Partial Wavefunction Collapse Under Repeated Weak Measurement of a non-Conserved Observable

TL;DR

This paper explores a novel quantum measurement scenario where a secondary observable exhibits QND-like behavior despite the primary observable not commuting with the Hamiltonian, revealing new insights into quantum measurement dynamics.

Contribution

It introduces the concept of Auxiliary Observable QND measurement, showing how a secondary observable can display QND behavior even when the primary observable does not commute with the Hamiltonian.

Findings

Expectation value of the target observable is conserved.

Partial system collapse occurs with respect to the target observable.

Local measurements can reveal global spectral information.

Abstract

Two hallmarks of quantum non-demolition (QND) measurement are the ensemble-level conservation of the expectation value of the measured observable $A$ and the eventual, inevitable collapse of the system into some eigenstate of $A$. This requires that $A$ commutes with $H$, the system's Hamiltonian. In what we term "Auxiliary Observable QND" measurement, $A$ does not commute with $H$ and the above two characteristics clearly cannot be present as the system's dynamics prevent $\langle A \rangle$ from reaching a definite value. However, in this paper we find that under such a measurement QND behavior still arises, but is seen in the behavior of a secondary "target" observable we call $B$, with the condition that $B$ commutes with both $A$ and $H$. In such cases, the expectation value of $B$ is conserved and the system at least partially collapses with respect to eigenstates of $B$. We show as an example how this surprising result applies to a Heisenberg chain, where we demonstrate that local measurements on a single site can reveal information about the spectrum of an entire system, a finding which may be of practical use in experiments.