  In this paper, we shed light on relations between three concepts studied in
representations theory, algebraic geometry and quantum information theory.
First - spherical actions of reductive groups on projective spaces. Second -
secant varieties of homogeneous projective varieties, and the related notions
of rank and border rank. Third - quantum entanglement. Our main result concerns
the relation between the problem of the state reconstruction from its reduced
one-particle density matrices and the minimal number of separable summands in
its decomposition. More precisely, we show that sphericity implies that states
of a given rank cannot be approximated by states of a lower rank. We call
states for which such approximation is possible exceptional states. For three,
important from quantum entanglement perspective cases of distinguishable,
fermionic and bosonic particles, we also show that non-sphericity implies the
existence of exceptional states. Remarkably, the exceptional states belong to
non-bipartite entanglement classes. In particular, we show that the $W$-type
states and their appropriate modifications are exceptional states stemming from
the second secant variety for three cases above. We point out that the
existence of the exceptional states is a physical obstruction for deciding the
local unitary equivalence of states by means of the one-particle reduced
density matrices. Finally, for a number of systems of distinguishable particles
with known orbit structure we list all exceptional states and discuss their
possible importance in entanglement theory.
