slug: zhou-bash-guha-gagatsos-2023-transmissivity-beta-prior-refutation
bibkey: zhou2023transmissivity
doi: 10.1103/PhysRevResearch.5.043033
url: https://arxiv.org/abs/2304.05539v1
triage: theorem
motivation_gids:
  - D5/S3/Estimation/TransmissivityBetaPriorProbeRefutation.result

In-between-state optimality for transmissivity sensing with a beta prior

Problem

B. Zhou, B. A. Bash, S. Guha and C. N. Gagatsos, “Bayesian minimum mean square error for transmissivity sensing”, arXiv:2304.05539v1, Phys. Rev. Research 5, 043033 (2023), Section VI (p. 6):

The numerical results indicate that the optimal input states with non-integer mean photon number, have the form of Eq. (27), which means that for $\bar{n}\in \mathbb{N}$ the optimal states are Fock states.

and Section VII (p. 7):

We note that for the beta prior PDF, we were not able to prove analytically that the Fock states or the in-between states of Eq. (27) states are optimal, even though our numerical computations support such conjecture.

The beta prior is eq. (16), $P(\tau )={\tau }^{\alpha -1}(1-\tau {)}^{\beta -1}/\mathcal{B}(\alpha ,\beta )$ with $\alpha ,\beta >0$. The verbatim definitions are in the literature note. Issue #14698 preregisters the reading with the parameters free, the witness and the exact certificates.

Motivation

The paper gives the first Bayesian treatment of quantum-limited transmissivity estimation. Its conjectured optimal probe, the adjacent-Fock “in-between” state, would fix the optimal input for every beta prior at every mean energy; photon counting on that probe is studied as a practical measurement.

Gap

The source supports the beta-prior conjecture by sampling 200 random superpositions of the first five Fock states per energy at two priors, $\alpha =\beta =1$ and $\alpha =2,\beta =4$, and states that it could not prove it. The repository refuted the paper’s two-point-prior conjecture in the two-point dossier; that refutation does not address the beta prior. The literature check in #14698 found no settlement; not-found-in-searched-scope.

Route

Take $\alpha =3$, $\beta =1$ (so $P(\tau )=3{\tau }^2$) and $\bar{n}=1/4$. Every phased in-between state is ${\chi }_{\phi }=\frac{\sqrt{3}}{2}|0⟩+\frac{e^{i\phi }}{2}|1⟩$; the probe $\psi =\frac{\sqrt{14}}{4}|0⟩+\frac{\sqrt{2}}{4}|2⟩$ has the same norm and mean photon number. The moments ${\Gamma }_k={\int }_0^{1}P(\tau ){\tau }^k\rho (\tau )d\tau$ are integrals of polynomials in $\tau$ and $\sqrt{\tau }$. The source’s error of a Hermitian $H$ is $\delta_B(H)=\int_0^1P(\tau)\operatorname{tr}[\rho(\tau)(H-\tau)^2],d\tau =\operatorname{tr}(H^2\Gamma_0)-2\operatorname{tr}(H\Gamma_1)+\operatorname{tr}\Gamma_2$, and its MMSE is the minimum over $H$. For a Hermitian $B$ with ${\Gamma }_{0}B+B{\Gamma }_0=2{\Gamma }_1$, ${\delta }_B(H)-(\mathrm{tr}{\Gamma }_2-\mathrm{tr}(B{\Gamma }_1))=\mathrm{tr}((H-B){\Gamma }_0(H-B))\ge 0$, with equality at $H=B$. The certificates

$$B_{\chi }=\left(\begin{array}{cc}216/305& 7\sqrt{3}/183\\ 7\sqrt{3}/183& 204/305\end{array}\right)(\text{phase-conjugated for }{\chi }_{\phi }),B_{\psi }=\left(\begin{array}{ccc}8/11& 0& 2\sqrt{7}/77\\ 0& 2/3& 0\\ 2\sqrt{7}/77& 0& 20/33\end{array}\right)$$

give MMSE $167/4575$ for every ${\chi }_{\phi }$ and $2/55$ for $\psi$; the gap is $7/50325>0$.

Falsifier

The universal claim is false if one admissible prior and one normalized competitor at a positive mean energy strictly beat every in-between phase; the witness above is such a pair. A comparison with one phase, or with photon counting alone, would not refute the claim.

