Precision measurements at the interface between unitary and non-unitary encoding

We investigate precision scaling at the interface between unitary and non-unitary encoding under generalized noise including single-particle and collective dephasing and decay. Using linear response theory and the error propagation formula, we derive analytic precision expressions for both the unitary parameter Ω and the dissipation strength γ. For unitary encoding, when the observable commutes with a Hermitian noise operator, the optimal encoding time is independent of N, yielding the Heisenberg limit ΔΩpropto 1 / N; otherwise the precision degrades to the standard quantum limit or ceases to improve with N. For non-unitary encoding, when \[hat{A}, hat{O}\] = 0, the precision is insensitive to intrinsic dynamics and encoding time, scaling as Δγpropto sqrt{γ/ expval*{hat{L}^dagger hat{L}}}. Notably, for collective decay, the Dicke state reaches the Heisenberg limit Δγpropto 1 / N, demonstrating that entanglement can enhance non-unitary estimation. Our results provide a unified framework and practical guidance for designing quantum metrology protocols in noisy environments.