Fractional Conformal Map, Qubit Dynamics and the Leggett-Garg Inequality

Abstract

Any pure state of a qubit can be geometrically represented as a point on the extended complex plane through stereographic projection. By employing successive conformal maps on the extended complex plane, we can generate an effective discrete-time evolution of the pure states of the qubit. This work focuses on a subset of analytic maps known as fractional linear conformal maps. We show that these maps serve as a unifying framework for a diverse range of quantum-inspired conceivable dynamics, including (i) unitary dynamics,(ii) non-unitary but linear dynamics and (iii) non-unitary and non-linear dynamics where linearity (non-linearity) refers to the action of the discrete time evolution operator on the Hilbert space. We provide a characterization of these maps in terms of Leggett-Garg Inequality complemented with No-signaling in Time (NSIT) and Arrow of Time (AoT) conditions.

Figures (2)

• Figure 1: Fractional Conformal Maps in two dimension and the Lüders bound
• Figure 2: Optimal LG $K_3$ values for a qubit undergoing dynamics induced by discrete FLC maps $f_{12}$, $f_{23}$, and $f_{13}$ (refer to Appendix-\ref{['AppC2']}). (a) Left diagram corresponds to the maps $f_{12}(z) = \frac{\alpha z + \beta}{\beta z + \alpha}$ and $f_{23}(z) = \frac{\gamma z + \delta}{\delta z + \gamma}$. (b) Right diagram corresponds to the maps $f_{12}(z) = \frac{\alpha z + \beta}{\beta^{*} z + \alpha^{*}}$ and $f_{23}(z) = \frac{\gamma z + \delta}{\delta^{*} z + \gamma^{*} }$. In both cases, the plot is independent of the initial qubit state on the Bloch sphere.