A Simple Algorithm for Best Separable State

2026-08-10

AI Summary

This work addresses the Best Separable State (BSS) problem: maximizing the acceptance probability of a quantum measurement over unentangled states, which is equivalent to maximizing $\langle x \otimes y, M(x \otimes y) \rangle$ over unit vectors $x, y$, where $0 \preceq M \preceq I$. Focusing on the perfect completeness case (i.e., $\mathrm{BSS}(M) = 1$), we introduce a streamlined sum-of-squares (SoS) relaxation rounding framework that extends global correlation rounding techniques and incorporates a novel “fixing lemma” of independent interest in LP/SDP rounding, high-dimensional probability, and statistical physics. Our method either finds a solution of value at least $1 - \varepsilon$ in time $n^{O(\sqrt{n/\varepsilon})}$, or achieves value $q/n$ in time $n^{O(\sqrt{q})}$, significantly improving upon prior algorithms.

Abstract

We study the best separable state problem (BSS), which asks for the maximum acceptance probability of a quantum measurement over unentangled states. In classical terms, the goal is to maximize $\langle(x \otimes y), M (x \otimes y)\rangle$ over unit vectors $x,y$ where $0 \preceq M \preceq I$; we call this value $\mathrm{BSS}(M)$. We study $\mathrm{BSS}$ in the "perfect completeness" regime, where given $M$ such that $\mathrm{BSS}(M) = 1$ the goal is to find the best possible solution $x,y$ -- this generalizes the problem of finding a rank-one matrix as close as possible to a given subspace of $\mathbb{R}^{n \times n}$ guaranteed to contain a rank-one matrix. The strongest known algorithmic guarantees for this problem are: (1) an algorithm which finds a solution with value $1-\varepsilon$ in time $\exp(\sqrt{n} (\log n)^{O(1)} / \varepsilon^2)$, due to Barak, Kothari, and Steurer, and (2) an algorithm which finds a solution with value $q/n$ in time roughly $n^{O(q)}$, due to Bhattiprolu, Ghosh, Guruswami, Lee, and Tulsiani. We give a much simpler approach to rounding the SoS relaxation, generalizing the canonical "global correlation rounding" technique, and obtain a better running time. Given $M$ with $\mathrm{BSS}(M) = 1$, our algorithm finds a solution with value $1-ε$ in time $n^{O(\sqrt{n/\varepsilon})}$, and a solution of value $q/n$ in time $n^{O(\sqrt q)}$. Using the same techniques, we prove a new variant of the "pinning lemma", a measure-decomposition theorem widely used in LP/SDP rounding, high-dimensional probability, and statistical physics, which we believe is of independent interest.

Problem

• Best Separable State
• Quantum Measurement
• Unentangled States
• Rank-one Matrix
• Perfect Completeness

Innovation

• Sum-of-Squares relaxation
• global correlation rounding
• best separable state
• pinning lemma
• quantum measurement optimization