Semidefinite programming formulations for the completely bounded norm of a tensor

We show that a certain tensor norm, the completely bounded norm, can be expressed by a semidefinite program. This tensor norm recently attracted attention in the field of quantum computing, where it was used by Arunachalam, Bri\"{e}t and Palazuelos for characterizing the quantum query complexity of Boolean functions. Combined with their results, we obtain a new characterization of the quantum query complexity through semidefinite programming. Using the duality theory of semidefinite programming we obtain a new type of certificates for large query complexity. We show that our class of certificates encompasses the linear programming certificates corresponding to the approximate degree of a Boolean function.