Fixed points in de Finetti hierarchies

De Finetti theorems convert permutation symmetry into approximate mixtures of product states and thereby justify a wide range of reductions in classical and quantum statistics.

In this work we study de Finetti hierarchies in which the feasible states are additionally constrained to be fixed points of quantum channels, a condition that subsumes invariance under arbitrary compact symmetry groups.

Combining the mean-ergodic theorem with the structure theory of conditional expectations, we prove a tight bound on the entanglement-assisted classical capacity of the dual of a conditional expectation, block-wise distortion bounds for informationally complete measurements adapted to fixed point algebras, and an exact type-based refinement of the chain rule for permutation-invariant states.

From these tools we derive several de Finetti theorems: a double-sided extension theorem with $O\left(\sqrt{\log n}/n\right)$ convergence, an interpolation theorem whose dimension dependence is governed solely by the block structure of the fixed point algebras and which recovers the dimension-independent classical behavior for maximal tori, and a Bose-symmetric variant.

Exploiting Schur-Weyl duality and Gelfand-Tsetlin bases, we further show that the rounding scheme producing certifiably good separable inner approximations for (constrained) separability problems can be implemented in time polynomial in $1/\epsilon$ for fixed local dimensions, complementing the known efficient outer hierarchies.

Applications to bilinear optimization under symmetries and to approximate quantum error correction are discussed.