note that though Ref. [@univUDD] focused on single-qubit decoherence in a bath, Eq. (\[Tn1\]) was proved therein for arbitrary Hermitian operators $C$ and $Z$. This motivated us to investigate under what conditions the unitary evolution operator of a controlled two-qubit system plus a bath can assume the same form as Eq. (\[oldun\]).

Decoherence Suppression in Two-qubit Systems
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Quantum coherence is often characterized by the magnitude of the off-diagonal matrix elements of the system density operator after tracing over the bath. In single-qubit cases, the transverse polarization then measures the coherence and the longitudinal polarization measures the population difference. Such a perspective is often helpful so long as its representation-dependent nature is well understood. In two-qubit systems or general multi-level systems, the concept of quantum coherence becomes more ambiguous because there are many off-diagonal matrix elements of the system density operator. Clearly then, to have a general and convenient coherence measure will be important for extending decoherence suppression studies beyond single-qubit systems.

Here we define a generalized polarization operator to characterize a certain type of coherence. Specifically, associated with an arbitrary pure state $|\Psi\rangle$ of our quantum system, we define the following polarization operator, $$\begin{aligned}
{\cal P}_{|\Psi \rangle} \equiv 2|\Psi\rangle\langle 