Quantify degradation caused by mode-dependent couplings

Quantify how the \(N\ln2\) collective-information advantage degrades when mode-dependent Lamb–Dicke factors break permutation symmetry, and determine whether an effective ion number \(N_{\rm eff}\) governs the resulting scaling.

Background

The analysis is restricted to the permutation-symmetric Dicke sector J=N/2J=N/2, a restriction justified by the symmetric initial state and uniform coupling. Realistic mode-dependent Lamb–Dicke factors break this symmetry.

Because the claimed collective law relies on equal sharing of a collective coordinate, the authors identify the effect of symmetry breaking as a central unresolved issue. The proposed future direction is to determine whether the collective advantage can still be described using an effective size NeffN_{\rm eff}.

References

Mode-dependent Lamb--Dicke factors break the symmetry, and since the law follows from equal sharing of the collective coordinate, we expect it to degrade as the symmetry does. Quantifying that degradation is the most important open question.

— 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”