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 $\approx O( (N-1)^{-1} 2^{N+1} d^{N+1} )$.