Researchers have developed new versions of quantum de Finetti theorems, applicable to both quantum states and channels, and valid for any distance measure. These theorems, which traditionally approximate the k-system marginals of permutation-invariant states of n-systems by mixtures of independent and identically distributed (iid) states, are now presented in a stronger form of operator inequalities. This implies that the error bounds are valid for every Schatten norm and for every quantum Rényi divergence satisfying data processing.

In the standard case, for a fixed local dimension, the new k/n error bound in max-relative entropy significantly improves upon the previously 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⁾:Aⁿ→Bⁿ, with d_A=dim A, it is proven that the k-system reduced channel is CP-dominated by a mixture of tensor-power channels with an error of O(k/√n) and a polynomial dependence on the local dimensions. This addresses a question previously raised by Berta et al. Furthermore, under the no-signalling condition, an exponential channel de Finetti theorem is proven, 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)(2d_A⁴k³/(nr²))⁽ʳ⁺¹⁾/² and decays exponentially with n for a suitable choice of parameters r and k.