Noise and Disturbance of Qubit Measurements: An Information-Theoretic Characterisation

TL;DR

This paper develops and analyzes information-theoretic measures for noise and disturbance in qubit measurements, deriving tight uncertainty relations and exploring optimal measurement strategies for joint estimation and disturbance tradeoffs.

Contribution

It introduces invariant, state-independent definitions for noise and disturbance, derives tight uncertainty relations for incompatible qubit observables, and investigates optimal measurement configurations.

Findings

Set of noise-noise values is convex.

Set of noise-disturbance values is not convex.

Four-outcome measurements are needed for optimal joint noise estimation.

Abstract

Information-theoretic definitions for the noise associated with a quantum measurement and the corresponding disturbance to the state of the system have recently been introduced [F. Buscemi et al., Phys. Rev. Lett. 112, 050401 (2014)]. These definitions are invariant under relabelling of measurement outcomes, and lend themselves readily to the formulation of state-independent uncertainty relations both for the joint estimate of observables (noise-noise relations) and the noise-disturbance tradeoff. Here we derive such relations for incompatible qubit observables, which we prove to be tight in the case of joint estimates, and present progress towards fully characterising the noise-disturbance tradeoff. In doing so, we show that the set of obtainable noise-noise values for such observables is convex, whereas the conjectured form for the set of obtainable noise-disturbance values is not. Furthermore, projective measurements are not optimal with respect to the joint-measurement noise or noise-disturbance tradeoffs. Interestingly, it seems that four-outcome measurements are needed in the former case, whereas three-outcome measurements are optimal in the latter.