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arXiv:2608.02590

Optimal Quantum de Finetti Theorems via Argmax Rounding

Fernando Granha Jeronimo, Pei Wu, Haochen Xu

quant-phcs.CCSubmitted Aug 3, 2026• Announced Aug 4, 2026• 0 citations
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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.

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Quantum de Finetti theorems for states and channels in any distance measure

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De Finetti theorems convert permutation symmetry into approximate mixtures of product states and thereby justify a wide range of reductions in classical and quantum statistics. In this work we study de Finetti hierarchies in which the feasible states are additionally constrained to be fixed points of quantum channels, a condition that subsumes invariance under arbitrary compact symmetry groups. Combining the mean-ergodic theorem with the structure theory of conditional expectations, we prove a tight bound on the entanglement-assisted classical capacity of the dual of a conditional expectation, block-wise distortion bounds for informationally complete measurements adapted to fixed point algebras, and an exact type-based refinement of the chain rule for permutation-invariant states. From these tools we derive several de Finetti theorems: a double-sided extension theorem with $O\left(\sqrt{\log n}/n\right)$ convergence, an interpolation theorem whose dimension dependence is governed solely by the block structure of the fixed point algebras and which recovers the dimension-independent classical behavior for maximal tori, and a Bose-symmetric variant. Exploiting Schur-Weyl duality and Gelfand-Tsetlin bases, we further show that the rounding scheme producing certifiably good separable inner approximations for (constrained) separability problems can be implemented in time polynomial in $1/ε$ for fixed local dimensions, complementing the known efficient outer hierarchies. Applications to bilinear optimization under symmetries and to approximate quantum error correction are discussed.

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Extendibility of Bosonic Gaussian States

Ludovico Lami, Sumeet Khatri, Gerardo Adesso +1 more

Extendibility of bosonic Gaussian states is a key issue in continuous-variable quantum information. We show that a bosonic Gaussian state is k-extendible if and only if it has a Gaussian k-extension, and we derive a simple semidefinite program, whose size scales linearly with the number of local modes, to efficiently decide k-extendibility of any given bosonic Gaussian state. When the system to be extended comprises one mode only, we provide a closed-form solution. Implications of these results for the steerability of quantum states and for the extendibility of bosonic Gaussian channels are discussed. We then derive upper bounds on the distance of a k-extendible bosonic Gaussian state to the set of all separable states, in terms of trace norm and Rényi relative entropies. These bounds, which can be seen as "Gaussian de Finetti theorems," exhibit a universal scaling in the total number of modes, independently of the mean energy of the state. Finally, we establish an upper bound on the entanglement of formation of Gaussian k-extendible states, which has no analogue in the finite-dimensional setting.

