Information-Geometric Quantum Process Tomography of Single Qubit Systems

We establish an exact information-geometric inequality that remains valid regardless of the underlying dynamics, encompassing both Markovian and non-Markovian evolutions within the mixed-state domain. This inequality can be viewed as an extension of thermodynamic speed limits, which are typically formulated as inequalities. For single qubits, we show that this inequality saturates into a strict equality because the density matrix belongs to the quantum exponential family with the Pauli matrices serving as sufficient statistics. From a practical perspective, this identity enables a non-iterative linear regression approach to continuous-time quantum process tomography, bypassing the local minima issues common in non-linear optimization. We demonstrate the efficiency of this method by estimating the Hamiltonian and dissipation parameters of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation. Numerical simulations confirm the validity of this geometric estimator and highlight the necessity of error mitigation near the pure-state boundary where the inverse metric becomes singular.