Noise and Disturbance in Quantum Measurements: An Information-Theoretic Approach

We introduce information-theoretic definitions for noise and disturbance in quantum measurements and prove a state-independent noise-disturbance tradeoff relation that these quantities have to satisfy in any conceivable setup. Contrary to previous approaches, the information-theoretic quantities we define are invariant under the relabelling of outcomes and allow for the possibility of using quantum or classical operations to “correct” for the disturbance. We also show how our bound implies strong tradeoff relations for mean square deviations.

Figure 1

Noise with respect to $X$ is measured by the ability to guess correctly, from the measurement outcome $M=m$, in which eigenstate $|{\psi }^x⟩$ of $X$ the system was initially prepared. The guessed value $\widehat{x}$ will, in general, be some function $f(m)$ of the measurement outcome. Optimization over $f$ is allowed.

Figure 2

Disturbance with respect to $Z$ is measured by the ability to guess correctly, from the outcome variable $M$ and the “disturbed” output quantum system $S^{\prime }$, in which eigenstate $|{\varphi }^z⟩$ of $Z$ the system was initially prepared. It is permitted to apply a quantum operation $\mathcal{E}$ to attempt to correct or minimize any disturbance, prior to a measurement of $Z$ (with outcome $\widehat{z}$). Optimization over $\mathcal{E}$ is allowed.