  There are several instances where quantum anomalies of continuous and
discrete classical symmetries play an important role in fundamental physics.
Examples come from chiral anomalies in the Standard Model of fundamental
interactions and gravitational anomalies in string theories. Their generic
origin is the fact that classical symmetries may not preserve the domains of
quantum operators like the Hamiltonian. In this work, we show by simple
examples that anomalous symmetries can often be implemented at the expense of
working with mixed states having non-zero entropies. In particular there is the
result on color breaking by non-abelian magnetic monopoles. This anomaly can be
rectified by using impure states. We also argue that non-abelian groups of
twisted bundles are always anomalous for pure states sharpening an earlier
argument of Sorkin and Balachandran. This is the case of mapping class groups
of geons indicating that "large" diffeos are anomalous for pure states in the
presence of geons. Nevertheless diffeo invariance may be restored by using
impure states. This work concludes with examples of these ideas drawn from
molecular physics.
  The above approach using impure states is entirely equivalent to restricting
all states to the algebra of observables invariant under the anomalous
symmetries. For anomalous gauge groups such as color, this would mean that we
work with observables singlet under global gauge transformations. For color,
this will mean that we work with color singlets, a reasonable constraint.