Evidence

D5/S3/Estimation/TransmissivityBetaPriorProbeRefutation.lean defines betaDensity, momentBeta, the Fock space FockSpace (${\ell }^2(\mathbb{N})$, Mathlib lp) with basis e, deltaB (eq. (2) for a bounded self-adjoint operator on it), MMSE (its infimum over such operators), the compression compression to the output span and its matrix risk deltaBMatrix, and claim; it proves deltaBMatrix_eq_moments, compression_le and result : ¬ claim, and reuses the frozen meanPhoton, outputState and inBetween and the public lemmas source_coefficient and sourceKraus of TransmissivityTwoPointProbeRefutation, which this delivery extracts there from local proofs (with sqrt_power). The axiom closure of result is exactly propext, Classical.choice and Quot.sound; there is no sorry or native_decide. The module statement is sha256:bee31b46ccee1dec94f1276515d16d19200d73f07e03a9a1687b3d5504d76e44; result is sha256:5fa291fa964c3f4a20d87b08ee8bfa0897b4ae3fa65e8b3104e8e65b2e8b4610 and claim is sha256:bc6d9af4ecf96bd3693cacd03e733601e457edfc2f6fb02f9fc5d3bd4611a092. The Freeze event is sha256:2ffe47a2c05ed8796fcea7ab53888f2ce433174ee4de485649472f005b06b1ef; its project-level prerequisite is the re-pinned node sha256:5f016c1168ccdd32fa689bddf160653279b2afbd621505c835d5950cad1e9750 of TransmissivityTwoPointProbeRefutation.

Triage

Tier 1: a conjecture of a 2023 paper, retained in its journal version, preregistered in #14698 before any Lean. theorem; resolution refuted. result, compression_le and deltaBMatrix_eq_moments are bind-only (Parseval expansion of $⟨(H-\tau )e_j,(H-\tau )e_i⟩$ in ${\ell }^2$, nonnegativity of positive semidefinite quadratic forms, monotonicity and linearity of the interval integral, exact Gamma values and interval integrals, matrix normalization and square completion); admission basis open-problem-resolution. Utility kind=certified-instance; basis=refutes. There is no digestion atom.

What the refutation shows

• Proved by result: at $\alpha =3,\beta =1$, $\bar{n}=1/4$, the vacuum–two-photon probe $\psi$ has strictly smaller MMSE than every phase of the adjacent vacuum–one-photon state, at the same mean energy. Adjacent photon-number support is therefore not sufficient for optimality under every beta prior.
• Computed, not formalized (exact rational arithmetic, #14698): the same comparison fails for the in-between state also at $\alpha =3,\beta =1$, $\bar{n}=1/8$ (gap $199/1554300$) and at the interior prior $\alpha =6,\beta =3$, $\bar{n}=1/4$ (gap $46510177/2452113546300$). That prior vanishes at both endpoints and has $\alpha ,\beta >2$, so the failure does not depend on a prior that stays positive at $\tau =1$, as $3{\tau }^2$ does. At the source’s priors $\alpha =\beta =1$ and $\alpha =2,\beta =4$, $\bar{n}=1/4$, the same $\psi$ is worse than the in-between state, consistent with the source’s figure.
• Computed, not formalized (floating-point scan over the first five Fock states): the in-between state was the best sampled probe at the source’s two priors for $\bar{n}\in \{1/8,1/4,1/2,3/4,3/2\}$, and was beaten at $\alpha =3,\beta =1$ for $\bar{n}\in \{1/8,1/4\}$, where the best sampled probe at $\bar{n}=1/4$ was $\frac{\sqrt{15}}{4}|0⟩+\frac{1}{4}|4⟩$. Priors concentrated near $\tau =1$ favor spreading a small energy budget onto a higher Fock component.
• Open: the region of $(\alpha ,\beta ,\bar{n})$ in which an in-between state is optimal, the optimal probe where it is not, and the source’s integer-energy statement that Fock states are optimal for the beta prior (its Fock-state MMSE, eq. (39), is not challenged here).
• Source scope: conclusions that select the in-between probe by the refuted universal claim lose that justification for priors such as $\alpha =3,\beta =1$; the fixed-probe risks and photon-counting results of the paper are unaffected.

ASSUMED-UNVERIFIED

MMSE ranges over bounded self-adjoint operators on ${\ell }^2(\mathbb{N})$; unbounded estimator observables are outside it (their eigenvalue estimates can be clipped to $[0,1]$ without increasing the error; not formalized). The literature check in #14698 is bounded and does not establish worldwide novelty or priority.