Hybrid Schrödinger-Liouville and projective dynamics

TL;DR

The paper reframes quantum measurement as ex ante dynamics on a refined quantum-classical state space, unifying continuous Schrödinger-Liouville evolution with measurement-induced updates via the Diósi-Poulin equation. It identifies conditions under which unitary, Lindbladian, and projective measurement dynamics arise as special cases, and provides an exactly solvable measurement implementation that exhibits exponential convergence to measurement statistics. A key contribution is the explicit construction of a projective-measurement dynamics within a single differential equation, and the demonstration of non-inertial measurement effects that deviate from instantaneous axiomatic predictions. This framework paves the way for port-Hamiltonian/control-theoretic approaches to quantum systems with measurements and classical-quantum interfaces, and suggests new avenues for modeling interactions with gravity and dynamical measurement devices.

Abstract

Quantum dynamics provides the arguably most fundamental example of hybrid dynamics: As long as no measurement takes place, the system state is governed by the Schrödinger-Liouville differential equation, which is however interrupted and replaced by projective dynamics at times when measurements take place. We show how this alternatingly continuous and projective evolution can be cast in form of one single differential equation for a refined state space manifold and thus be made amenable to standard port-theoretic analysis and control techniques.

Figures (2)

• Figure 1: Temporal evolution of the Bloch vector $\vec{r}$ of a qubit in the pure initial state $\hat{\rho}(0)$ corresponding to $\vec{r}(0)=(1/2,0,0)$ at the beginning of the non-inertial dynamically implemented measurement specified by (\ref{['eqn:rotM']}) and (\ref{['eqn:rotn']}), for various angular velocities $\omega$ and the choice $\gamma=1$ for the numerical solution. Since $r_3(t)=0$ throughout, the entire evolution takes place in the shown equatorial plane of the Bloch ball.
• Figure 2: Probability $p(1,t)$ for dynamically modelled non-inertial measurement apparatuses, under the same conditions and with corresponding colour coding as in Fig. \ref{['fig:Blochtrajectory']}, to yield the measurement result $+1$ if it is read off at time $t$. While curves for $\omega$ not equal, but very close to zero (not shown), almost reach probability one, all of these finally converge to probability $0.5$ and not $1$.