Symmetry reduction for testing $k$-block-positivity via extendibility

We study the problem of testing $k$-block-positivity via symmetric $N$-extendibility by taking the tensor product with a $k$-dimensional maximally entangled state. We exploit the unitary symmetry of the maximally entangled state to reduce the size of the corresponding semidefinite programs (SDP). For example, for $k=2$, the SDP is reduced from one block of size $2^{N+1}{d}^{N+1}$ to $\lfloor \frac{N+1}{2}\rfloor$ blocks of size $\approx O((N-1{)}^{-1}{2}^{N+1}{d}^{N+1})$.