Problem 16 Goal — Complete MUB dimension six

There are no parameters or hypotheses: this is a closed existence assertion over the standard complex and real numbers. It asserts that there exists a function $B:\{0,\ldots,6\}\to\mathbb C^{6\times6}$, giving seven matrices $B_0,\ldots,B_6$, such that, for every $r\in\{0,\ldots,6\}$, $B_r^\dagger B_r=I_6$, where $B_r^\dagger$ is the conjugate transpose. Entrywise, for every $i,j\in\{0,\ldots,5\}$, this means $\sum_{k=0}^{5}\overline{(B_r)_{ki}}(B_r)_{kj}=\delta_{ij}$, with $\delta_{ij}=1$ when $i=j$ and $0$ otherwise; thus the six columns have exactly the identity matrix as their Gram matrix. In addition, for every ordered pair $r,s\in\{0,\ldots,6\}$ with $r\ne s$, and for every $i,j\in\{0,\ldots,5\}$, it requires $\left|\sum_{k=0}^{5}\overline{(B_r)_{ki}}(B_s)_{kj}\right|^2=\frac16$, where $|z|^2=(\operatorname{Re}z)^2+(\operatorname{Im}z)^2$; the equality is exact and includes both $i=j$ and $i\ne j$. The distinct-index condition makes this latter requirement vacuous when $r=s$, although the Gram-matrix condition still applies separately to every $B_r$. The witness $B$ is not asserted to be unique. Because the dimensions are fixed at six and seven, no zero-dimensional or empty-index case occurs, and the denominator is the nonzero real number $6$. “Complete” contributes no additional maximality clause: beyond the existence of these seven indexed matrices with the stated properties, the declaration does not separately assert that no eighth such basis exists, classify all such collections, or identify collections up to permutation, phase, or another equivalence.