The sum-of-squares hierarchy on the sphere, and applications in quantum information theory

Abstract

We consider the problem of maximizing a homogeneous polynomial on the unit sphere and its hierarchy of Sum-of-Squares (SOS) relaxations. Exploiting the polynomial kernel technique, we obtain a quadratic improvement of the known convergence rate by Reznick and Doherty & Wehner. Specifically, we show that the rate of convergence is no worse than $O(d^2/{\ell }^2)$ in the regime $\ell \ge \Omega (d)$ where $\ell$ is the level of the hierarchy and $d$ the dimension, solving a problem left open in the recent paper by de Klerk & Laurent (arXiv:1904.08828). Importantly, our analysis also works for matrix-valued polynomials on the sphere which has applications in quantum information for the Best Separable State problem. By exploiting the duality relation between sums of squares and the DPS hierarchy in quantum information theory, we show that our result generalizes to nonquadratic polynomials the convergence rates of Navascués, Owari & Plenio.

References in corpus (10)

• Quantum entanglement
• A complete family of separability criteria
• Distinguishing separable and entangled states
• One-and-a-half quantum de Finetti theorems
• Hypercontractivity, Sum-of-Squares Proofs, and their Applications
• Separability and distillability in composite quantum systems -a primer-
• The power of symmetric extensions for entanglement detection
• Convergence of SDP hierarchies for polynomial optimization on the hypersphere
• A quasipolynomial-time algorithm for the quantum separability problem
• Mixed-state entanglement and quantum communication

Cited by in corpus (18)

• Semidefinite programming relaxations for quantum correlations
• Quantum NETwork: from theory to practice
• Homogenization for polynomial optimization with unbounded sets
• Degree bounds for Putinar's Positivstellensatz on the hypercube
• Near-optimal analysis of Lasserre's univariate measure-based bounds for multivariate polynomial optimization
• Geometric Rényi Divergence and its Applications in Quantum Channel Capacities
• On Łojasiewicz Inequalities and the Effective Putinar's Positivstellensatz
• A convex form that is not a sum of squares
• Symmetry-Based Quantum Circuit Mapping
• Revealing hidden physical nonclassicality with nonnegative polynomials
• Harmonic Hierarchies for Polynomial Optimization
• On the Effective Putinar's Positivstellensatz and Moment Approximation
• An Inexact Projected Gradient Method with Rounding and Lifting by Nonlinear Programming for Solving Rank-One Semidefinite Relaxation of Polynomial Optimization
• A hierarchy of eigencomputations for polynomial optimization on the sphere
• A refinement of Reznick's Positivstellensatz with applications to quantum information theory
• Moment-sos and spectral hierarchies for polynomial optimization on the sphere and quantum de Finetti theorems
• Optimality conditions for homogeneous polynomial optimization on the unit sphere
• Semidefinite Relaxations of Products of Nonnegative Forms on the Sphere