Asymptotic validity of the reduced-density expression for large N when n and/or m diverge

Establish that, in the ancilla-mediated simulation of the quantum Zeno effect where a system qubit q0 with Hamiltonian H evolves for total time T via N=mn steps of duration τ=T/N with interleaved controlled gates C_U(0,b) acting on m identical ancillas initially in state |α⟩ and repeated over n cycles, the reduced density matrix ρ_f of q0 satisfies in the large-N limit the asymptotic expression ρ_f = [[|⟨ξ|0⟩|^2, B⟨0|ξ⟩⟨ξ|1⟩], [B*⟨1|ξ⟩⟨ξ|0⟩, |⟨ξ|1⟩|^2]] + o(1), with B = (⟨α|U^n|α⟩)^m exp[−T(⟨0|H|0⟩ − ⟨1|H|1⟩)/ħ], not only in the two regimes proven in the paper (fixed m with n→∞ under det(U^{⊗m}−I)≠0, and n=1 with m→∞ under det(U−I)≠0) but also whenever n→∞, m→∞, or both tend to infinity.

Background

The paper analyzes a quantum circuit that simulates quantum Zeno dynamics by repeatedly coupling a system qubit to ancillas via controlled unitaries. The total evolution time T is split into N=mn steps of duration τ=T/N. They derive, under certain conditions, an asymptotic expression for the reduced density matrix of the system qubit: ρ_f has Zeno-projected diagonals, and its off-diagonal term is multiplied by B = (⟨α|U^n|α⟩)^m exp[−T(⟨0|H|0⟩−⟨1|H|1⟩)/ħ].

The authors rigorously establish this expression in two cases: (i) fixed number of ancillas m with the number of cycles n→∞, assuming det(U^{⊗m}−I)≠0; and (ii) a single cycle n=1 with m→∞, assuming det(U−I)≠0. They conjecture that the same large-N expression should hold more generally when n, m, or both grow, but they currently lack a proof. Proving this would unify the asymptotics across regimes and strengthen the theoretical link between ancilla-mediated control, bang-bang decoupling, and Zeno dynamics.