Unitary-accessible coherence-erasure distance: Exact qubit solution and a tight qutrit bound

Abstract

We introduce the unitary-accessible coherence-erasure distance, which measures the minimum unitary-control cost required to transform a quantum state into an incoherent state without changing its spectrum. The target states are restricted to incoherent states on the same unitary orbit, leading to a geometry different from conventional state-space proximity. We establish inequalities relating orbit-restricted quantum transport, Bures geometry, and the unitary-control distance. These quantities coincide for pure states but can differ for mixed states. For qubits, we derive an exact expression and show that, in the nondegenerate case, the coherence-erasure cost depends on the eigenbasis orientation rather than on the eigenvalues. For nondegenerate qutrits, we reduce the problem to an optimization over the monomial unitary group. We derive the tight universal bound $\arccos[(2\sqrt2-1)/4]$ and construct an explicit eigenbasis that attains it. The resulting distance also determines the minimum coherence-erasure time under bounded unitary driving, giving it a direct operational meaning.