Conditional probabilities in quantum-optical settings

Alfredo Luis

REVIEW 3 major objections 3 minor 59 references

Conditional probabilities in quantum-optical settings

T0 review · 3 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The paper shows that conditioning on one variable in a joint noisy measurement of complementary observables can reduce, or even eliminate, uncertainty in the conjugate variable, and that the resulting conditional distribution generally cann

desk verdict The math checks out and the CV variance result is a real, citable advance, but the paper overstates the no-go: it only rules out a Born form with the fixed marginal POVM, not Born-rule conditioning in general. read the letter →

arxiv 2607.21100 v1 pith:IHDFGXEI submitted 2026-07-23 quant-ph

Alfredo Luis This is my paper

classification quant-ph

keywords conditionalprobabilitiesjointmeasurementscomplementaryobservablesdoublehomodynedetectionsqueezedQfunctionpositiveoperatorvaluedmeasuresstatereductionqubitmeasurement

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 argues that conditional probabilities extracted from a joint noisy measurement of two complementary observables behave unlike ordinary quantum state-reduction statistics. Learning the value of one variable can reduce, or even completely eliminate, uncertainty in the conjugate variable, and the resulting conditional distribution generally cannot be represented as tr[ρ_y ΔX(x)] for any physical state ρ_y using the marginal POVM. The argument runs through two canonical quantum-optical settings: a squeezed Husimi Q function realized by double homodyne detection, and a noisy joint qubit measurement. In the Gaussian case, the conditional x-variance falls below the measurement's added noise Σxx for a vacuum input with moderate squeezing; in the qubit case, an optimal joint POVM makes the conditional probability exactly 1. If correct, this means conditioning on a measurement outcome is not equivalent to a collapse to some reduced state, and the effect is an operational signature of correlations induced by the measurement apparatus itself.

What carries the argument

The central object is the joint POVM (positive operator-valued measure, the generalized-measurement object assigning a positive operator to each outcome) describing a noisy simultaneous measurement of complementary observables X and Y. For the continuous-variable case this is Δ(x,y) = (1/π)D(x,y)ρ0D†, a displaced squeezed vacuum generating a squeezed Q function; its Gaussian statistics have covariance Φ = Γ + Σ. The load-bearing identity is the lower bound σ²_{x|y} ≥ Σxx, which any Born-form conditional p~(x|y)=tr[ρ_y ΔX(x)] must satisfy. Violating this bound is what proves that no physical reduced state can reproduce the conditional statistics. For qubits, the analogous POVM has three noise

What would settle it

A double-homodyne experiment on a vacuum input with θ=π/4 and squeezing |r|≥0.6585: if the empirically estimated conditional variance of x given y is not below Σxx, or if a physical state ρ_y reproducing the conditional distribution through the marginal POVM is found, the paper's central no-go claim is refuted.

Watch

Extended reading notes

Core claim

For the squeezed-Q joint measurement Δ(x,y) = (1/π)D(x,y)ρ0D† with a squeezed-vacuum reference, the conditional distribution p~_{x|y}(x|y) is Gaussian with variance σ²_{x|y} = (ΦxxΦyy − Φxy²)/Φyy. Because σ²_{x|y} can fall below Σxx — for a vacuum input with θ=π/4 and |r|≥0.6585 — the inequality σ²_{x|y} ≥ Σxx that a Born-form representation would require is violated, so no physical state ρ_y satisfies p~(x|y)=tr[ρ_y ΔX(x)] with the marginal POVM ΔX(x). The qubit analogue uses the POVM Δ(x,y) = ¼(σ0+xγXσX+yγYσY+xyγXYσZ), and under the optimality condition γX²+γY²+γXY²=1 the conditional probability reaches p~max(x|y)=1. In both settings the non-Born behavior is driven by measurement-induced c

Load-bearing premise

The no-go conclusion rests on requiring that the conditional x-statistics after a y outcome come from a fixed marginal POVM via some physical state; if the state-update map is allowed to depend on the outcome itself, the non-Born result becomes a representation choice rather than a physical impossibility.

