Path distance of a quantum unitary evolution
Jie‐Hui Huang, Shan-Shan Dong, Guang-Long Chen +3 more
The Hamiltonian and evolution time of a quantum unitary system completely determine a unitary operator, which turns an initial state ${\ensuremath{\rho}}_{0}$ into its time-evolved state ${\ensuremath{\rho}}_{t}$ and may also turn another state ${\ensuremath{\sigma}}_{0}$ into its time-evolved state ${\ensuremath{\sigma}}_{t}$. Although the two pairs of states $({\ensuremath{\rho}}_{0},{\ensuremath{\rho}}_{t})$ and $({\ensuremath{\sigma}}_{0},{\ensuremath{\sigma}}_{t})$ are governed by the same Hamiltonian and experience equal evolution periods, their state distances are not necessarily equal to each other. Specifically, the evolution operator contains information about the evolution other than that reflected by two specific states connected through it. As a matter of fact, two pairs of states with different state distances may have equal path distances. Here we propose a geometric measure for the path distance contained in a quantum unitary evolution, which depends only on the unitary operator itself and is independent of any pair of states connected through it. This path distance meets the basic requirements for a good measure of distance and is valid during the whole evolution process, no matter how long the evolution time is. This path distance is bounded by the path length of the evolution from above and by the Bures angle between the two states connected through it from below. It has potential applications in quantum information processing, such as providing another way to study the optimal control between two quantum states. We finally present the form of Hamiltonian to realize a unitary operator on demand.
Physical Review A•Aug 7, 2023•3 citations