Derive the collective prefactor in the resolved-variance law

Derive why the optimal balanced two-outcome collective measurement on the spin resolves an asymptotic between-outcome variance of \(\mathrm{Var}_{\rm bet}^{\rm coll}\to(\ln 2/4)N\), rather than the Gaussian median-split value \(N/(2\pi)\), thereby establishing the \(N\ln 2\) prefactor theoretically.

Background

The paper derives a weak-charging resolved-variance law in which the daemonic gain is proportional to the variance of the collective coordinate resolved by the measurement. A balanced single-ion measurement resolves exactly $1/4$, whereas numerical results indicate that the balanced collective measurement resolves approximately (ln⁡2/4)N(\ln 2/4)N.

A Gaussian median-split argument instead predicts N/(2π)N/(2\pi), which the numerical data exclude. The authors therefore leave unresolved the classical-statistical explanation for the observed ln⁡2\ln 2 coefficient; resolving it would convert the numerically supported scaling law into a theorem.

References

The prefactor is not derived. Proposition~\ref{prop:linear} fixes the scaling but not the constant, which requires knowing how much of \mathrm{Var}[J_x] a balanced collective split resolves. Numerically \mathrm{Var}{\rm bet}{\rm coll}\to(\ln2/4)N, giving /\to N\ln2. The natural estimate---J_x is a sum of N independent \pm1/2 variables, hence asymptotically Gaussian with \sigma{2}=N/4, and a median split of a Gaussian resolves (2/\pi)\sigma{2}---predicts \mathrm{Var}{\rm bet}{\rm coll}=(1/2\pi)N, which is 9\% below the measured value and clearly excluded by the data (Sec.~\ref{sec:law}). Identifying the correct classical statistics problem, and thereby deriving \ln2, is the main theoretical question left open by this work.

— One Bit of Collective Information Is Worth N ln 2 Bits of Local Information in a Many-Body Quantum Battery  (2609.04730 - B et al., 4 Sep 2026) in Section 3, subsection “The linear collective advantage”; also identified as C2 in the Contributions

The law is clean over 4\le N\le24, but we cannot exclude a crossover at larger N.

— One Bit of Collective Information Is Worth N ln 2 Bits of Local Information in a Many-Body Quantum Battery  (2609.04730 - B et al., 4 Sep 2026) in Section 7, subsection “Limitations”

That one bit outperforms a complete readout per bit invites an optimisation over the number and placement of outcomes; we do not know whether the return per bit is always maximised at two outcomes.

— One Bit of Collective Information Is Worth N ln 2 Bits of Local Information in a Many-Body Quantum Battery  (2609.04730 - B et al., 4 Sep 2026) in Section 8, subsection “Future work”