Positive semidefinite quadratic forms on unitary matrices

Letf(x1,…,xn)=∑i,j=1nαijxixj,aij=aji∈Rbe a real quadratic form such that the trace of the Hermitian matrixf(V1,…,Vn):=∑i,j=1nαijVi∗Vjis nonnegative for all unitary 2n×2n2n×2n matrices V1,…,VnV1,…,Vn. We prove that f(U1,…,Un)f(U1,…,Un) is positive semidefinite for all unitary matrices U1,…,UnU1,…,Un of arbitrary size m×mm×m.