On the Existence of Stationary States

Krišjānis Prūsis, Jevgēnijs Vihrovs, Thomas G. Wong

When classically searching a database, having additional correct answers makes the search easier. For a discrete-time quantum walk searching a graph for a marked vertex, however, additional marked vertices can make the search harder by causing the system to approximately begin in a stationary state, so the system fails to evolve. In this paper, we completely characterize the stationary states, or 1-eigenvectors, of the quantum walk search operator for general graphs and configurations of marked vertices by decomposing their amplitudes into uniform and flip states. This infinitely expands the number of known stationary states and gives an optimization procedure to find the stationary state closest to the initial uniform state of the walk. We further prove theorems on the existence of stationary states, with them conditionally existing if the marked vertices form a bipartite connected component and always existing if nonbipartite. These results utilize the standard oracle in Grover's algorithm, but we show that a different type of oracle prevents stationary states from interfering with the search algorithm.

Wolfgang Ernst Friederich Pauli

J. Y. Zeng, Yian Lei

The intimate connection between the Berry phase and the nonstationarity of a quantum state is discussed. For a quantum system with time-independent Hamiltonian $H$, there may exist stationary states. It is shown that for a stationary state, the Berry phase always vanishes, and only for a nonstationary state after a cyclic evolution might the Berry phase appear. For time-dependent $H$, there exists no exactly stationary state, hence the Berry phase may appear.

Ketill Ingólfsson

Yu. S. Tyupkin, A. S. Shvart︠s︡

Franco Fagnola, Roberto B. Quezada

We analyze the two-photon absorption and emission process and characterize the stationary states at zero and positive temperature. We show that entangled stationary states exist only at zero temperature and, at positive temperature, there exists infinitely many commuting invariant states satisfying the detailed balance condition.