Optimal-speed unitary quantum time evolutions and propagation of light with maximal degree of coherence

It is recognized that Grover arrived at his original quantum search algorithm inspired by his comprehension of the interference of classical waves originating from an array of antennas. It is also known that quantum-mechanical characterization of electromagnetic radiation is isomorphic to the treatment of the orientation of a spin-$1/2$ particle. In this paper, motivated by Grover's original intuition and starting from this mathematical equivalence, we present a quantitative link between the geometry of time-independent optimal-speed Hamiltonian evolutions on the Bloch sphere and the geometry of intensity-preserving propagation of light with maximal degree of coherence on the Poincaré sphere. Finally, identifying interference as the fundamental physical ingredient underlying both physical phenomena, we propose that our work can provide in retrospect a quantitative geometric background underlying Grover's powerful intuition.

Figure 1

(a) The Bloch sphere for pure quantum states of a single qubit. A point $P=P\left(\theta \text{,}\phi \right)$ on the surface of the Bloch sphere is defined by the Bloch vector $\vec{r}\stackrel{\text{def}}{=}\left(r_x\text{,}{r}_y\text{,}{r}_z\right)=(rsin\theta cos\phi \text{,}rsin\theta sin\phi \text{,}rcos\theta )$ with $‖\vec{r}‖=1$. Mixed states are specified by $‖\vec{r}‖\le 1$, with the origin representing a maximally mixed state. The angles $\theta$ and $\phi$ are the polar and the azimuthal angles, respectively. (b) The Poincaré sphere of unit radius for the polarized state of a beam of light. A point $P=P\left(\beta \text{,}\chi \right)$ on the surface of the Poincaré sphere is defined by the vector $\vec{s}\stackrel{\text{def}}{=}\left(S_1\text{,}{S}_2\text{,}{S}_3\right)=(S_{0}cos2\beta cos2\chi \text{,}{S}_{0}cos2\beta sin2\chi \text{,}{S}_{0}sin2\beta )$ where $\vec{S}\stackrel{\text{def}}{=}\left(S_0\text{,}{S}_1\text{,}{S}_2\text{,}{S}_3\right)$ with $S_1^2+S_2^2+S_3^2=S_0^2\equiv 1$ is the Stokes vector specified by the four Stokes parameters $\left\{S_0\text{,}{S}_1\text{,}{S}_2\text{,}{S}_3\right\}$. The parameter $S_0$ denotes the total intensity of the beam, while the remaining three parameters $S_1, S_2$, and $S_3$ specify the polarization state of the beam. A partially polarized beam of light is specified by $S_0\le 1$, with the origin being a completely unpolarized beam. Finally, $\beta$ and $\chi$ are the ellipticity and the orientation angles, respectively.