Hu Mengyao, Chen Lin, Sun Yize
School of Mathematical Sciences, Beihang University, Beijing 100191, People's Republic of China.
International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, People's Republic of China.
Proc Math Phys Eng Sci. 2020 Mar;476(2235):20190754. doi: 10.1098/rspa.2019.0754. Epub 2020 Mar 25.
Constructing four six-dimensional mutually unbiased bases (MUBs) is an open problem in quantum physics and measurement. We investigate the existence of four MUBs including the identity, and a complex Hadamard matrix (CHM) of Schmidt rank three. The CHM is equivalent to a controlled unitary operation on the qubit-qutrit system via local unitary transformation ⊗ and ⊗ . We show that and have no zero entry, and apply it to exclude constructed examples as members of MUBs. We further show that the maximum of entangling power of controlled unitary operation is log 3 ebits. We derive the condition under which the maximum is achieved, and construct concrete examples. Our results describe the phenomenon that if a CHM of Schmidt rank three belongs to an MUB then its entangling power may not reach the maximum.
构造四个六维相互无偏基（MUBs）是量子物理与测量领域的一个开放问题。我们研究包含单位矩阵以及一个施密特秩为三的复哈达玛矩阵（CHM）的四个MUBs的存在性。通过局部酉变换(I\otimes I)和(I\otimes\sigma_x)，CHM等同于量子比特 - 量子三态系统上的一个受控酉操作。我们证明(I\otimes\sigma_x)没有零元素，并将其用于排除作为MUBs成员的构造示例。我们进一步表明受控酉操作的最大纠缠能力为(\log 3) 量子比特。我们推导了达到最大值的条件，并构造了具体示例。我们的结果描述了这样一种现象：如果一个施密特秩为三的CHM属于一个MUB，那么它的纠缠能力可能无法达到最大值。
