Quantum evolution

Lin-Feng Du, Li‐Guo Qin, Qin Guo +2 more

The average energy and energy spread of a quantum system are two important factors that affect a system's evolution speed, which is usually defined as the changing rate of the distance between a quantum state and the state during its evolution. Alternatively, the evolution speed of a quantum system could also be studied based on its evolution path. It is found that if we define the evolution speed of a quantum unitary system as the changing rate of the path distance between a quantum state and the state during its evolution, the instantaneous evolution speed is determined by the Hamiltonian's eigenvalues, rather than its expectation value or variance imposed on any quantum state. More specifically, the range of the Hamiltonian's eigenvalues determines the evolution speed of a quantum system through the relation ${\ensuremath{\omega}}_{H}=\frac{1}{2}({\ensuremath{\lambda}}_{max}\ensuremath{-}{\ensuremath{\lambda}}_{min})$, which is independent of the quantum state. Given an initial state ${\ensuremath{\rho}}_{0}$ in a quantum unitary system, the minimum time required to evolve it to a target state ${\ensuremath{\rho}}_{\ensuremath{\tau}}$ is ${\ensuremath{\tau}}_{min}=\frac{{\mathcal{L}}_{min}({\ensuremath{\rho}}_{0},{\ensuremath{\rho}}_{\ensuremath{\tau}})}{{\ensuremath{\omega}}_{H}}$, where ${\mathcal{L}}_{min}({\ensuremath{\rho}}_{0},{\ensuremath{\rho}}_{\ensuremath{\tau}})$ is the path distance contained in the shortest evolution from ${\ensuremath{\rho}}_{0}$ to ${\ensuremath{\rho}}_{\ensuremath{\tau}}$. This lower bound of the evolution time is tight and achievable in a finite-energy system.

Giulio Chiribella, Erik M. Aurell, Karol Życzkowski

A cornerstone of quantum mechanics is the characterisation of symmetries provided by Wigner's theorem. Wigner's theorem shows that every symmetry of the quantum state space must be either a unitary transformation, or a antiunitary transformation. Here we extend Wigner's theorem from quantum states to quantum evolutions, including both the deterministic evolution associated to the dynamics of closed systems, and the stochastic evolution associated to quantum measurements. We prove that every symmetry of the space of quantum evolutions can be decomposed into two state space symmetries that are either both unitary or both antiunitary. Building on this result, we show that it is impossible to extend the time reversal symmetry of unitary quantum dynamics to a symmetry of the full set of quantum evolutions. Our no-go theorem implies that any time symmetric formulation of quantum theory must either restrict the set of the allowed evolutions, or modify the operational interpretation of quantum states and processes. We propose one such modification, wherein the evolutions are restricted to a suitable set of time symmetric operations, including both unitary evolution and the state reductions associated to projective measurements.

Alex Khang, Kali Charan Rath

Lian-Ao Wu

Recently we showed that quantum cyclic evolution occurs, provided the initial wave function was one of the eigenfunctions of the evolution operator, whatever the concrete form of the Hamiltonian [Phys. Rev. A 50, 5317 (1994)]. This paper will show that, for some specific Hamiltonians, cyclic evolution may occur, even if the initial wave function is not one of the eigenfunctions of the evolution operator. The general conditions for the occurrence of cyclic evolution is derived. Several interesting examples of quantum evolution, also including noncyclic evolution, and some specific geometric phases are given. \textcopyright{} 1996 The American Physical Society.

Peter Murphy