Convex semidefinite tensor optimization and quantum entanglement

TL;DR

This work extends the cone of positive semidefinite matrices to PSD tensors to model separable quantum states and studies convex optimization over this cone, noting that while a smooth factorization-based parameterization exists, the PSD tensor cone is not self-dual and membership testing is NP-hard. To address computational intractability, the authors develop an iterative refinement framework (IR) that alternates lifted ADMM (LADMM) for heuristic solutions with cutting planes to certify bounds, and a spatial branch-and-bound (sBB) method that provides a linear minimization oracle using convex relaxations derived from DPS-type outer approximations and McCormick/RLT inequalities. The framework yields both upper and lower bounds on the optimal value and enables analysis of the white-noise mixing threshold, a key entanglement measure, with numerical experiments demonstrating effectiveness on benchmark instances. By combining problem-specific relaxations with general-purpose optimization techniques, the paper bridges convex optimization over tensor cones and quantum entanglement testing, offering scalable tools that can be extended to operator convex/concave functions and other tensor-based quantum information problems.

Abstract

The cone of positive-semidefinite (PSD) matrices is fundamental in convex optimization, and we extend this notion to tensors, defining PSD tensors, which correspond to separable quantum states. We study the convex optimization problem over the PSD tensor cone. While this convex cone admits a smooth reparameterization through tensor factorizations (analogous to the matrix case), it is not self-dual. Moreover, there are currently no efficient algorithms for projecting onto or testing membership in this cone, and the semidefinite tensor optimization problem, although convex, is NP-hard. To address these challenges, we develop methods for computing lower and upper bounds on the optimal value of the problem. We propose a general-purpose iterative refinement algorithm that combines a lifted alternating direction method of multipliers with a cutting-plane approach. This algorithm exploits PSD tensor factorizations to produce heuristic solutions and refine the solutions using cutting planes. Since the method requires a linear minimization oracle over PSD tensors, we design a spatial branch-and-bound algorithm based on convex relaxations and valid inequalities. Our framework allows us to study the white-noise mixing threshold, which characterizes the entanglement properties of quantum states. Numerical experiments on benchmark instances demonstrate the effectiveness of the proposed methods.