  We prove a new kind of quantum de Finetti theorem for representations of the
unitary group U(d). Consider a pure state that lies in the irreducible
representation U_{mu+nu} for Young diagrams mu and nu. U_{mu+nu} is contained
in the tensor product of U_mu and U_nu; let xi be the state obtained by tracing
out U_nu. We show that xi is close to a convex combination of states Uv, where
U is in U(d) and v is the highest weight vector in U_mu. When U_{mu+nu} is the
symmetric representation, this yields the conventional quantum de Finetti
theorem for symmetric states, and our method of proof gives near-optimal bounds
for the approximation of xi by a convex combination of product states. For the
class of symmetric Werner states, we give a second de Finetti-style theorem
(our 'half' theorem); the de Finetti-approximation in this case takes a
particularly simple form, involving only product states with a fixed spectrum.
Our proof uses purely group theoretic methods, and makes a link with the
shifted Schur functions. It also provides some useful examples, and gives some
insight into the structure of the set of convex combinations of product states.
