We prove that for any ε>0 it is NP-hard to approximate the
non-commutative Grothendieck problem to within a factor 1/2+ε,
which matches the approximation ratio of the algorithm of Naor, Regev, and
Vidick (STOC'13). Our proof uses an embedding of ℓ2 into the space of
matrices endowed with the trace norm with the property that the image of
standard basis vectors is longer than that of unit vectors with no large
coordinates
