Geometric quantum speed limits for Markovian dynamics in open quantum systems

Abstract

We study theoretically the geometric quantum speed limits (QSLs) of open quantum systems under Markovian dynamical evolution. Three types of QSL time bounds are introduced based on the geometric inequality associated with the dynamical evolution from an initial state to a final state. By illustrating three types of QSL bounds at the cases of presence or absence of system driving, we demonstrate that the unitary part, dominated by system Hamiltonian, supplies the internal motivation for a Markovian evolution which deviates from its geodesic. Specifically, in the case of unsaturated QSL bounds, the parameters of the system Hamiltonian serve as the eigen-frequency of the oscillations of geodesic distance in the time domain and, on the other hand, drive a further evolution of an open quantum system in a given time period due to its significant contribution in dynamical speedup. We present physical pictures of both saturated and unsaturated QSLs of Markovian dynamics by means of the dynamical evolution trajectories in the Bloch sphere which demonstrates the significant role of system Hamiltonian even in the case of initial mixed states. It is further indicated that whether the QSL bound is saturated is ruled by the coummutator between the Hamiltonian and reduced density matrix of the quantum system. Our study provides a detailed description of QSL times and reveals the effects of system Hamiltonian on the unsaturation of QSL bounds under Markovian evolution.

Realizing quantum speed limit in open system with a PT -symmetric trapped-ion qubit

“…Despite the well-established limit for the unitary dynamics, there does not exist a single standard for the QSL in open systems, due to the complexity of the coupling between the system and the environment. In recent years, intensive theoretical studies have been devoted to explore how fast an open quantum system can evolve [16][17][18][19][20][21][22][23][24][25][26][27]. One of the particular examples is that for a non-Hermitian PT -symmetric system, the state transfer can speedup and the flipping time will approach to an infinitesimal time scale without violating the time-energy uncertainty principle [28,29].…”

Quantum dynamical speedup for correlated initial states

“…In recent years, due to the fact that it is difficult to access an initial pure state in practical scenarios, studying the QSL for mixed initial states has been the subject of some works [20,21]. In addition to providing different bounds, some works have been done on the issue of QSL, such as the dependence of QSL on the initial state [22], and many other works [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41].…”

Phase Covariant Channel: Quantum Speed Limit of Evolution

“…It is also known that non‐Markovianity is not always required to speed up quantum evolution. [ 31–33 ] From the practical point of view, $$\tau _{QSL}$$ finds many applications in a wide range of fields. [ 34 ] The phase‐covariant map describes the physical processes involving absorption, emission, and pure dephasing.…”