  We study the separability problem in mixtures of Dicke states i.e., the
separability of the so-called Diagonal Symmetric (DS) states. First, we show
that separability in the case of DS in $C^d\otimes C^d$ (symmetric qudits) can
be reformulated as a quadratic conic optimization problem. This connection
allows us to exchange concepts and ideas between quantum information and this
field of mathematics. For instance, copositive matrices can be understood as
indecomposable entanglement witnesses for DS states. As a consequence, we show
that positivity of the partial transposition (PPT) is sufficient and necessary
for separability of DS states for $d \leq 4$. Furthermore, for $d \geq 5$, we
provide analytic examples of PPT-entangled states. Second, we develop new
sufficient separability conditions beyond the PPT criterion for bipartite DS
states. Finally, we focus on $N$-partite DS qubits, where PPT is known to be
necessary and sufficient for separability. In this case, we present a family of
almost DS states that are PPT with respect to each partition but nevertheless
entangled.