Editorial extensions

If this is right

Conditioning on a joint-measurement outcome is not equivalent to a projective or Lüders-style state reduction; the posterior statistics of the conjugate variable need not come from any physical state via the marginal POVM.
In double-homodyne detection on a vacuum input with θ=π/4 and |r|≥0.6585, the conditional x-variance is predicted to drop below the detector's added noise Σxx, a directly observable signature.
For qubits, an optimal joint POVM makes x perfectly predictable from y, with p(x|y)=1, for the pure state s=(xγX, -yγY, xyγXY).
The effect is driven by measurement-induced correlations (Σxy or γXY) that disappear in the marginals; therefore joint statistics carry information beyond what marginal measurements can reveal.
In the fully classical limit, where the reference field has no fluctuations, the effect vanishes, indicating that the non-Born behavior is intrinsically quantum and stems from the squeezed-vacuum reference.

Reading between the lines

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

If the state-update rule for the conditioned variable is allowed to depend on the recorded outcome y — for example, by using the full Kraus operators of Δ(x,y) rather than the fixed marginal POVM ΔX(x) — the 'no Born form' conclusion becomes a representation choice rather than a physical impossibility; the operational content is that the marginal POVM alone cannot serve as the update rule.
The continuous-variable infimum σ²_{x|y} = 1/(16Σyy) suggests a practical data-processing strategy: deliberately engineering correlated measurement noise (via a squeezed reference) and post-selecting on y could suppress uncertainty in the variable of interest below the detector's intrinsic noise floor.
The same formalism could be applied to other conjugate pairs — time-frequency or number-phase — wherever a joint POVM exists, predicting analogous non-Born conditional statistics.
The qubit result p(x|y)=1 hints at a deterministic 'readout' of one dichotomic observable conditioned on the other when the noise parameters are optimal, which could be relevant for communication or cryptographic protocols built from joint measurements.

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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

Short version: the core derivation is sound, and the specific no-go is real, but the headline interpretation is too strong. Equations (4.5)–(4.8) are correct: for vacuum input and a squeezed reference, the conditional variance can drop below the intrinsic noise Σ_xx, so no physical ρ_y can satisfy p~(x|y) = tr[ρ_y Δ_X(x)] with the marginal POVM fixed. The qubit maximization leading to p_max = 1 in (8.14) also checks out. That is a clean extension of the earlier qubit paper [37], and the continuous-variable double-homodyne example is new.

What the paper does well: the inputs (r, θ, γ) are declared, not fitted; the Gaussian calculation is explicit and reproducible; the double-homodyne realization is plausibly connected; and the observation that conditioning can reduce uncertainty below the measurement's intrinsic noise is a useful counterexample to naive intuition.

The soft spot is interpretive, and it is load-bearing for the abstract and conclusions. The paper phrases the result as if conditioning in joint measurements cannot be given any Born-form or state-reduction account. What is actually proven is narrower: with the marginal POVM held fixed, no state reproduces the conditional distribution. But the conditional distribution is itself a Born-rule probability from the full joint POVM, tr[ρ Δ(x,y)] / tr[ρ Δ_Y(y)], and the usual Kraus update for the joint measurement, coarse-grained over x, gives the same x-statistics with an outcome-dependent POVM. So the claim that typical rules of state reduction fail depends on requiring the reduced state to work with Δ_X alone. That is a legitimate target, but it is not the only notion of state reduction, and the paper should say so explicitly. As written, a reader could come away thinking quantum conditional probabilities are not Born-rule, which is not what the equations show.

Minor issues: the state-reduction discussion in Sec. IV is qualitative, and Sec. VI assumes the idealized setup realizes exactly the POVM (2.1). The classical/semiclassical section is illustrative rather than quantitative. These are minor.

Bottom line: a solid, mostly correct paper within its subfield. The central variance result is new and reproducible, and the paper deserves refereeing, though the revision should qualify the no-go and engage with the conditional-POVM/instrument formulation. I would bring it to a reading group and would cite the CV result.

