Measuring Entanglement via SICs and 2-designs

We consider measuring entanglement via the classical quadratic R\'{e}nyi entropy of joint probability distributions over the measurement outcomes associated with tensor products of elements of local positive operator valued measures (POVMs). We examine the case of pure $d\times d$ bipartite quantum states and identical local POVMs. In this case, we prove that if the local POVMs are rank 1, then the classical quadratic R\'{e}nyi entropy of such a distribution (denoted by $H$) is independent of the underlying Schmidt bases if and only if the local POVMs are equivalent to spherical 2-designs. We also prove that if the local POVMs admit a cardinality equal to the composite Hilbert space dimension, then $H$ is independent of the underlying Schmidt bases if and only if the local POVMs are symmetric informationally complete POVMs of arbitrary rank. We show that different degrees of entanglement correspond to distinct spheres within the corresponding joint probability simplexes. Furthermore, we derive a separability criterion for mixed isotropic quantum states in terms of probabilities for outcomes of generalized quantum measurements constructed from tensor products of elements of local POVMs formed from spherical 2-designs.