对称信息完全测度的李代数意义
The Lie Algebraic Significance of Symmetric Informationally Complete Measurements
摘要：
对称信息完全正算子值测度（SIC-POVM）的例子已经在小于或等于67的每个维度上被构造。然而，它仍然是一个悬而未决的问题，它们是否存在于所有有限的维度。SIC-POVM通常被认为是量子态空间中的高度对称结构。然而，它的元素同样可以被看作是李代数gl（d，C）的基。在本文中，我们将研究由此产生的结构常数，这是计算的痕迹的三重产品的SIC-POVM元素，它原来，表征的SIC-POVM到酉等价。我们表明，结构常数有许多显着的性质。特别是，我们表明，存在一个SIC-POVM在维d是等价的存在的一个特定的结构中的伴随表示的gl（d，C）。我们希望，通过这种方式将问题从量子态空间转化为李代数问题，可能有助于使存在性问题变得易于处理。
英文摘要：
Examples of symmetric informationally complete positive operator valued measures (SIC-POVMs) have been constructed in every dimension less than or equal to 67. However, it remains an open question whether they exist in all finite dimensions. A SIC-POVM is usually thought of as a highly symmetric structure in quantum state space. However, its elements can equally well be regarded as a basis for the Lie algebra gl(d,C). In this paper we examine the resulting structure constants, which are calculated from the traces of the triple products of the SIC-POVM elements and which, it turns out, characterize the SIC-POVM up to unitary equivalence. We show that the structure constants have numerous remarkable properties. In particular we show that the existence of a SIC-POVM in dimension d is equivalent to the existence of a certain structure in the adjoint representation of gl(d,C). We hope that transforming the problem in this way, from a question about quantum state space to a question about Lie algebras, may help to make the existence problem tractable.
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