Upper bound and shareability of quantum discord based on entropic uncertainty relations

By using the quantum-memory-assisted entropic uncertainty relation (EUR), we derive a computable tight upper bound for quantum discord, which applies to an arbitrary bipartite state. Detailed examples show that this upper bound is tighter than other known bounds in a wide regime. Furthermore, we show that for any tripartite pure state, the quantum-memory-assisted EUR imposes a constraint on the shareability of quantum correlations among the constituent parties. This conclusion amends the well-accepted result that quantum discord is not monogamous.

Figure 1

Upper bounds of QD for ${\rho }_{\mathrm{PP}}$ of Eq. (7) with (a) $d=2$, $u_1=2\sqrt{2}/3$, $u_2=1/3$ and (b) $d=3$, $u_1=\sqrt{7}/3$, $u_{2,3}=1/3$. The dash-dotted and the dashed lines are the exact results of QD and its upper bounds given by Eq. (6) [the stars denote critical points after which $S\left({\rho }_A\right)$ in Eq. (6) dominates], while the solid red, blue, and green lines (from bottom to top) are those given by ${\Lambda }_{\alpha }$ [Eqs. (8) and (10)] with $\alpha =T$, $M$, and $F$, respectively.