Excluding the existence of four MUBs in $\bbC^6$ is an open problem inquantum information. We investigate the number of product vectors in the set offour mutually unbiased bases (MUBs) in dimension six, by assuming that the setexists and contains a product-vector basis. We show that in most cases thenumber of product vectors in each of the remaining three MUBs is at most two.We further construct the exceptional case in which the three MUBs respectivelycontain at most three, two and two product vectors. We also investigate thenumber of vectors mutually unbiased to an orthonormal basis.
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机译：排除$ \ bbC ^ 6 $中存在四个MUB是量子信息中的一个开放问题。我们假设维度集存在并包含一个乘积向量，研究维度6中四个互不偏基（MUB）集合中乘积向量的数量。我们显示出在大多数情况下，其余三个MUB中的每一个的乘积向量的数量最多为两个。我们进一步构造了例外情况，其中三个MUB分别包含最多三个，两个和两个乘积向量。我们还研究了正交基础上相互无偏的向量的数量。
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