An orthogonal representation [1] of a graph $G=(V,E)$ on $n$ vertices $\{1,2,...,n\}$ is an assignment of (complex) unit vectors $v_{1},v_{2},...,v_{n}$ to the vertices of $G$ such that $\langle v_{i},v_{j}\rangle =0$ if and only if $\{i,j\}\in E(G)$.

An orthogonal matrix representation (see, e.g., [2]) of $G$ is an assignment of unitary matrices $U_{1},U_{2},...,U_{n}$ to the vertices of $G$ such that $[U_{i}^{\dagger }U_{j}]_{k,k}=0$, for every $k$, if and only if $% \{i,j\}\in E\left( G\right) $.

Is it true that if there is an orthogonal representation of a certain dimension then there is always an orthogonal matrix representation of the same dimension?

[1] L. Lovász, On the Shannon Capacity of a Graph, IEEE Trans. Inf. Theory, 25 (1):1-7 (1979).

[2] P. J. Cameron, A. Montanaro, M. Newmann, S. Severini, A. Winter, On the quantum chromatic number of a graph, http://arxiv.org/abs/quant-ph/0608016