Referee Report

3 major / 3 minor

Summary. The paper studies conditional probabilities obtained from joint noisy measurements of conjugate quadratures in two settings: double homodyne detection with a squeezed-vacuum reference (the 'squeezed Q' POVM) and a qubit joint POVM. For Gaussian system states it derives the conditional variance σ²_{x|y} and shows that for certain reference states (e.g., vacuum input, θ=π/4, |r|≥0.6585) this variance drops below the intrinsic noise Σ_xx of the marginal POVM. It concludes that no physical state ρ_y can reproduce p~(x|y) via the Born rule with the marginal POVM, Eq. (4.4). In the qubit case it finds that the conditional probability can reach the value 1 for an optimal POVM, and asserts that no ρ_y exists when γ_X≠1. The paper interprets these findings as showing that typical state-reduction rules do not apply to this kind of conditioning.

Significance. The technical content is valuable: it provides explicit counterexamples to the expectation that any conditional distribution obtained from a joint measurement can be represented as a fixed state measured by the marginal POVM of the conditioning observable. The Gaussian variance inequality and the qubit p_max=1 result are striking and of direct interest to quantum measurement theory. The paper is self-contained in the Gaussian part, gives a practical implementation in Sec. VI, and the main algebra in Secs. II-IV checks out. However, the broader claim about state reduction is representation-dependent, and one intermediate qubit formula is incorrect, so the paper needs revision before the interpretive conclusions can be accepted as stated.

major comments (3)

[Sec. IV, Eq. (4.4); abstract] The no-go result is representation-dependent. The paper proves that there is no physical state ρ_y and fixed marginal POVM ΔX(x) satisfying Eq. (4.4). However, the conditional distribution is itself a legitimate Born-rule expression from the joint POVM: p~(x|y)=tr[ρΔ(x,y)]/tr[ρΔY(y)]. This is already a standard quantum-mechanical conditional probability, and the effective x-measurement after conditioning on y is y-dependent, not the marginal ΔX. The conclusion that the conditional distribution 'cannot be derived from typical rules of state reduction' (abstract, Sec. IV, Sec. IX) overreaches. What is established is only that the marginal POVM does not provide the correct post-measurement x-statistics. The abstract and conclusions should be revised to state this precisely.
[Sec. VIII, Eq. (8.10)] The maximization leading to Eq. (8.10) is incorrect. Starting from (8.7), the inequality (8.9) bounds only the numerator, but the denominator 1+yγ_Y s_Y must be included in the variation. The correct maximum over s_Y is p_max = 1/2(1 + sqrt(γ_X²+γ_XY²)/sqrt(1−γ_Y²)), not the expression with the factor in the numerator. For example, with γ_X=0.5, γ_Y=0.5, γ_XY=0, the paper's (8.10) gives 0.716, while the state (8.11) substituted into (8.7) gives 0.789. This internal inconsistency must be fixed; the final optimal-POVM result (8.14) is correct, but the general formula and derivation need correction.
[Sec. VIII, after Eq. (8.16)] The claim that no physical ρ_y can reproduce the conditional probability via the marginal POVM when γ_X≠1 is stated without proof, with a reference to [37]. Since this is a central assertion of the qubit section, a direct argument should be included: from (8.16), tr[ρ_y ΔX(x)] ≤ (1+γ_X)/2 for any ρ_y, while the conditional distribution with the optimal state (8.15) reaches 1 when (8.13) holds. This is a simple proof and should be provided rather than deferred to a reference.

minor comments (3)

[Sec. V, Eqs. (5.1)-(5.3)] The derivation of the minimum and infimum is compressed; a short sketch of the algebra leading from (4.3) to (5.2)-(5.3) would improve readability and checkability.
[Sec. IV, wording] The phrase 'The answer in general is negative' is acceptable because a family of counterexamples is given, but it could be misread as 'for all states.' Consider rephrasing to 'there exist states for which no such ρ_y exists.'
[Sec. VII] The semiclassical discussion correctly restricts attention to the domain where the Wigner function is nonnegative, but it would be helpful to state explicitly that this restricts the values of r and θ for which the analysis applies.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central nonrepresentability result follows from declared POVM assumptions and explicit inequalities; self-citations are background only.

