  Noether's theorem is a fundamental result in physics stating that every
symmetry of the dynamics implies a conservation law. It is, however, deficient
in several respects: (i) it is not applicable to dynamics wherein the system
interacts with an environment, and (ii) even in the case where the system is
isolated, if the quantum state is mixed then the Noether conservation laws do
not capture all of the consequences of the symmetries. To address these
deficiencies, we introduce measures of the extent to which a quantum state
breaks a symmetry. Such measures yield novel constraints on state transitions:
for nonisolated systems, they cannot increase, while for isolated systems they
are conserved. We demonstrate that the problem of finding nontrivial asymmetry
measures can be solved using the tools of quantum information theory.
Applications include deriving model-independent bounds on the quantum noise in
amplifiers and assessing quantum schemes for achieving high-precision
metrology.
