论文标题
三种数值方法可以使用铃铛不等式找到相互无偏基的基础
Three numerical approaches to find mutually unbiased bases using Bell inequalities
论文作者
论文摘要
相互无偏的基础对应于量子信息理论中非常有用的测量对。在最小的综合维度（六）中，众所周知，存在三到七个互无偏的基础，有数十年历史的猜想，称为Zauner的猜想，表明最多存在三个。在这里，我们通过为每对整数$ n，d \ ge 2 $构建铃bell不等式来解决Zauner的猜想，这些整数$ n，d \ ge 2 $在该维度中存在$ n $ mubs时，可以在尺寸$ d $中最大程度地违反。因此，我们将Zauner的猜想变成了一个优化问题，我们通过三种数值方法来解决该问题：远见优化，非线性半决赛编程和Monte Carlo Techniques。所有三种方法都在低维度中正确识别已知情况，并且所有方法都表明，在第六维度中不存在四个互无偏的碱基，所有这些基础都在数值上优化了相应的铃铛不平等。此外，这些数值优化器似乎与第六维中的“四个最遥远的碱基”一致，通过数值优化[P. Raynal，X.Lü，B.-G。 Englert，物理。 Rev. A，83 062303（2011）]。最后，蒙特卡洛的结果表明，最多三个MUB在尺寸十中存在。
Mutually unbiased bases correspond to highly useful pairs of measurements in quantum information theory. In the smallest composite dimension, six, it is known that between three and seven mutually unbiased bases exist, with a decades-old conjecture, known as Zauner's conjecture, stating that there exist at most three. Here we tackle Zauner's conjecture numerically through the construction of Bell inequalities for every pair of integers $n,d \ge 2$ that can be maximally violated in dimension $d$ if and only if $n$ MUBs exist in that dimension. Hence we turn Zauner's conjecture into an optimisation problem, which we address by means of three numerical methods: see-saw optimisation, non-linear semidefinite programming and Monte Carlo techniques. All three methods correctly identify the known cases in low dimensions and all suggest that there do not exist four mutually unbiased bases in dimension six, with all finding the same bases that numerically optimise the corresponding Bell inequality. Moreover, these numerical optimisers appear to coincide with the "four most distant bases" in dimension six, found through numerically optimising a distance measure in [P. Raynal, X. Lü, B.-G. Englert, Phys. Rev. A, 83 062303 (2011)]. Finally, the Monte Carlo results suggest that at most three MUBs exist in dimension ten.