full rationale

The paper's claimed results are not circular. The squeezed-Q POVM (2.1) and qubit POVM (8.1) are stated as inputs, and the state parameters (r, θ, γ) are not fitted to the target conclusion. The key nonrepresentability result is derived: assuming a reduced state ρ_y could reproduce the conditional statistics via the fixed marginal POVM (4.4), the conditional variance would have to be at least the marginal kernel variance Σxx; the vacuum example violates this for |r|≥0.6585 (Eqs. 4.5–4.8), so no such ρ_y exists. This is a genuine mathematical consequence, not an assumed premise. Similarly, the qubit bound (8.10) and the p_max=1 result (8.14) follow from an explicit optimization over system states for a fixed POVM, not from fitting the answer into the input. The self-citations [27,28,57] and [37] supply context or prior related work; the qubit result from [37] is rederived in Eqs. (8.7)–(8.15), so the argument does not rest on an unverified self-citation. The broader wording that the conditional distribution 'cannot be derived from typical rules of state reduction' is an interpretive extension of the proven statement about the fixed marginal POVM; it may overreach, but that is a scope/correctness concern, not circularity, because the mathematical derivation does not presuppose that broader claim.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The derivation relies only on standard quantum-optics tools and explicitly stated POVM parameters; no fitted constants. The tunable POVM parameters (r, θ, γ) are inputs, not inferential free parameters.

free parameters (2)

Squeezing parameter r and rotation angle θ of reference squeezed vacuum = θ=π/4, |r| ≥ 0.6585 for violation example
Settings of the POVM (2.1), not fitted to data; chosen to exhibit the non-Born-representability effect.
Qubit POVM noise coefficients γ_X, γ_Y, γ_XY = optimal case γ_X²+γ_Y²+γ_XY²=1
Positivity-constrained POVM parameters; not fitted to data.

assumptions (6)

standard math Born rule p~(x,y)=tr[ρ∆(x,y)] and Bayes rule define conditional probabilities.
Used in Eq. (2.3) and (4.1).
standard math Wigner-function convolution: p~ = W_S * W_0, with marginals as convolutions.
Used in Eqs. (3.1), (3.2), (6.5).
domain assumption Rotated squeezed vacuum is a physical state with det Σ = 1/16.
Eqs. (2.4)-(2.7); this provides the noise kernel.
domain assumption Double-homodyne beam-splitter setup realizes the POVM (2.1).
Sec. VI, Eqs. (6.1)-(6.5); if the realization failed, the continuous-variable example would be purely formal.
standard math Qubit POVM positivity requires γ_X²+γ_Y²+γ_XY²≤1.
Eq. (8.2), used for the maximization bound.
domain assumption Gaussian pure system states satisfy ΓxxΓyy−Γxy²=1/16.
Used in Sec. V to minimize the conditioned variance.

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Cite this review

Pith. "Pith review of Conditional probabilities in quantum-optical settings." pith.science (2026). https://pith.science/paper/IHDFGXEI

@misc{pith2026260721100,
  author       = {Pith},
  title        = {Pith review of: Conditional probabilities in quantum-optical settings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IHDFGXEI}},
  note         = {Machine review of arXiv:2607.21100}
}

read the original abstract

We examine conditional probabilities derived from the joint noisy measurement of two complementary observables. We show that conditioning on one variable can lead to reduction of uncertainty in the other one, even completely eliminating uncertainty. We show that the conditional distribution cannot, in general, be represented in Born form using the marginal positive operator valued measure and any physical system state, nor can it be derived from typical rules of state reduction. We examine these issues in two basic quantum-optical schemes. These are double homodyne detection and a qubit measurement.

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