On the existence of quantum anticliques for non-commutative operator graphs

In quantum information theory, operator systems $V$ are called non-commutative operator graphs. The talk will be devoted to the question of existence of orthogonal projections $P$, $\mathop{\mathrm{dim}}P>1$, with the property $\mathop{\mathrm{dim}}PVP=1$. Such projections are sometimes called quantum anticliques.