  We show that two density operators of mixed quantum states are in the same
local unitary orbit if and only if they agree on polynomial invariants in a
certain Noetherian ring for which degree bounds are known in the literature.
This implicitly gives a finite complete set of invariants for local unitary
equivalence. This is done by showing that local unitary equivalence of density
operators is equivalent to local ${\rm GL}$ equivalence and then using
techniques from algebraic geometry and geometric invariant theory. We also
classify the SLOCC polynomial invariants and give a degree bound for generators
of the invariant ring in the case of $n$-qubit pure states. Of course it is
well known that polynomial invariants are not a complete set of invariants for
SLOCC.
