Speaker
Description
We develop quantitative de Finetti representation theorems beyond standard quantum settings, driven by the principle that permutation symmetry enforces approximate independence at finite extension level. First, using a GPT-motivated notion of relative entropy (via an integral representation) we define mutual information for general convex state spaces and prove a uniform monogamy bound for multipartite extensions: for permutation-invariant states, the total mutual information is bounded by a constant depending only on the A-system. This yields an information-theoretic finite de Finetti theorem for convex bodies, asserting that the $AB$-marginal of an $n$-extendible (max-tensor) state is close to a separable (min-tensor) state.
Second, for constrained separability problems arising in quantum information, we establish constrained de Finetti theorems compatible with additional linear marginal/fixed-point constraints, including a Bose-symmetric variant that operates directly on symmetric-subspace-supported extensions. These results provide quantitative trace-norm closeness of constrained symmetric (or Bose-symmetric) extensions to convex mixtures of constrained product states, enabling de Finetti control in settings where extremal decompositions into pure products are unavailable. The work is based on arXiv:2507.12326, 2507.12302 and 2601.15184.