arXiv 1402.73302014-02-28quant-phmath-phmath.COmath.MP

在2维和3维中生成SIC-POVM的一些非标准方法

Some non-standard ways to generate SIC-POVMs in dimensions 2 and 3

Gary McConnell

备注：: 10 pages, 1 figure

摘要：

对称信息完备正算符值测度（SIC-POVM）的概念作为量子系统的一种最优测度基而出现在物理学中。然而，它们的存在性问题等价于\emph{complex equiangular lines}的极大集的存在性问题。也就是说，给定一个维数为$d$的复希尔伯特空间，在跨越这些线的单位向量之间的内积都有一个共同的绝对值的意义上，人们能找到的最大数量的（复）线是多少？最大集合将由所有具有共同角度$\arccos{\frac{1}{\sqrt{d+1}}}$的$d^2$线组成。同样的问题在实际情况中也被提出来了，并得到了部分答案。但在撰写本文时，在真实或复杂的情况下还没有发现统一的理论结果：一些零星的低维数值构造已经转化为代数解，但除此之外，我们知之甚少。它是明确的，这样的结构总是出现的轨道下的Weyl（或广义泡利）组的行动，某些基准向量。本文提出了几种新的最低维构造方法（$d=2$和$d=3$）。我们应该提到的是，这样构造的SIC-POVM都是酉等价于先前已知的SIC-POVM。

英文摘要：

The notion of Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) arose in physics as a kind of optimal measurement basis for quantum systems. However the question of their existence is equivalent to that of the existence of a maximal set of \emph{complex equiangular lines}. That is to say, given a complex Hilbert space of dimension $d$, what is the maximal number of (complex) lines one can find which all make a common (real) angle with one another, in the sense that the inner products between unit vectors spanning those lines all have a common absolute value? A maximal set would consist of $d^2$ lines all with a common angle of $\arccos{\frac{1}{\sqrt{d+1}}}$. The same question has been posed in the real case and some partial answers are known. But at the time of writing no unifying theoretical result has been found in the real or the complex case: some sporadic low-dimensional numerical constructions have been converted into algebraic solutions but beyond this very little is known. It is conjectured that such structures always arise as orbits of certain fiducial vectors under the action of the Weyl (or generalised Pauli) group. In this paper we point out some new construction methods in the lowest dimensions ($d=2$ and $d=3$). We should mention that the SIC-POVMs so constructed are all unitarily equivalent to previously known SIC-POVMs.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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