arXiv:1904.02692•Physical Review Letters•Jul 31, 2019•19 citations
We prove optimal finite quantum de Finetti upper bounds. Given a bosonic state ρN∈D(SymN(Cd))ρ_N\in D(\mathrm{Sym}^N(\mathbb C^d))ρN​∈D(SymN(Cd)), there is a probability measure ννν 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)O(d/N)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, ttt-site marginals satisfy O(td/N)O(t\sqrt d/N)O(td​/N) bosonic and O(td/N)O(td/N)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)\varepsilon\in(0,1)ε∈(0,1), we construct a channel with input dimension D=exp⁡(Oε(dlog⁡d))=exp⁡(o(d))D=\exp(O_\varepsilon(\sqrt d\log d))=\exp(o(d))D=exp(Oε​(d​logd))=exp(o(d)) whose outputs are ε\varepsilonε-close to separable states of local dimension ddd and whose image contains every such separable state, thereby refuting Watrous's disentangler conjecture. We also obtain deterministic exp⁡(O~(d/ε))\exp(\widetilde O(\sqrt d/\varepsilon))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)Θ(N^{-1/2})Θ(N−1/2) when the dimension may grow.
Standard finite quantum de Finetti theorems approximate the kkk-system marginals of permutation-invariant states of nnn-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/nk/nk/n error bound in max-relative entropy improves on the previously best known k2/nk^2/nk2/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⊗nN^{(n)}:A^{\otimes n}\to B^{\otimes n}N(n):A⊗n→B⊗n, where dA=dim⁡Ad_A=\dim AdA​=dimA, we prove that the kkk-system reduced channel is CP-dominated by a mixture of tensor-power channels with error O(k/n)O(k/\sqrt{n})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 rrr defects, the representation error is at most poly(n)(2dA4k3/(nr2))(r+1)/2\mathrm{poly}(n)\bigl(2d_A^4k^3/(nr^2)\bigr)^{(r+1)/2}poly(n)(2dA4​k3/(nr2))(r+1)/2 and decays exponentially in nnn for a suitable choice of parameters rrr and kkk.
De Finetti theorems convert permutation symmetry into approximate mixtures of product states and thereby justify a wide range of reductions in classical and quantum statistics. In this work we study de Finetti hierarchies in which the feasible states are additionally constrained to be fixed points of quantum channels, a condition that subsumes invariance under arbitrary compact symmetry groups. Combining the mean-ergodic theorem with the structure theory of conditional expectations, we prove a tight bound on the entanglement-assisted classical capacity of the dual of a conditional expectation, block-wise distortion bounds for informationally complete measurements adapted to fixed point algebras, and an exact type-based refinement of the chain rule for permutation-invariant states. From these tools we derive several de Finetti theorems: a double-sided extension theorem with O(log⁡n/n)O\left(\sqrt{\log n}/n\right)O(logn​/n) convergence, an interpolation theorem whose dimension dependence is governed solely by the block structure of the fixed point algebras and which recovers the dimension-independent classical behavior for maximal tori, and a Bose-symmetric variant. Exploiting Schur-Weyl duality and Gelfand-Tsetlin bases, we further show that the rounding scheme producing certifiably good separable inner approximations for (constrained) separability problems can be implemented in time polynomial in 1/ε1/ε1/ε for fixed local dimensions, complementing the known efficient outer hierarchies. Applications to bilinear optimization under symmetries and to approximate quantum error correction are discussed.
We develop an argmax principle for analyzing sum-of-squares relaxations of optimization problems over the unit sphere. Given a feasible pseudo-expectation, we form a polynomial of high-order pseudo-moments, such as Φk(u)=E~⟨x,u⟩2kΦ_k(u)=\widetilde{\mathbb E}\langle x,u\rangle^{2k}Φk​(u)=E⟨x,u⟩2k. Our guiding principle is that its maximizers are rounding candidates: their local and global optimality conditions reveal the reweighed pseudo-expectation inequalities governing SoS convergence. This viewpoint unifies several problems previously analyzed by rather different techniques. We obtain three results. First, for Best Separable State, we give a degree-O(n/ε)O(\sqrt{n/ε})O(n/ε​) SoS analysis for approximating hsep(P)h_{\mathrm{sep}}(P)hsep​(P) in the perfect-completeness regime, improving and simplifying Barak, Kothari and Steurer (STOC'17). The dependence is essentially tight for inverse-linear gap under the Exponential-Time Hypothesis, matching hardness from QMA(2)\mathrm{QMA}(2)QMA(2) protocols. Second, for the matrix 2→42\to42→4 norm, degree-O(n/ε)O(\sqrt n/ε)O(n​/ε) SoS gives a multiplicative (1+ε)(1+ε)(1+ε) approximation. Barak et al. (STOC'12) previously gave a comparable-time constant-gap decision algorithm; our result gives a multiplicative guarantee and extends to a family of p→qp\to qp→q norms with even qqq. Finally, for degree-ddd polynomial optimization, we recover the convergence theorem of Bhattiprolu et al. (FOCS'17) with a shorter, more direct proof: degree-kkk SoS gives approximation ratio Od((n/k)d/2−1)O_d((n/k)^{d/2-1})Od​((n/k)d/2−1). The paper introduces no new relaxation. Instead, the high-moment argmax gives a common way to read an SoS solution, unifying previously separate convergence analyses and yielding sharper bounds or simpler proofs.
