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.
Categories: quant-ph