Optimal Quantum de Finetti Theorems via Argmax Rounding
2026-08-03 • Computational Complexity
Computational Complexity
AI summaryⓘ
The authors prove the best possible upper bounds for approximating certain quantum states called bosonic states using a method known as the quantum de Finetti theorem. They show how close these states are to simpler states by measuring their difference and find precise formulas for this closeness depending on the size and dimension of the system. Their work resolves previous open questions about how these bounds depend on the system's dimension and extends to more general quantum states. The techniques involve advanced mathematical programming and lead to applications that challenge earlier conjectures and improve certain quantum state testing algorithms.
quantum de Finetti theorembosonic statesquantum state approximationexchangeable statessemidefinite programmingsum-of-squares roundingseparability testingBest Separable State problemHilbert--Schmidt distanceWatrous's disentangler conjecture
Authors
Fernando Granha Jeronimo, Pei Wu, Haochen Xu
Abstract
We prove optimal finite quantum de Finetti upper bounds. Given a bosonic state $ρ_N\in D(\mathrm{Sym}^N(\mathbb C^d))$, there is a probability measure $ν$ on the unit sphere such that \[ \left\| ρ_N^{(2)}-\int |u\rangle\langle u|^{\otimes 2}\,dν(u) \right\|_1 \le \frac{\sqrt{d-1}}{N-1}. \] 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 $O(t\sqrt d/N)$ 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 $\varepsilon\in(0,1)$, we construct a channel with input dimension $D=\exp(O_\varepsilon(\sqrt d\log d))=\exp(o(d))$ whose outputs are $\varepsilon$-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 $\exp(\widetilde O(\sqrt d/\varepsilon))$-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.