, Eq. (11)] The normalization coefficient λ in Eq. (11) is derived using Eqs. (13)–(14), but the definition of P is not motivated; consider showing the overlap calculation in a footnote for clarity.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the output state and all computed nonclassicality measures follow from the stated von Neumann interaction and postselection, with no fitted quantity or self-citation chain doing the work.

full rationale

The derivation chain is self-contained. The starting point is the PCS input state |γ,δ> of Eq. (1); the von Neumann interaction Hint = g σx ⊗ Px in Eq. (6) gives the joint evolution in Eq. (9); postselecting |ψf> = |H> yields the exact output state |Ψ> in Eq. (10), with normalization λ in Eq. (11) and the weak value ⟨σx⟩w in Eq. (12). All subsequent quantities—quadrature squeezing parameters Q1,Ψ and Q2,Ψ (Eqs. (20)-(21)), sum squeezing (Eq. (26)), correlation functions (Eqs. (27), (29)-(30)), HZ and EPR correlations (Eqs. (31), (33)), the joint Wigner function, fidelity, and teleportation fidelity—are computed from |Ψ> using the expectation values in Appendix A. No parameter is fitted to data, and no computed enhancement is used to define an input quantity; the enhancement claims are model-derived consequences for a postselected subensemble. The self-citations [41-43] and [60-61] are contextual references to prior weak-measurement and pointer-state techniques, not load-bearing proofs of the central enhancement result, and no uniqueness or ansatz is imported from them. The paper may have correctness or robustness issues—for instance, Appendix A contains sums with apparent negative factorials and the g^(2) definitions in Eqs. (29)-(30) appear dimensionally/numerically questionable—but those are errors or unverified computations, not circular reductions. The central claim therefore does not reduce by construction to its inputs.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The model uses standard quantum optics inputs: PCS, von Neumann interaction, and postselection. No new entities are introduced. The free parameters are the PCS amplitude γ, the coupling strength Γ, the PND δ, and the weak-value angle α; all are chosen by hand to exhibit the claimed enhancement, not fitted to data.

free parameters (5)
  • gamma = 0.5, 1.5, 10 (chosen for plots)
    PCS amplitude; the quantitative enhancement results depend on this choice, and the regime of weak coupling with |γ|<1 is invoked in the limiting-case discussion.
  • Gamma (coupling strength) = 0.3, 0.5, 0.7, 1
    Measurement strength; enhancements are shown for these selected values, with degradation at larger Γ in some figures.
  • delta (PND) = 0, 1, 2
    Photon number difference between modes; entanglement enhancement only appears for δ≥2, so the central claim is conditional on δ.
  • alpha (weak value angle) = 8π/9 (w≈5.671 or 5.761)
    Chosen to give a large anomalous weak value; the claimed enhancements require w>1, so this choice is load-bearing.
  • vartheta = 0
    Set to zero to make the weak value real; the imaginary part is not explored.
assumptions (4)
  • standard math The evolution operator for the von Neumann interaction is exp(-i∫Hdt) = 1/2[(I+σx)D(Γ/2)+(I-σx)D†(Γ/2)]
    Standard result for this Hamiltonian; used to derive Eq. (9).
  • domain assumption The weak value defined by pre- and post-selection is valid for arbitrary coupling strength Γ, not only for weak coupling
    The paper claims this citing Refs [60,61]; it underpins the exact treatment for all Γ.
  • ad hoc to paper The expectation values of operators under the output state are given by the infinite sums in Appendix A
    These expressions are stated without derivation and are the numerical engine of the paper; a typo in g^(2) suggests other expressions could be flawed.
  • standard math The pair coherent state has zero first-order moments ⟨a⟩=⟨b⟩=0 for real γ
    Used implicitly in simplifying displaced-state expectation values; follows from the twin-Fock structure of PCS.

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Cite this review

Pith. "Pith review of Enhancement of non-Gaussianity and nonclassicality of pair coherent states with postselected von Neumann measurement." pith.science (2026). https://pith.science/paper/XGFKKJHT

@misc{pith2026241212824,
  author       = {Pith},
  title        = {Pith review of: Enhancement of non-Gaussianity and nonclassicality of pair coherent states with postselected von Neumann measurement},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XGFKKJHT}},
  note         = {Machine review of arXiv:2412.12824}
}
read the original abstract

We investigate the effects of postselected von Neumann measurements on the nonclassical properties of pair coherent states (PCS). We calculated key quantum characteristics, such as squeezing, photon statistics, and entanglement between the two PCS modes. Our results demonstrate that postselected von Neumann measurements enhance both the non-Gaussianity and nonclassicality of PCS. These findings are validated by analyzing the scaled joint Wigner function across various system parameters. The theoretical optimization scheme offers an alternative approach for improving PCS-based quantum information efficiency and facilitates practical implementations in quantum technologies.

Figures

Figures reproduced from arXiv: 2412.12824 by the authors.

Figure 1
Figure 1. (a) Schematic diagram of weak measurement (WM) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Quadrature squeezing Q1,Ψ as a function of the state parameter γ. The thick green line corresponds to the coupling strength parameter Γ = 0, representing the case where the state is the initial PCS |ϕ⟩. The solid gray line represents Γ = 0.3, while the purple, blue, and red dashed lines correspond to Γ = 0.5, Γ = 0.7, and Γ = 1, respectively. For these calculations, we take δ = 0, α = 8π 9 and ϑ = 0 . the F2 directi… view at source ↗
Figure 4
Figure 4. (b) presents the variation Sab,Ψ (ϖ) as a function of the weak value parameter α for different values of Γ, while keeping γ = 0.5 fixed. As indicated in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figures from the paper (8 more)
Figure 5
Figure 5. Figure 5: SOCC g (2) ab,Ψ(0) as a function of the state parameter γ for different coupling strength parameters Γ. The green thick curve corresponds to the initial |ϕ⟩ case (Γ = 0). Here, we take weak value as ⟨σx⟩w = 5.761 and the other parameters are the same as in [PITH_FULL_…
Figure 6
Figure 6. Figure 6: Second-order correlation as a function of the [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: HZ correlation between two modes of the final MD [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 9
Figure 9. Figure 9: Cuts in the scaled two-mode Wigner function. (a [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: Joint Wigner function of the final MD state [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: Fidelity F as a function of the state parameter γ. Here, we take weak value as ⟨σx⟩w = 5.761 and other parameters are the same as [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 12
Figure 12. Figure 12: The fidelity of teleportation of coherent state for [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]
Figure 13
Figure 13. Figure 13: The success probability of postselection of the final MD state |Ψ⟩ as a function weak value parameter α for different coupling strength parameter Γ. Here, we take δ = 0, γ = 2 and other parameters are the same as [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]

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