A geometric Fano--Procrustes framework for purification-based distances and quantum channels analysis
In this work we reformulate the Uhlmann purification-overlap optimization and develop a purification-based geometric framework for the analysis of mixed qubit states and qubit channels. Using the Fano representation of two-qubit pure states, a purification is described in terms of the Bloch vector of the system, the ancilla Bloch vector, and a real correlation matrix. For a fixed one-qubit mixed state, the freedom in the choice of purification can be parametrized by proper rotations acting on the ancillary degrees of freedom. As a result, the optimization over purifications entering the definition of the metric \(D_N\) introduced in Ref.~\cite{Lamberti2009} is reduced to an orthogonal Procrustes problem on the Lie group \(SO(3)\). This reduction yields not only the maximal purification overlap, but also the optimal rotation relating the purification frames. From this rotation we define a purification misalignment angle \(Θ\), which provides geometric information not contained in scalar fidelity-based distinguishability measures. The formalism is applied to representative qubit channels, including depolarizing, bit-flip, phase-flip, amplitude-damping channels, and an imperfect quantum NOT gate. For symmetry-adapted evolutions preserving the Bloch-vector direction, the optimal rotation is trivial and \(Θ=0\), whereas noncollinear channel actions generate a nonzero misalignment. The pair \((D_N,Θ)\) therefore separates the magnitude of the maximal purification overlap from the geometric reorientation of the optimal purification frames. Since the optimal Procrustes rotation can be lifted to a local unitary acting on the ancilla, the construction also provides an operational interpretation of the optimal purification in terms of an ancilla-side transformation.