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 ${\bf \Delta K}(\Pi_{M}\to \Pi_{S})$ and ${\bf \Delta K}(\Pi'_{M}\to \Pi'_{S})$ quantify an enhancement of ability to predict results of the complementary projective measurements $\Pi_{S},\Pi'_{S}$ on the qubit $S$ from the projective measurements $\Pi_{M},\Pi'_{M}$ performed on the qubit $M$. For any state $\rho_{SM}$ and for arbitrary $\Pi_{S},\Pi'_{S}$ and $\Pi_{M},\Pi'_{M}$, the knowledge excesses satisfy the following inequality ${\bf \Delta K}^{2}(\Pi_{M}\to \Pi_{S})+{\bf \Delta K}^{2} (\Pi'_{M}\to \Pi'_{S})\leq (B_{max}/2)^2$, 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 $\Pi_{S},\Pi'_{S}$ and $\Pi_{M},\Pi'_{M}$ are sufficient to saturate the inequality.