bibkey: len2022quantum authors: Yink Loong Len, Tuvia Gefen, Alex Retzker, Jan Kołodyński year: 2022 title: “Quantum metrology with imperfect measurements” doi: 10.1038/s41467-022-33563-8 url: https://arxiv.org/abs/2109.01160v2 claim: “For a pure encoded probe state and an imperfect measurement, the quantum Fisher information is multiplied by gamma = max over orthogonal pairs of sum_x Re<xi_perp|M_x|xi>^2 / <xi|M_x|xi>. The supplement conjectures that for every classical noise channel M_x = sum_i p(x|i) Pi_i applied independently to N probes, an optimal pair is a cat pair cos(theta)|j>^N + sin(theta)|k>^N, -sin(theta)|j>^N + cos(theta)|k>^N.” strata_touched:

• D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation license: citation-only triage: anchor

Quantum metrology with imperfect measurements

Yink Loong Len, Tuvia Gefen, Alex Retzker and Jan Kołodyński, Nature Communications 13, 6971 (2022); arXiv:2109.01160v2. Quotations are from the arXiv v2 source.

The noisy Fisher coefficient (Lemma 1, Eq. (gamma_M)):

\gamma_\cM=\max_{\ket{\xi},\ket{\xi_{\perp}}} \sum_x \frac{\mathrm{Re}\!\left\{\langle\xi_{\perp}|M_{x}|\xi\rangle\right\}^{2}}{\langle\xi|M_{x}|\xi\rangle}

and its multi-probe form (Supplement, Eq. (AgammaN)) with M_{\xvec}=M_{x_1}\otimes M_{x_2}\otimes\cdots\otimes M_{x_N} (Eq. (AMxvec)) and \VPhi\ket{\psi^N_{\pm}}=\ket{\zeta^N},\ket{\zeta^N_\perp}, where $|{\psi }_{\pm }^N⟩=\frac{1}{\sqrt{2}}(|{\psi }^N⟩\pm |{\psi }_{\perp }^N⟩)$ (Eq. (AVPhichoice)).

The conjecture (Supplement, end of “A note on optimality”):

In fact this intuition together with some numerical evidence leads us to conjecture that for any classical noise channel, $M_x={\sum }_{i}p\left(x|i\right){\Pi }_i$ that is applied independently on each of the $N$ probes, the optimal $|{\zeta }^N⟩,|{\zeta }_{\perp }^N⟩$ take the form of ``cat states“: $|\zeta^{N} \rangle =\cos\left(\theta\right)|j\rangle^{\otimes N}+\sin\left(\theta\right)|k\rangle^{\otimes N}$, $|\zeta_{\perp}^{N} \rangle =-\sin\left(\theta\right)|j\rangle^{\otimes N}+\cos\left(\theta\right)|k\rangle^{\otimes N}$, where $ \ket{j}, \ket{k}$ can be found numerically for just a single probe, and $\theta$ depends on $N$ and should be found numerically.

The encoding reads “optimal” as maximizing Eq. (AgammaN) over all orthonormal pairs $\zeta ,{\zeta }_{\perp }$ for channels with all $p(x|i)>0$, where a maximizing pair exists, with $V_{\Phi }\psi =(\zeta +{\zeta }_{\perp })/\sqrt{2}$ and $V_{\Phi }{\psi }_{\perp }=(\zeta -{\zeta }_{\perp })/\sqrt{2}$, and states the conjecture as the existence of some cat pair with $j\ne k$ attaining that maximum.

Verified locator

• DOI: https://doi.org/10.1038/s41467-022-33563-8 (Nature Communications 13, 6971 (2022); the conjecture is in Supplementary Note 5, printed p. 11, as reported by the literature seat).
• URL: https://arxiv.org/abs/2109.01160v2 (source QMwIM-v2.tex, md5 d618d71e36ac84cb2be4e09e2335e219): Lemma 1 (l. 324–337), Eq. (AgammaN) (l. 1478–1482), Eq. (AMxvec) (l. 1483–1485), the definition of ${\psi }_{\pm }^N$ (l. 1488), Eq. (AVPhichoice) (l. 1507–1510), “A note on optimality” (l. 1550–1567), the conjecture (l. 1558–1567).