Cancellation of off-diagonal channel contributions in multi-channel Lindbladians

Determine whether the off-diagonal contributions in the sum of Lindblad terms, \(\sum_\alpha \hat L_\alpha^\dagger \hat A_{mn}\hat L_\alpha\), can cancel between different jump operators so that a multi-channel model excluded by the per-operator criterion nevertheless remains immune to the reference-based readout.

Background

The paper establishes an eigen-operator criterion for the data-ruler readout: the measured transition operator must evolve proportionally to itself under the adjoint generator. For individual Lindblad jump operators and a non-degenerate Hamiltonian, the stated condition requires each jump operator to be diagonal in the energy basis or proportional to a single transition operator, together with the commutation condition for its number operator.

Section S11.6 identifies an unresolved extension to multiple noise channels. Although the per-jump-operator condition excludes certain models, the full Lindblad generator contains a sum over channels. In principle, off-diagonal terms generated by distinct jump operators could cancel in this sum, preserving the eigen-operator property and hence the readout’s immunity. The paper reports that no such cancellation was found but does not rule it out.