The Tracial Hahn-Banach Theorem, Polar Duals, Matrix Convex Sets, and Projections of Free Spectrahedra
The Tracial Hahn-Banach Theorem, Polar Duals, Matrix Convex Sets, and Projections of Free Spectrahedra
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DOI：
10.4171/jems/707
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发表时间：
2014-07
影响因子：
2.6
通讯作者：
J. Helton;I. Klep;S. McCullough
中科院分区：
文献类型：
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作者：
J. Helton;I. Klep;S. McCullough
本文研究了矩阵凸集，并引入了它们的迹类似，我们称之为压缩迹凸集。在这两种情况下，完全正（CP）映射起着核心作用：单位CP映射的情况下，矩阵凸集和迹保持CP（CPTP）映射的情况下，压缩tracial凸集。CPTP映射，也称为量子通道，是量子信息理论中的基本对象。自由凸性与线性矩阵不等式L（x）= A_0 + A_1x_1+…+ A_gx_g > 0及其矩阵凸解集{ X：L（X）是半正定的}，称为自由谱面.矩阵凸集的Effros-Winkler Hahn-Banach分离定理指出，矩阵凸集是具有算子系数的线性矩阵不等式的解集。部分出于CP插值问题，我们开发的基础上凸分析和对偶的tracial设置，包括tracial类似物的Effros-Winkler定理。一个g+h变量的自由谱面体到g变量的投影是一个称为自由谱滴的矩阵凸集。作为一类，自由谱滴比自由谱面更一般，但同时也比一般的矩阵凸集更易处理。此外，许多矩阵凸集可以从上面的自由谱滴近似。这里建立了谱滴及其极坐标的一些基本结果。例如，自由谱滴的自由极对偶也是自由谱滴。对于在自由谱点上为正的自由多项式，我们也给出了一个Positivstellenstrance。
This article investigates matrix convex sets and introduces their tracial analogs which we call contractively tracial convex sets. In both contexts completely positive (cp) maps play a central role: unital cp maps in the case of matrix convex sets and trace preserving cp (CPTP) maps in the case of contractively tracial convex sets. CPTP maps, also known as quantum channels, are fundamental objects in quantum information theory. Free convexity is intimately connected with Linear Matrix Inequalities (LMIs) L(x) = A_0 + A_1 x_1 + ... + A_g x_g > 0 and their matrix convex solution sets { X : L(X) is positive semidefinite }, called free spectrahedra. The Effros-Winkler Hahn-Banach Separation Theorem for matrix convex sets states that matrix convex sets are solution sets of LMIs with operator coefficients. Motivated in part by cp interpolation problems, we develop the foundations of convex analysis and duality in the tracial setting, including tracial analogs of the Effros-Winkler Theorem. The projection of a free spectrahedron in g+h variables to g variables is a matrix convex set called a free spectrahedrop. As a class, free spectrahedrops are more general than free spectrahedra, but at the same time more tractable than general matrix convex sets. Moreover, many matrix convex sets can be approximated from above by free spectrahedrops. Here a number of fundamental results for spectrahedrops and their polar duals are established. For example, the free polar dual of a free spectrahedrop is again a free spectrahedrop. We also give a Positivstellensatz for free polynomials that are positive on a free spectrahedrop.