In-between-state optimality for Bayesian transmissivity sensing

B. Zhou, B. A. Bash, S. Guha and C. N. Gagatsos, “Bayesian minimum mean square error for transmissivity sensing”, arXiv:2304.05539v1, Section V, p. 4, state:

For the two-point prior PDF (15), we verify that for integer n̄ the optimal state is the Fock state with the same photon number. For real n̄, our numerical results support the conjecture that the optimal state has the form of the state (27), up to a phase, i.e., |Φ′ₙ̄⟩ = e^{iφn̂}|Φₙ̄⟩.

Issue #11653 preregisters this Tier 1 conjecture, the quantified reading and the witness. The two-point prior is $P(\tau)=q\delta(\tau-\tau_0)+(1-q)\delta(\tau-\tau_1)$, with $q,\tau_0,\tau_1\in[0,1]$.

A pure input has finite Fock coefficients $d_n$, Hilbert norm one and mean photon number $\sum_n n|d_n|^2=\bar n>0$. The loss branch acts as $K_l|n\rangle=\sqrt{\binom nl\tau^{n-l}(1-\tau)^l}|n-l\rangle$. The output is $\rho(\tau)=\sum_l K_l|\psi\rangle\langle\psi|K_l^\dagger$, and $\Gamma_k=q\tau_0^k\rho(\tau_0)+(1-q)\tau_1^k\rho(\tau_1)$. For finite positive semidefinite effects summing to the identity and real estimates, the risk is $\sum_j\operatorname{Re}\operatorname{tr}(E_j(x_j^2\Gamma_0-2x_j\Gamma_1+\Gamma_2))$. Lean MMSE is the infimum of these risks over every finite outcome count on that input's finite output span.

Equations (27)–(29) define an adjacent-Fock superposition with coefficients $c=\sqrt{1-\lceil\bar n\rceil+\bar n}$ and $a=\sqrt{1-c^2}$. Equation (33) multiplies its coefficient at photon number $n$ by $e^{i\phi n}$. The quantified claim asserts that, for every admissible prior, positive mean energy and finite normalized competitor at that energy, some phase of the in-between state has no larger finite-POVM MMSE.

The literature search recorded in #11653 reports no settlement in its checked arXiv and citation scope; one journal full text was inaccessible. That record is source-reported, not a new exhaustive literature check or a priority claim.

Identifying the finite-outcome quantity with the source's MMSE over arbitrary measurements requires the following compression argument: if $P$ projects onto the output support, then $\operatorname{tr}(\rho E)=\operatorname{tr}(\rho PEP)$ because $\rho=P\rho P$. Compressed effects sum (or integrate) to the identity on that span. A finite POVM on the span extends to full Fock space by assigning $1-P$ to one outcome, without changing the output statistics. ASSUMED-UNVERIFIED: this compression argument and the equality with the arbitrary-outcome source MMSE are not kernel-checked in this module. In particular, preserving outcome statistics does not by itself formalize the passage from arbitrary outcome spaces to the infimum over finite POVMs. The kernel-checked settlement concerns exactly the preregistered finite-POVM claim; the identification with the full source reading retains this boundary.

Inside result, the operator variance of any finite measurement is expressed as a sum of positive semidefinite sandwiches. For a Hermitian Sylvester certificate $B$ with $\Gamma_0B+B\Gamma_0=2\Gamma_1$, square completion gives $risk\geq\operatorname{Re}\operatorname{tr}(\Gamma_2-B\Gamma_1)$. The spectral projections of $B$, estimated by its eigenvalues, attain this bound with finitely many outcomes.

Take $q=1/2$, $\tau_0=4/9$, $\tau_1=1$, $\bar n=1/2$. All phases of $\chi_\phi=(|0\rangle+e^{i\phi}|1\rangle)/\sqrt2$ have finite-POVM MMSE $1625/23976$. The competitor $\psi=(\sqrt3/2)|0\rangle+(1/2)|2\rangle$ has Hilbert norm one, mean energy $1/2$ and finite-POVM MMSE $110575/1674432$. The gap is $107725/61953984>0$.

The certificates are

$$B_\chi=\frac1{1332}\begin{pmatrix}787&145\145&937\end{pmatrix},\qquad B_\psi=\frac1{93024}\begin{pmatrix}62971&0&4293\sqrt3\0&41344&0\4293\sqrt3&0&68965\end{pmatrix}.$$

Phase conjugation supplies the certificate for every $\chi_\phi$.

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

D5/S3/Estimation/TransmissivityTwoPointProbeRefutation.lean exposes meanPhoton, outputState, momentState, finitePOVM, bayesianRisk, MMSE, inBetween, claim and result : ¬ claim. The Kraus-coefficient lemmas sqrt_power, source_coefficient and sourceKraus are public, also used by the beta-prior refutation D5/S3/Estimation/TransmissivityBetaPriorProbeRefutation; the measurement arguments and the fixed-instance facts are local have terms. result: proof_shape: bind-only; escape_witness: none; admission_basis: open-problem-resolution (#11653; Refuted). The spectral construction applies Mathlib's spectral theorem with finite-sum and matrix normalization; the lower bound applies upstream positive-semidefinite facts and square completion. Utility is a certified instance refuting the designated claim. The Scribe resolution is Refuted and cites the Library note zhou2023transmissivity.

Tier 1: an externally published conjecture with a preregistered finite-POVM reading. No atom or coverage step is associated with this settlement.

What the settlement shows

• Proved in this module: adjacent photon-number support is not sufficient for optimality at non-integer mean energy. At the stated prior, the vacuum–two-photon competitor beats every phase of the adjacent vacuum–one-photon state, with exactly the same mean energy.
• Proved within the result's derivation: both witness risks are finite-POVM optima, certified by square completion and attained by spectral measurements. The failure concerns the probe family, rather than a suboptimal measurement of the in-between state. These are proof-internal facts, not separate public theorems.
• Proved within the result's derivation: the positive gap is $107725/61953984$. The lower bound holds for every finite POVM and all real estimates, and the in-between risk is independent of its phase.
• Open: global optimality of the competitor, the source's integer-energy optimality statement and the parameter region in which an adjacent-Fock probe is optimal. None is settled by result. The beta-prior conjecture is refuted separately in the beta-prior dossier.
• Open source scope: conclusions that select an in-between probe by the refuted universal optimality claim lose that justification. Results about that fixed probe's risk remain compatible with the witness. No other source theorem is refuted here; full-space arbitrary-outcome identification retains the Gap boundary.

The written compression argument and equality between the finite-POVM MMSE and the source's arbitrary-measurement MMSE are not kernel-checked results of this module. The literature check in #11653 is source-reported and bounded, including its inaccessible journal full text. Information-escape registration is paused under CLAUDE.md section 3.9.