On the orthogonal rank of Cayley graphs and impossibility of quantum round elimination

Abstract: After Bob sends Alice a bit, she responds with a lengthy reply. At the cost of a factor of two in the total communication, Alice could just as well have given the two possible replies without listening and have Bob select which applies to him. Motivated by a conjecture stating that this form of "round elimination" is impossible in exact quantum communication complexity, we study the orthogonal rank and a symmetric variant thereof for a certain family of Cayley graphs. The orthogonal rank of a graph is the smallest number $d$ for which one can label each vertex with a nonzero $d$-dimensional complex vector such that adjacent vertices receive orthogonal vectors. We show an exp$(n)$ lower bound on the orthogonal rank of the graph on ${0,1}^n$ in which two strings are adjacent if they have Hamming distance at least $n/2$. In combination with previous work, this implies an affirmative answer to the above conjecture.