Information geometry aspects of minimum entropy production paths from quantum mechanical evolutions

We present an information geometric analysis of entropic speeds and entropy production rates in geodesic evolution on manifolds of parametrized quantum states. These pure states emerge as outputs of suitable $\mathrm{su}\left(2\text{;}\mathbb{C}\right)$ time-dependent Hamiltonian operators used to describe distinct types of analog quantum search schemes. The Riemannian metrization on the manifold is specified by the Fisher information evaluated along the parametrized squared probability amplitudes obtained from analysis of the temporal quantum mechanical evolution of a spin-$1/2$ particle in an external time-dependent magnetic field that specifies the $\mathrm{su}\left(2\text{;}\mathbb{C}\right)$ Hamiltonian model. We employ a minimum action method to transfer a quantum system from an initial state to a final state on the manifold in a finite temporal interval. Furthermore, we demonstrate that the minimizing (optimum) path is the shortest (geodesic) path between the two states and, in particular, minimizes also the total entropy production that occurs during the transfer. Finally, by evaluating the entropic speed and the total entropy production along the optimum transfer paths in a number of physical scenarios of interest in analog quantum search problems, we show in a clear quantitative manner that to a faster transfer there corresponds necessarily a higher entropy production rate. Thus we conclude that lower entropic efficiency values appear to accompany higher entropic speed values in quantum transfer processes.

Figure 1

Illustrative depiction of the behavior of the Fisher information $\mathcal{F}\left(\theta \right)$ versus $\theta$ [plot (c)] defined in terms of the transition probabilities $p_w\left(\theta \right)$ (dashed line) and $p_{w_{\perp }}\left(\theta \right)$ (solid line) [plot (b)]. These probabilities, in turn, emerge from the specific external magnetic field configuration $\vec{B}={\vec{B}}_{\perp }+{\vec{B}}_{\parallel }$ with ${\vec{B}}_{\perp }=B_x\widehat{x}+B_y\widehat{y}$ [plot (a)] that characterizes the $\mathrm{su}\left(2\text{;}\mathbb{C}\right)$ Hamiltonian model being considered. In plots (b) and (c), we set $\mathrm{\Gamma }/\hslash \lambda =\pi /2$ and $\lambda =1$. In plot (a), the two-dimensional parametric plot of the transverse magnetic field components, we set $\left|{\omega }_0\right|=10\pi$ and $\lambda =1$. Furthermore, for simplifying normalization purposes, we also set $\mathrm{\Gamma }/{\mu }_{\text{Bohr}}=1$ with ${\mu }_{\text{Bohr}}$ denoting the Bohr magneton in plot (a). All physical quantities are assumed to be suitably expressed in terms of the MKSA unit system. Finally, the $\mathrm{su}\left(2\text{;}\mathbb{C}\right)$ Hamiltonian model considered in this figure corresponds to the fourth scenario studied in the paper.

Figure 2

Illustrative depiction of the behavior of the Fisher information $\mathcal{F}\left(\theta \right)$ versus $\theta$ [plot (a)], the entropic speed $v_E\left(\xi \right)$ versus $\xi$ [plot (b)], and the entropic efficiency ${\eta }_E\left(\xi \right)$ versus $\xi$ [plot (c)]. Dotted, dashed, thin solid, and thick solid lines in plots (a), (b), and (c) correspond to constant, oscillatory, exponential decay, and power law decay of the Fisher information, respectively. In all plots, we impose $\mathrm{\Gamma }/\left(\hslash \lambda \right)=\pi /2, \lambda =1/\pi , {\theta }_0=1$, and ${\dot{\theta }}_0=1$. Finally, all physical quantities are assumed to be suitably expressed in terms of the MKSA unit system.

Figure 3

Plot of the two-dimensional parametric region $\mathcal{P}$ where the exponential decay strategy outperforms the power law decay strategy in terms of higher entropic speed values. The black region $\mathcal{P}$ tends to vanish as the values of $\lambda$ become sufficiently large.