  We describe a connection between entanglement and designs. It involves the
conical 2-designs introduced in a previous paper. These are a generalization of
projective 2-designs which includes full sets of arbitrary rank mutually
unbiased measurements (MUMs) and arbitrary rank symmetric informationally
complete measurements (SIMs), as well as the more familiar MUBs and SICs. We
show that a POVM is a conical 2-design if and only if there exists what we call
a regular entanglement monotone whose restriction to the pure states is a
function of the norm of the probability vector. In that case the concurrence is
such a monotone. We also generalize and develop previous work on designs and
entanglement detection.
