  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.
