Geng Isabelle Jianing, Golubeva Kimberly, Gour Gilad
Department of Mathematics and Statistics, Institute for Quantum Science and Technology, University of Calgary, Alberta T2N 1N4, Canada.
Phys Rev Lett. 2021 Mar 12;126(10):100401. doi: 10.1103/PhysRevLett.126.100401.
Symmetric informationally complete (SIC) positive operator valued measures (POVMs) are a class of quantum measurements which, in addition to being informationally complete, satisfy three conditions: (1) every POVM element is rank one, (2) the Hilbert-Schmidt inner product between any two distinct elements is constant, and (3) the trace of each element is constant. The third condition is often overlooked, since it may give the impression that it follows trivially from the second. We show that this condition cannot be removed, as it leads to two distinct values for the trace of an element of the POVM. This observation has led us to define a broader class of measurements which we call semi-SIC POVMs. In dimension two, we show that semi-SIC POVMs exist, and we construct the entire family. In higher dimensions, we characterize key properties and applications of semi-SIC POVMs, and note that the proof of their existence remains open.
对称信息完备（SIC）正算子值测量（POVM）是一类量子测量，除了信息完备之外，还满足三个条件：（1）每个POVM元素的秩为一；（2）任意两个不同元素之间的希尔伯特 - 施密特内积是常数；（3）每个元素的迹是常数。第三个条件常常被忽视，因为它可能给人一种它可从第二个条件轻易推出的印象。我们证明这个条件不能被去掉，因为它会导致POVM元素的迹出现两个不同的值。这一观察结果促使我们定义了一类更广泛的测量，我们称之为半SIC POVM。在二维情况下，我们证明了半SIC POVM的存在，并构建了整个族。在更高维度中，我们刻画了半SIC POVM的关键性质和应用，并指出其存在性的证明仍然是开放的。
