Approximate quantum non-demolition measurements

TL;DR

This paper investigates how well finite-dimensional quantum systems can approximate ideal quantum nondemolition measurements for waveform estimation, revealing that better approximations require large energy or system size.

Contribution

It demonstrates that finite-dimensional approximations of QND measurements improve slowly with system size, highlighting the energy and dimensionality requirements for effective approximations.

Findings

Estimation errors decrease as a power law with increasing dimension.

Good QND approximations demand large energy or system size.

Results likely extend to truncated oscillators or spin systems.

Abstract

With the advent of gravitational wave detectors employing squeezed light, quantum waveform estimation---estimating a time-dependent signal by means of a quantum-mechanical probe---is of increasing importance. As is well known, backaction of quantum measurement limits the precision with which the waveform can be estimated, though these limits can in principle be overcome by "quantum nondemolition" (QND) measurement setups found in the literature. Strictly speaking, however, their implementation would require infinite energy, as their mathematical description involves Hamiltonians unbounded from below. This raises the question of how well one may approximate nondemolition setups with finite energy or finite-dimensional realizations. Here we consider a finite-dimensional waveform estimation setup based on the "quasi-ideal clock" and show that the estimation errors due to approximating the QND condition decrease slowly, as a power law, with increasing dimension. As a result, we find that good QND approximations require large energy or dimensionality. We argue that this result can be expected to also hold for setups based on truncated oscillators or spin systems.