Information-Geometric Quantum Process Tomography of Single Qubit Systems

Abstract

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.

Figures (3)

• Figure 1: Time evolution of the Bloch vector components $a_i$. The horizontal axis represents the dimensionless time $t/T_0$, where $T_0$ is a constant with the dimension of time. The left panel shows ideal trajectories governed by the GKSL equation. The right panel shows trajectories perturbed by Gaussian white noise to mimic experimental errors. These results are obtained using the parameters $\bm{e} = (1.0, -0.6, 0.4)/T_0$ and $\bm{d} = (0.2, 0.3, 0.1)/T_0$, with the initial condition $\bm{a}(0) = (0.815, -0.007, 0.466)$.
• Figure 2: Convergence of the Hamiltonian parameter estimation. The estimated Hamiltonian parameters $e_i$ are plotted against the number of accumulated data points. The left and right panels correspond to the estimation results using the noiseless and noisy Bloch vector trajectories shown in Fig. \ref{['fig:bloch_vec']}, respectively. The symbols denote the estimated parameters as follows: triangles for $T_0 e_1$, circles for $T_0 e_2$, and crosses for $T_0 e_3$.
• Figure 3: Convergence of the dissipation parameter estimation. The estimated dissipation parameters $d_i$, corresponding to the relaxation rates, are plotted against the number of accumulated data points. Similar to Fig. \ref{['fig:parameters_fit1']}, the left and right panels show the estimation using the pure GKSL dynamics and the perturbed data, respectively. The symbols denote the estimated parameters as follows: triangles for $T_0 d_1$, circles for $T_0 d_2$, and crosses for $T_0 d_3$.