The sum-of-squares hierarchy of semidefinite programs has become a common tool for algorithm design in theoretical computer science, including problems in quantum information. In this work we study a connection between a Hermitian version of the SoS hierarchy, related to the quantum de Finetti theorem, and geometric quantization of compact Kähler manifolds (such as complex projective space CPd\mathbb{C}P^{d}CPd, the set of all pure states in a (d+1)(d + 1)(d+1)-dimensional Hilbert space). We show that previously known HSoS rounding algorithms can be recast as quantizing an objective function to obtain a finite-dimensional matrix, finding its top eigenvector, and then (possibly nonconstructively) rounding it by using a version of the Husimi quasiprobability distribution. Dually, we recover most known quantum de Finetti theorems by doing the same steps in the reverse order: a quantum state is first approximated by its Husimi distribution, and then quantized to obtain a separable state approximating the original one. In cases when there is a transitive group action on the manifold we give some new proofs of existing de Finetti theorems, as well as some applications including a new version of Renner's exponential de Finetti theorem proven using the Borel--Weil--Bott theorem, and hardness of approximation results and optimal degree-2 integrality gaps for the basic SDP relaxation of \textsc{Quantum Max-ddd-Cut} (for arbitrary ddd). We also describe how versions of these results can be proven when there is no transitive group action. In these cases we can deduce some error bounds for the HSoS hierarchy on complex projective varieties which are smooth.
We prove quantitative lower bounds on the semidefinite extension complexity of the set of separable quantum states on Cd⊗Cd\mathbb{C}^d\otimes\mathbb{C}^dCd⊗Cd. We consider semidefinite programs (SDPs) that approximate the maximum acceptance probability of a measurement over separable states, the optimization problem underlying QMA(2). In the extended-formulation model of Harrow, Natarajan, and Wu (HNW), all measurements share a common feasible region and an objective-independent embedding of product states that exactly reproduces their acceptance probabilities. For every 0<θ<2/70<θ<2/70<θ<2/7, there are constants cθ,aθ>0c_θ,a_θ>0cθ​,aθ​>0 such that, for sufficiently large ddd, any such SDP with uniform additive error 0<a≤aθ0<a\le a_θ0<a≤aθ​ has size at least dcθmin⁡{a−1/3,dθ}d^{c_θ\min\{a^{-1/3},d^θ\}}dcθ​min{a−1/3,dθ}. The bound applies at sufficiently small constant error, is superpolynomial in ddd whenever a=o(1)a=o(1)a=o(1), and becomes dΩ(dθ)d^{Ω(d^θ)}dΩ(dθ) when a≤d−3θa\le d^{-3θ}a≤d−3θ, improving HNW's quasipolynomial bound at inverse- square error. The same bound holds for any SDP-representable convex set of states that contains all separable states and lies within trace distance aaa of them, giving a quantitative counterpart to Fawzi's theorem that the separable set has no exact semidefinite representation. Our proof combines the quantitative pseudo-density theorem of Lee, Raghavendra, and Steurer with explicit block-positive operators and Chebyshev amplification. Our main results are supported by Lean proofs.
Extendibility of bosonic Gaussian states is a key issue in continuous-variable quantum information. We show that a bosonic Gaussian state is k-extendible if and only if it has a Gaussian k-extension, and we derive a simple semidefinite program, whose size scales linearly with the number of local modes, to efficiently decide k-extendibility of any given bosonic Gaussian state. When the system to be extended comprises one mode only, we provide a closed-form solution. Implications of these results for the steerability of quantum states and for the extendibility of bosonic Gaussian channels are discussed. We then derive upper bounds on the distance of a k-extendible bosonic Gaussian state to the set of all separable states, in terms of trace norm and Rényi relative entropies. These bounds, which can be seen as "Gaussian de Finetti theorems," exhibit a universal scaling in the total number of modes, independently of the mean energy of the state. Finally, we establish an upper bound on the entanglement of formation of Gaussian k-extendible states, which has no analogue in the finite-dimensional setting.