bibkey: cenni2022gaussianthermometry authors: M. F. B. Cenni; L. Lami; A. Acín; M. Mehboudi year: 2022 title: “Thermometry of Gaussian quantum systems using Gaussian measurements” doi: null url: https://arxiv.org/abs/2110.02098v4 claim: “Section 3.2, Eq. (47), conjectures that for product thermal states the optimal joint Gaussian measurement has the same Fisher information as the optimal local measurement.” strata_touched:

• D5/S3/Estimation/GaussianThermometryJointMeasurementRefutation license: citation-only triage: anchor

Thermometry of Gaussian quantum systems using Gaussian measurements

M. F. B. Cenni, L. Lami, A. Acín and M. Mehboudi, arXiv:2110.02098v4 (2022).

Page 6, §3.2.2, Eq. (36), defines the zero-displacement Gaussian-measurement Fisher information by

“${\mathcal{F}}^{\mathrm{C}}(\mathbf{\sigma };{\mathbf{\sigma }}_s^M)\equiv {\mathcal{F}}^{\mathrm{C}}(0,\mathbf{\sigma };{\mathbf{\sigma }}_s^M)$” and “{\cal F}^{\rm C}({\bm d},{\bm \sigma};{\bm \sigma}_s^{M}) = \partial_{T} {\bm d}^{T}({\bm \sigma} + {\bm \sigma}_s^{M})^{-1}\partial_{T}{\bm d} + \frac{1}{2}\Tr \left[\left(({\bm \sigma} + {\bm \sigma}_s^{M})^{-1} \partial_{T}{\bm \sigma}\right)^2 \right].”

For a thermal mode the paper writes

“If the state is at thermal equilibrium, then $\nu =\coth (\omega /2T)$.”

The open statement in §3.2, Eq. (47), is quoted verbatim:

“${\mathcal{F}}^{\mathrm{C}}({\oplus }_k{\mathbf{\sigma }}_k;{\mathbf{\sigma }}_{\max }^M)\stackrel{?}{=}{\mathcal{F}}^{\mathrm{C}}({\oplus }_k{\mathbf{\sigma }}_k;{\oplus }_k{\mathbf{\sigma }}_{k,\max }^M)$”

“Based on these observations we conjecture this is generally true, however, a rigorous proof is missing currently.”

The formal encoding uses D5.S3.Observer.Fluctuation.ThermalCoefficientFloor.coth, thermalNuDeriv, thermalCov, thermalCovDeriv, IsGaussianMeasurementCov, fisherC, localCov, localFisher, and claim in the refutation module. The two-mode certificate uses T = 1, frequencies (ln 3, 2 ln 3) (encoded equivalently as (2 artanh(1/2), 4 artanh(1/2))), and the explicit symplectic covariance from the module.

Verified locator

• arXiv: https://arxiv.org/abs/2110.02098v4, p. 6 §3.2.2, Eq. (36); §3.2, Eq. (47), source version 2022-06-27.