Error-disturbance relations in mixed states

Heisenberg's uncertainty principle was originally formulated in 1927 as a quantitative relation between the "mean error" of a measurement of one observable and the disturbance thereby caused on another observable. Heisenberg derived this famous relation under an additional assumption on quantum measurements that has been abandoned in the modern theory, and its universal validity was questioned in a debate on the sensitivity limit to gravitational-wave detectors in 1980s. A universally valid form of the error-disturbance relation was shown to be derived in the modern framework of general quantum measurements in 2003. We have experienced a considerable progress in theoretical and experimental study of error-disturbance relations in the last decade. In 2013 Branciard showed a new stronger form of universally valid error-disturbance relations, one of which is proved tight for spin measurements carried out in "pure" states. Nevertheless, a recent information-theoretical study of error-disturbance relations has suggested that Branciard relations can be considerably strengthened for measurements in mixed states. Here, we show a method for strengthening Branciard relations in mixed states and derive several new universally valid and stronger error-disturbance relations in mixed states. In particular, it is proved that one of them gives an ultimate error-disturbance relation for spin measurements, which is tight in any state. The new relations will play an important role in applications to state estimation problems including quantum cryptographic scenarios.