A continuous-time diffusion limit theorem for dynamical decoupling and intrinsic decoherence

Abstract

We discuss a few mathematical aspects of random dynamical decoupling, a key tool procedure in quantum information theory. In particular, we place it in the context of discrete stochastic processes, limit theorems and CPT semigroups on matrix algebras. We obtain precise analytical expressions for expectation and variance of the density matrix and fidelity over time in the continuum-time limit depending on the system Lindbladian, which then lead to rough short-time estimates depending only on certain coupling strengths. We prove that dynamical decoupling does not work in the case of intrinsic (i.e., not environment-induced) decoherence, and together with the above-mentioned estimates this yields a novel method of partially identifying intrinsic decoherence.

Distinguishing decoherence from alternative quantum theories by dynamical decoupling

“…It is fascinating to contemplate that in the vast experimental evidence for dynamical decoupling such AQTs have already been discovered. The analysis of such experiments requires a detailed mathematical analysis, parts of which we have provided in [8] and parts of it remain to be done in future.…”

“…Furthermore to distinguish extrinsic and intrinsic decoherence we need bounds. Using a central limit theorem, such bounds are developed in [8] for the expectation of the decoupling error¯ , while here we will focus on specific examples. The decoupling error = tr{(1 1 − Λ t,n ) † (1 1 − Λ t,n )}/d 2 compares the free evolution under random dynamical decoupling with the identity operation.…”

“…The decoupling error = tr{(1 1 − Λ t,n ) † (1 1 − Λ t,n )}/d 2 compares the free evolution under random dynamical decoupling with the identity operation. In the limit ∆t → 0, keeping the total time t fixed, the decoupling error becomes [8]…”

“…Our work is based on a very simple idea, namely that dynamical decoupling [7] -a popular method to suppress quantum noise -only works for systems which are truly coupled to environments [8], but not for systems which have intrinsic noise terms, as arriving from axiomatic modifications of Schrödinger's equation [6,[9][10][11].…”

Dynamical decoupling and homogenization of continuous variable systems

“…Before we develop dynamical decoupling for quadratic Hamiltonians in infinite-dimensional quantum systems, we first review the concept of dynamical decoupling for finite-dimensional quantum systems, focussing on the group-based approach [1,26]. Consider a finite-dimensional quantum system, say of dimension n ∈ N, with Hilbert space C n and Hamiltonian H. The idea of dynamical decoupling is to rapidly rotate the quantum system by means of classical fields in order to average the system-environment coupling in H to zero.…”