Problem 16 Milestone — Three MUB dimension six

There exists a family $B=(B_r)_{r\in\{0,1,2\}}$ of three $6\times6$ complex matrices, with every row and column indexed by $\{0,1,2,3,4,5\}$, such that the following two conditions hold. First, for every $r\in\{0,1,2\}$, $(B_r)^\dagger B_r=I_6$, where $(B_r)^\dagger$ is the conjugate transpose and $I_6$ is the $6\times6$ identity matrix; equivalently, for every pair of column indices $i,j\in\{0,\ldots,5\}$, $\sum_{k=0}^{5}\overline{(B_r)_{ki}}(B_r)_{kj}$ equals $1$ when $i=j$ and $0$ when $i\ne j$. This equation is exactly the formal orthonormal-basis condition used here. Second, for every ordered pair $r,s\in\{0,1,2\}$ with $r\ne s$, and for every $i,j\in\{0,\ldots,5\}$, the complex number $((B_r)^\dagger B_s)_{ij}=\sum_{k=0}^{5}\overline{(B_r)_{ki}}(B_s)_{kj}$ has squared modulus exactly $1/6$; explicitly, its real part squared plus its imaginary part squared is the real number $1/6$. The second condition therefore covers all six ordered pairs of distinct labels, including both orders of each unordered pair, and all $36$ transition entries for each such pair; it imposes no transition-amplitude condition when $r=s$, although the first condition still applies to every $B_r$. The assertion has no additional parameters, hypotheses, typeclass assumptions, or uniqueness requirement. Its index sets have fixed positive sizes three and six, so no empty-family, zero-dimensional, or division-by-zero case is included; the implication guarded by $r\ne s$ is vacuous only on the diagonal pairs $r=s$.