A Characterization of Projective Unitary Equivalence of Finite Frames and Applications
A Characterization of Projective Unitary Equivalence of Finite Frames and Applications
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有限框架射影酉等价的刻画及应用
DOI：
10.1137/15m1042140
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
S. Waldron
中科院分区：
文献类型：
--
作者：
Tuan;S. Waldron
有限紧框架的许多应用(例如，在量子信息论中使用SICS和相互无偏基(MUB)，以及用于分析受擦除影响的信号的调和框架)仅依赖于直到射影么正等价的向量。众所周知，内积空间中的两个有限向量序列酉等价当且仅当它们各自的内积(Gramian矩阵)相等。本文给出了内积空间中两个向量(直线)序列的射影酉等价性的一个相应结果，即有限个(Bargmann)射影(酉)不变量等价。这一结果等价于求某一矩阵的一阶矩阵完成。我们给出了一个从这些射影不变量的一个小子集恢复向量序列(直到射影酉等价性)的算法，并将其应用于SICS，MUB和调和框架。我们还将我们的结果推广到向量的射影相似性。
Many applications of finite tight frames (e.g., the use of SICs and mutually unbiased bases (MUBs) in quantum information theory and harmonic frames for the analysis of signals subject to erasures) depend only on the vectors up to projective unitary equivalence. It is well known that two finite sequences of vectors in inner product spaces are unitarily equivalent if and only if their respective inner products (Gramian matrices) are equal. Here we present a corresponding result for the projective unitary equivalence of two sequences of vectors (lines) in inner product spaces, i.e., that a finite number of (Bargmann) projective (unitary) invariants are equal. This result is equivalent to finding a rank-one matrix completion of a certain matrix. We give an algorithm to recover the sequence of vectors (up to projective unitary equivalence) from a small subset of these projective invariants and apply it to SICs, MUBs, and harmonic frames. We also extend our results to the projective similarity of vectors.