Problem 16 definitions — Existence of complete sets of mutually unbiased bases

For any $6\times6$ complex matrix $U=(U_{ki})$, with $k,i\in\{0,\ldots,5\}$, $\operatorname{IsOrthonormalBasis6}(U)$ is the proposition $U^{*}U=I_6$, where $U^{*}$ is the conjugate transpose and $I_6$ is the $6\times6$ identity matrix. Entrywise, this requires, for every $i,j\in\{0,\ldots,5\}$, $\sum_{k=0}^{5}\overline{U_{ki}}\,U_{kj}=1$ when $i=j$, and $\sum_{k=0}^{5}\overline{U_{ki}}\,U_{kj}=0$ when $i\ne j$. Thus all thirty-six entrywise equalities are required, and every sum has exactly six terms; there is no empty-index or zero-dimensional case. The definition contains no separately stated condition $UU^{*}=I_6$, invertibility condition, or spanning condition, and it defines a property of $U$ without asserting that any such matrix exists.

For any two $6\times6$ complex matrices $U=(U_{ki})$ and $V=(V_{kj})$, $\operatorname{MutuallyUnbiased6}(U,V)$ is the proposition that, for every $i,j\in\{0,\ldots,5\}$, the squared complex modulus of the $(i,j)$-entry of $U^{*}V$ is exactly $1/6$; explicitly, $\left|\sum_{k=0}^{5}\overline{U_{ki}}\,V_{kj}\right|^{2}=1/6$. Here the squared modulus of a complex number $z$ means $(\operatorname{Re}z)^2+(\operatorname{Im}z)^2$, with no square root. All thirty-six choices of $i,j$ are included, each sum has six terms, and the denominator is the fixed nonzero real number $6$. No hypothesis in this definition requires either $U$ or $V$ to satisfy $\operatorname{IsOrthonormalBasis6}$, so this predicate is defined for arbitrary pairs of $6\times6$ complex matrices; it defines a property and makes no existence assertion.

For any function $B$ assigning a $6\times6$ complex matrix $B_r$ to each label $r\in\{0,\ldots,6\}$, $\operatorname{IsCompleteMUB6}(B)$ is the conjunction of the following two requirements. First, for every $r\in\{0,\ldots,6\}$, $(B_r)^{*}B_r=I_6$, meaning that for every $i,j\in\{0,\ldots,5\}$, $\sum_{k=0}^{5}\overline{(B_r)_{ki}}\,(B_r)_{kj}$ equals $1$ if $i=j$ and $0$ if $i\ne j$. Second, for every ordered pair $r,s\in\{0,\ldots,6\}$, if $r\ne s$, then for every $i,j\in\{0,\ldots,5\}$, $\left|\sum_{k=0}^{5}\overline{(B_r)_{ki}}\,(B_s)_{kj}\right|^{2}=1/6$. The second requirement therefore covers all forty-two ordered pairs of unequal labels, equivalently each of the twenty-one unordered pairs in both orders. When $r=s$, its implication has a false antecedent and imposes no transition-amplitude condition, although the first requirement still applies to $B_r$. The function $B$ is not explicitly required to be injective or its matrices explicitly required to be distinct beyond what the displayed conditions entail. The declaration merely defines this property of a seven-indexed family: it does not assert that such a family exists, quantify over an eighth matrix, exclude extension by additional matrices, or state any maximality property.