Universally Valid Error-Disturbance Relations in Continuous Measurements

TL;DR

This paper derives new error-disturbance relations for continuous quantum measurements in Fourier space, confirming tradeoffs and optimal disturbance levels in an optomechanical system, and highlighting limits imposed by quantum mechanics.

Contribution

It introduces universally valid error-disturbance inequalities for continuous measurements in Fourier space, extending previous discrete measurement results.

Findings

New EDR inequalities in Fourier space for continuous measurements

Confirmation of error-disturbance tradeoff and optimal disturbance strength

Inequalities are not saturated due to quantum limits

Abstract

In quantum physics, measurement error and disturbance were first naively thought to be simply constrained by the Heisenberg uncertainty relation. Later, more rigorous analysis showed that the error and disturbance satisfy more subtle inequalities. Several versions of universally valid error-disturbance relations (EDR) have already been obtained and experimentally verified in the regimes where naive applications of the Heisenberg uncertainty relation failed. However, these EDRs were formulated for discrete measurements. In this paper, we consider continuous measurement processes and obtain new EDR inequalities in the Fourier space: in terms of the power spectra of the system and probe variables. By applying our EDRs to a linear optomechanical system, we confirm that a tradeoff relation between error and disturbance leads to the existence of an optimal strength of the disturbance in a joint measurement. Interestingly, even with this optimal case, the inequality of the new EDR is not saturated because of doublely existing standard quantum limits in the inequality.