Quantum de Finetti theorems for states and channels in any distance measure
Abstract
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=dim A, 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.
Related
- Exponential de Finetti Theorems for Fermionic Gaussian States
- De Finetti Structures in Networks of Quantum Systems
- Optimal Diamond Distance Estimation for Unitary Channels
- Irreversibility of quantum field theory in de Sitter: The C, F and A theorems
- Error-Correction Transitions in Finite-Depth Quantum Channels
BibTeX
@misc{ye2026quantum,
title = {{Quantum de Finetti theorems for states and channels in any distance measure}},
author = {Liuhang Ye and Bjarne Bergh and Nilanjana Datta},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40343},
eprint = {2609.40343},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40343},
}
RIS
TY - UNPB
TI - Quantum de Finetti theorems for states and channels in any distance measure
AU - Liuhang Ye
AU - Bjarne Bergh
AU - Nilanjana Datta
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40343
UR - https://arxiv.org/abs/2609.40343
AB - 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 k^2/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 d_A=dim A, 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)(2d_A^4k^3/(nr^2))^(r+1)/2 and decays exponentially in n for a suitable choice of parameters r and k.
ER -