CODE from Mutually unbiased bases containing a complex Hadamard matrix of Schmidt rank three. 6 November 2019 26 February 2020.

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 I₂⊗V and I₂⊗W. We show that V and W 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 log₂3 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.

构建四组六维相互无偏基（mutually unbiased bases）是量子物理与量子测量领域的公开问题。我们研究了包含恒等基与一类施密特秩为3的复哈达玛矩阵（complex Hadamard matrix）在内的四组MUB的存在性。该复哈达玛矩阵可通过局域幺正变换I₂⊗V与I₂⊗W，对应于量子比特-量子三能级系统上的受控幺正操作。我们证明了V与W均无零元素，并利用该结论将部分已构造的矩阵排除在MUB的候选成员之外。我们进一步证明，该受控幺正操作的最大纠缠能力为log₂3 纠缠比特（ebits）。我们推导了达到该最大值的条件，并构造了具体示例。我们的研究结果揭示了如下现象：若一类施密特秩为3的复哈达玛矩阵属于某组MUB，则其纠缠能力未必能达到前述最大值。