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

Abstract

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 $\hat{I}\text{c◯}$ and the dissipation strength ${\hat{I}}^3$. For unitary encoding, when the observable commutes with a Hermitian noise operator, the optimal encoding time is independent of $N$, yielding the Heisenberg limit $\hat{I}”\hat{I}\text{c◯}\propto 1/N$; 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 $Δγ\propto \sqrt{γ/ \expval*{\hat{L}^\dagger \hat{L}}}$. Notably, for collective decay, the Dicke state reaches the Heisenberg limit $\hat{I}”{\hat{I}}^3\propto 1/N$, 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.

11 pages, 10 figures