Analytical framework and reservoir engineering of coherence freezing in single-qubit systems

Abstract

We present a unified, time-dependence–agnostic affine-Bloch framework for freezing of single-qubit quantum coherence in open systems. From the affine map of completely positive trace-preserving dynamics, we derive compact necessary and sufficient conditions for constancy of both the ${\ell }_1$-norm and the relative entropy of coherence, yielding explicit geometric “freeze manifolds” in Bloch space. Specializing to four canonical noisy channels—Pauli (unital, diagonal), depolarizing (isotropic, unital), amplitude damping (AD), and generalized amplitude damping (GAD)—we show that nontrivial freezing occurs only for Pauli subchannels that exactly preserve a Cartesian axis; otherwise, freezing is restricted to incoherent (z-axis) states, independently of the specific time-dependence. Within strictly incoherent (SIO) classes, freezing of the relative entropy of coherence implies measure-independent freezing. Beyond analysis, we introduce two design templates that enforce exact freezing by making target off-diagonal operators fixed points of the Heisenberg generator: (A) a projector-based, ancilla-free construction and (B) a decoherence-free-subspace (DFS) scheme via collective symmetry. Numerical case studies corroborate the predicted hierarchy and the engineered-freeze protocols.