We give a method for computing asymptotic formulas and approximations for the volumes of spectrahedra, based on the maximum-entropy principle from statistical physics. The method gives an approximate volume formula based on a single convex optimization problem of minimizing $-\log \det P$ over the spectrahedron. Spectrahedra can be described as affine slices of the convex cone of positive semi-definite (PSD) matrices, and the method yields efficient deterministic approximation algorithms and asymptotic formulas whenever the number of affine constraints is sufficiently dominated by the dimension of the PSD cone. Our approach is inspired by the work of Barvinok and Hartigan who used an analogous framework for approximately computing volumes of polytopes. Spectrahedra, however, possess a remarkable feature not shared by polytopes, a new fact that we also prove: central sections of the set of density matrices (the quantum version of the simplex) all have asymptotically the same volume. This allows for very general approximation algorithms, which apply to large classes of naturally occurring spectrahedra. We give two main applications of this method. First, we apply this method to what we call the "multi-way Birkhoff spectrahedron" and obtain an explicit asymptotic formula for its volume. This spectrahedron is the set of quantum states with maximal entanglement (i.e., the quantum states having univariant quantum marginals equal to the identity matrix) and is the quantum analog of the multi-way Birkhoff polytope. Second, we apply this method to explicitly compute the asymptotic volume of central sections of the set of density matrices.
翻译:我们给出了一种方法,用于根据统计物理中的最大顺差原则计算光谱体积的无线公式和近似值。 该方法提供了一种基于单一的顺方优化问题的大致体积公式, 将美元- log\ det P$ 最小化在光谱中。 Spectrahedra 可以被描述为正半无线( PSD) 基质的正半无线( PSD) 基质的正方形锥形的偏角片片段, 而该方法可以产生高效的确定性近似算法和顺方形公式, 当偏差限制的数量充分受SDF 的多端特性控制时。 我们的方法受到巴维诺克和哈特伊根纳( Hartirgan) 的类似框架的启发。 然而, Spectrahedra 拥有一个与多面( points) 共性半无异性( Ped) 基质质矩阵的中央部分( 简单直径) 直径( 直径) 直径( 直径) 直径( 直径) 直径( 直径) 直径( 直径) 直径( 直径) 直径) 直径) 和直径(我们的直径) 。