The Set of Separable States has no Finite Semidefinite Representation Except in Dimension 3×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-6
The Set of Separable States has no Finite Semidefinite Representation Except in Dimension 3×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-6
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DOI：
10.1007/s00220-021-04163-2
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发表时间：
2019-05
影响因子：
2.4
通讯作者：
Hamza Fawzi
中科院分区：
文献类型：
--
作者：
Hamza Fawzi
给定整数，设 为希尔伯特空间上可分离状态的集合。众所周知，可分离状态集有一个使用半定规划的简单描述：它由具有正部分转置的状态集给出。在本文中，我们证明，对于较大的 nandm 值，该集合没有有限大小的半定规划描述。 Asis 是一个半代数集，这为 Helton-Nie 猜想提供了一个新的反例，该猜想最近被 Scheiderer 的突破性结果所反驳。与 Scheiderer 的方法相比，我们的证明是基本的，仅依赖于半代数集和函数的基本结果。
Given integers, letbe the set of separable states on the Hilbert space. It is well-known that forthe set of separable states has a simple description using semidefinite programming: it is given by the set of states that have a positive partial transpose. In this paper we show that for larger values ofnandmthe sethas no semidefinite programming description of finite size. Asis a semialgebraic set this provides a new counterexample to the Helton–Nie conjecture, which was recently disproved by Scheiderer in a breakthrough result. Compared to Scheiderer’s approach, our proof is elementary and relies only on basic results about semialgebraic sets and functions.