We give a simple proof of the exponential de Finetti theorem due to Renner. Like Renner's proof, ours combines the post-selection de Finetti theorem, the Gentle Measurement lemma, and the Chernoff bound, but avoids virtually all calculations, including any use of the theory of types.
Standard finite quantum de Finetti theorems approximate the k-system marginals of permutation-invariant states of n-systems by mixtures of independent and identically distributed (iid) states, usually in trace distance. We prove both standard and Renner's exponential de Finetti theorems in the stronger form of operator inequalities, implying bounds in every Schatten norm and for every quantum Rényi divergence satisfying data processing. In the standard case, for fixed local dimension, our k/n error bound in max-relative entropy improves on the previously best known k2/n scaling, even in the classical setting. The operator-inequality approach is particularly suited to study channel de Finetti representations because operator order between Choi states is equivalent to completely positive (CP) order between the underlying channels. For permutation-covariant channels N(n):A⊗n→B⊗n, where dA=dimA, we prove that the k-system reduced channel is CP-dominated by a mixture of tensor-power channels with error O(k/n) and polynomial dependence on the local dimensions, addressing a question raised by Berta et al. [Math. Program. 194, 781-829 (2022)]. Under the no-signalling condition, we also prove an exponential channel de Finetti theorem where the approximating mixture consists of Choi-almost-iid channels, whose normalized Choi states are almost-iid in the sense of Mazzola-Sutter-Renner. In the case of r defects, the representation error is at most poly(n)(2dA4k3/(nr2))(r+1)/2 and decays exponentially in n for a suitable choice of parameters r and k.
We prove optimal finite quantum de Finetti upper bounds. Given a bosonic state ρN∈D(SymN(Cd)), there is a probability measure ν on the unit sphere such that ρN(2)−∫∣u⟩⟨u∣⊗2dν(u)1≤N−1d−1. By purification, the bosonic theorem also gives the optimal O(d/N) upper bound for arbitrary exchangeable states. These results settle the dimension dependence left open by Christandl, König, Mitchison, and Renner (CMP 2007). The proof casts de Finetti approximation as sum-of-squares rounding and applies the argmax method of Jeronimo, Wu, and Xu (manuscript 2026).
More generally, t-site marginals satisfy O(td/N) bosonic and O(td/N) permutation-invariant bounds. Our proof formulates de Finetti approximation as the integrality gap of a symmetric-extension semidefinite program and rounds an optimum by the argmax principle. The sharp bounds have several consequences. For every fixed ε∈(0,1), we construct a channel with input dimension D=exp(Oε(dlogd))=exp(o(d)) whose outputs are ε-close to separable states of local dimension d and whose image contains every such separable state, thereby refuting Watrous's disentangler conjecture. We also obtain deterministic exp(O(d/ε))-time algorithms for explicit Best Separable State without perfect completeness and for trace-distance separability testing.
Finally, spectral truncation gives the first dimension-free bosonic de Finetti theorem in Hilbert--Schmidt distance, with the optimal rate Θ(N−1/2) when the dimension may grow.