Knowledge excess duality and violation of Bell inequalities

A constraint on two complementary knowledge excesses by maximal violation of Bell inequalities for a single copy of any mixed state of two qubits $S,M$ is analyzed. The complementary knowledge excesses ${\bf ΔK}(Π_{M}\to Π_{S})$ and ${\bf ΔK}(Π'_{M}\to Π'_{S})$ quantify an enhancement of ability to predict results of the complementary projective measurements $Π_{S},Π'_{S}$ on the qubit $S$ from the projective measurements $Π_{M},Π'_{M}$ performed on the qubit $M$. For any state $ρ_{SM}$ and for arbitrary $Π_{S},Π'_{S}$ and $Π_{M},Π'_{M}$, the knowledge excesses satisfy the following inequality ${\bf ΔK}^{2}(Π_{M}\to Π_{S})+{\bf ΔK}^{2} (Π'_{M}\to Π'_{S})\leq (B_{max}/2)^2$, where $B_{max}$ is maximum of violation of Bell inequalities under single-copy local operations (local filtering and unitary transformations). Particularly, for the Bell-diagonal states only an appropriate choice of the measurements $Π_{S},Π'_{S}$ and $Π_{M},Π'_{M}$ are sufficient to saturate the inequality.