Unification of Random Dynamical Decoupling and the Quantum Zeno Effect

Abstract

Periodic deterministic bang-bang dynamical decoupling and the quantum Zeno effect are known to emerge from the same physical mechanism. Both concepts are based on cycles of strong and frequent kicks provoking a subdivision of the Hilbert space into independent subspaces. However, previous unification results do not capture the case of random bang-bang dynamical decoupling, which can be advantageous to the deterministic case but has an inherently acyclic structure. Here, we establish a correspondence between random dynamical decoupling and the quantum Zeno effect by investigating the average over random decoupling evolutions. This protocol is a manifestation of the quantum Zeno dynamics and leads to a unitary bath evolution. By providing a framework that we call equitability of system and bath, we show that the system dynamics under random dynamical decoupling converges to a unitary with a decoupling error that characteristically depends on the convergence speed of the Zeno limit. This reveals a unification of the random dynamical decoupling and the quantum Zeno effect.

Optimal Convergence Rate in the Quantum Zeno Effect for Open Quantum Systems in Infinite Dimensions

“…Another advantage of our setup is the freedom it provides for choosing the Banach space X , which allows us to treat open quantum systems (X = T (H) the trace class operators over a Hilbert space) and closed quantum systems (X = H a Hilbert space) on the same footing. In the case of finite-dimensional closed quantum systems, Proposition 3.1 reduces to the following bound, which was independently proven in [19,Theorem 1] (up to a change of the numerical constant in the quadratic term from 5 2 to 2): Corollary 3.2. Let H be a Hilbert space, H ∈ B(H) be a Hermitian operator, and P ∈ B(H) a Hermitian projection.…”

“…However, the optimality of ( 2) was left open. 1 Main contributions In this paper, we achieve the optimal convergence rate O(n −1 ) of the Zeno sequence consistent with the finite-dimensional case [5] by providing an explicit bound which recently attracted interest in finite closed quantum systems [19,Theorem 1]. Moreover, we generalize the results of [2] in two complementary directions: In Theorem 5.1, we assume a special case of the uniform power convergence assumption on M , that is M n −P ≤ δ n for some δ ∈ (0, 1), and weaken the assumption on the semigroup to the uniform asymptotic Zeno condition inherited from the unitary setting of [35]: for t → 0 (1 − P )e tL P ∞ = O(t) and P e tL (1 − P ) ∞ = O(t).…”

Self-restricting Noise and Quantum Relative Entropy Decay

“…Historically, the quantum Zeno effect is usually stated as frequent measurements suppressing the time-evolution of a quantum system away from the basis of measurement [MS77]. A number of results that include versions of the generalized Zeno effect [BZ18, MW19, BFN + 20, BDS21, MR21], dynamical decoupling [FLP04,HBY21], and adiabatic theorems [Kat50, BFN + 19] show that in general, a quantum process that rapidly changes the state will modify the effective dynamics of a simultaneous, slower process. Frequent applications of a channel or a continuous process with detailed balance may suppress dynamics of a concurrent Hamiltonian.…”