Upper bound on our knowledge about noncommuting observables for a qubit system

A trade-off relation on our knowledge about two noncommuting observables of a qubit system in simultaneous measurement is formulated. The obtained inequality offers a quantitative information-theoretic representation of Bohr’s principle of complementarity, and can be interpreted as a trade-off relation on the asymptotic accuracy of the maximum-likelihood estimation of the probability distributions of observables.

Figure 1

(Color) Trade-off relation on the accuracy of noncommuting observables. $P$ and $Q$ indicate the regimes satisfying inequality (12) for the case of $\theta =\pi ∕6$. The regime $Q$ is accessible only through simultaneous measurement.