Counterexamples to Kalai's conjecture C

1 The First Counterexample Let ρ be a quantum state on the space (C) of n qudits (d-dimensional quantum systems). Denote by ρS the reduced state of ρ onto some subset S of the qudits. We partition S further into two nonempty subsets of qudits A and A := S\A (thus S must contain at least two qudits). Denote by Sep(A) the convex set of quantum states which are bipartite separable on S across the bipartition defined by A and A. Define the following function