Optimal Quantum de Finetti Theorems via Argmax Rounding

• URL: http://arxiv.org/abs/2608.02590v1
• Date: Mon, 03 Aug 2026 17:57:15 GMT
• Title: Optimal Quantum de Finetti Theorems via Argmax Rounding
• Abstract summary: We prove optimal finite quantum de Finetti upper bounds for bosonic states and arbitrary exchangeable states.<n>Our proof formulates de Finetti approximation as the integrality gap of a symmetric-extension semidefinite program.<n> spectral truncation gives the first dimension-free bosonic de Finetti theorem in Hilbert--Schmidt distance.
• Score: 42.55259086694773
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• Quantum Algorithms and Hardness for Point-Count Approximation over Finite Fields [1.4323566945483497]
  We study the approximation of the number of solutions of Laurentrms over finite fields.<n>Our first main result is a quantum algorithm that outputs $widehat(f)$ satisfying [ |widehatN(f) - N(f)| le varepsilon qn+s/2- ] with success probability $1-$.<n>As a second main result, we prove that the same problem becomes $#$P-hard under randomized-time Turing reductions.
  arXiv  Detail & Related papers  (2026-08-25T00:23:27Z)
• A Simple Algorithm for Best Separable State [5.5559749120901385]
  We study the best separable state problem (BSS), which asks for the maximum acceptance probability of a quantum measurement over unentangled states.<n>In classical terms, the goal is to maximize $langle(x otimes y), M (x otimes y)rangle$ over unit vectors $x,y$ where $0 preceq M preceq I$.
  arXiv  Detail & Related papers  (2026-08-10T19:04:15Z)
• An Argmax Principle for Sum-of-Squares Relaxations on the Sphere [42.540924632302925]
  We develop an argmax principle for analyzing sum-of-squares relaxations of optimization problems over the unit sphere.<n>Our guiding principle is that its maximizers are rounding candidates.
  arXiv  Detail & Related papers  (2026-08-03T17:58:00Z)
• The Condition-Number Barrier in Sparse Least Squares [77.64108812086542]
  In [AS21], Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in convex optimization cannot be improved by a sparse-time algorithm.<n>We establish their conjectured lower bound for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis.
  arXiv  Detail & Related papers  (2026-08-03T17:57:01Z)
• Robust quantum state certification and uncertainty principles for total influence [5.7531205491708945]
  We show that nonadaptive single-qubit Pauli measurements suffice to test whether an unknown $n$-qubit state $$ is $varepsilon$-close to or $O(varepsilon)$-far from an ideal target state $|rangle$.<n>The test uses $O(varepsilon-2log (1/)$ copies of $$ to achieve confidence $1-$, which is information-theoretically optimal even among protocols with arbitrary joint measurements.
  arXiv  Detail & Related papers  (2026-07-29T17:53:23Z)
• Nonasymptotic bounds for quantum purity amplification [0.9969485010222057]
  In quantum purity amplification, one is given $n$ copies of a noisy quantum state $in mathbbCd times d$ and asked to prepare $k$ copies of its principal eigenstate $|v_drangle$.<n>Several prior works have derived information-theoretically optimal algorithms for this problem, but the they prove are only shown in the regime as the number of samples $n$ tends to infinity.
  arXiv  Detail & Related papers  (2026-05-26T16:39:24Z)
• Quantum Algorithms and Lower Bounds for Finite-Sum Optimization [22.076317220348145]
  We give a quantum algorithm with complexity $tildeObig(n+sqrtd+sqrtell/mubig)$, improving the classical tight bound $tildeThetabig(n+sqrtnell/mubig)$. We also prove a quantum lower bound $tildeOmega(n+n3/4(ell/mu)1/4)$ when $d$ is large enough.
  arXiv  Detail & Related papers  (2024-06-05T07:13:52Z)
• Optimal and Efficient Algorithms for Decentralized Online Convex Optimization [51.00357162913229]
  Decentralized online convex optimization (D-OCO) is designed to minimize a sequence of global loss functions using only local computations and communications.<n>We develop a novel D-OCO algorithm that can reduce the regret bounds for convex and strongly convex functions to $tildeO(nrho-1/4sqrtT)$ and $tildeO(nrho-1/2log T)$.<n>Our analysis reveals that the projection-free variant can achieve $O(nT3/4)$ and $O(n
  arXiv  Detail & Related papers  (2024-02-14T13:44:16Z)
• The Approximate Degree of DNF and CNF Formulas [95.94432031144716]
  For every $delta>0,$ we construct CNF and formulas of size with approximate degree $Omega(n1-delta),$ essentially matching the trivial upper bound of $n. We show that for every $delta>0$, these models require $Omega(n1-delta)$, $Omega(n/4kk2)1-delta$, and $Omega(n/4kk2)1-delta$, respectively.
  arXiv  Detail & Related papers  (2022-09-04T10:01:39Z)
• Low-degree learning and the metric entropy of polynomials [44.99833362998488]
  We prove that any (deterministic or randomized) algorithm which learns $mathscrF_nd$ with $L$-accuracy $varepsilon$ requires at least $Omega(sqrtvarepsilon)2dlog n leq log mathsfM(mathscrF_n,d,|cdot|_L,varepsilon) satisfies the two-sided estimate $$c (1-varepsilon)2dlog
  arXiv  Detail & Related papers  (2022-03-17T23:52:08Z)
• Unique Games hardness of Quantum Max-Cut, and a conjectured vector-valued Borell's inequality [6.621324975749854]
  We show that the noise stability of a function $f:mathbbRn to -1, 1$ is the expected value of $f(boldsymbolx) cdot f(boldsymboly)$. We conjecture that the expected value of $langle f(boldsymbolx), f(boldsymboly)rangle$ is minimized by the function $f(x) = x_leq k / Vert x_leq k /
  arXiv  Detail & Related papers  (2021-11-01T20:45:42Z)
• An Optimal Separation of Randomized and Quantum Query Complexity [67.19751155411075]
  We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order $ellsqrtbinomdell (1+log n)ell-1,$ sum to at most $cellsqrtbinomdell (1+log n)ell-1,$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant.
  arXiv  Detail & Related papers  (2020-08-24T06:50:57Z)
• On the Complexity of Minimizing Convex Finite Sums Without Using the Indices of the Individual Functions [62.01594253618911]
  We exploit the finite noise structure of finite sums to derive a matching $O(n2)$-upper bound under the global oracle model. Following a similar approach, we propose a novel adaptation of SVRG which is both emphcompatible with oracles, and achieves complexity bounds of $tildeO(n2+nsqrtL/mu)log (1/epsilon)$ and $O(nsqrtL/epsilon)$, for $mu>0$ and $mu=0$
  arXiv  Detail & Related papers  (2020-02-09T03:39:46Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.