We study the question of how highly entangled two particles can be when also
entangled in a similar way with other particles on the complete graph for the
case of Werner, isotropic and Brauer states. In order to do so we solve
optimization problems motivated by many-body physics, computational complexity
and quantum cryptography. We formalize our question as a semi-definite program
and then solve this optimization problem analytically, using tools from
representation theory. In particular, we determine the exact maximum values of
the projection to the maximally entangled state and antisymmetric Werner state
possible, solving long-standing open problems. We find these optimal values by
use of SDP duality and representation theory of the symmetric and orthogonal
groups, and the Brauer algebra.Comment: Submitted to QIP202