Deriving tight error-trade-off relations for approximate joint measurements of incompatible quantum observables

The quantification of the measurement uncertainty aspect of Heisenberg's uncertainty principle-that is, the study of trade-offs between accuracy and disturbance, or between accuracies in an approximate joint measurement on two incompatible observables-has regained a lot of interest recently. Several approaches have been proposed and debated. In this paper we consider Ozawa's definitions for inaccuracies (as root-mean-square errors) in approximate joint measurements, and study how these are constrained in different cases, whether one specifies certain properties of the approximations-namely their standard deviations and/or their bias-or not. Extending our previous work [C. Branciard, Proc. Natl. Acad. Sci. USA 110, 6742 (2013)], we derive error-trade-off relations, which we prove to be tight for pure states. We show explicitly how all previously known relations for Ozawa's inaccuracies follow from ours. While our relations are in general not tight for mixed states, we show how these can be strengthened and how tight relations can still be obtained in that case.