Shortest evolution path between two mixed states and its realization

In a quantum unitary system, an initial state may have different paths to evolve to a target state, and the only requirement is that the initial and target states share the same eigenvalue spectrum. We focus on the evolution between two nondegenerate mixed states in this paper, and investigate the shortest evolution path between them. By minimizing the path distance contained in the unitary operator connecting the initial and target states over a series of phases, the shortest evolution path could be figured out. This minimum path distance has an analytical form in the single-qubit dynamical system, and its solution in the three- or higher-dimensional dynamical system could be obtained numerically. Based on the unitary operator associated with the shortest evolution path, a general form for the Hamiltonian to realize it is presented. Here we present another way to study quantum optimal control, which is based on the path distance between the initial state and its evolution state, rather than the state distance between them.

Figure 1

Numerical results of the path distance contained in the unitary operator (24) as a function of the two parameters $\delta {\varphi }_1$ and $\delta {\varphi }_2$. The maximum path distance $d\left(U_{\left\{\delta \varphi \right\}}\right)\simeq 2.0944$ occurs at $\delta {\varphi }_1=0$ and $\delta {\varphi }_2\simeq 2.4981$, and the minimum path distance $d\left(U_{\left\{\delta \varphi \right\}}\right)\simeq 1.2490$ occurs at $\delta {\varphi }_1=\delta {\varphi }_2\simeq -2.4981$. Each point in the curved surface corresponds to a unitary operator turning the initial state ${\rho }_0$ (17) to the target state ${\rho }_{\tau }$ (18), thus describes an evolution path from ${\rho }_0$ to ${\rho }_{\tau }$. Although these paths connect the same pair of states, they may contain different path distances.