Entanglement-assisted guessing of complementary measurement outcomes

Heisenberg's uncertainty principle implies that if one party (Alice) prepares a system and randomly measures one of two incompatible observables, then another party (Bob) cannot perfectly predict the measurement outcomes. This implication assumes that Bob does not possess an additional system that is entangled to the measured one; indeed, the seminal paper of Einstein, Podolsky, and Rosen (EPR) showed that maximal entanglement allows Bob to perfectly win this guessing game. Although not in contradiction, the observations made by EPR and Heisenberg illustrate two extreme cases of the interplay between entanglement and uncertainty. On the one hand, no entanglement means that Bob's predictions must display some uncertainty. Yet on the other hand, maximal entanglement means that there is no more uncertainty at all. Here we follow an operational approach and give an exact relation—an equality—between the amount of uncertainty as measured by the guessing probability and the amount of entanglement as measured by the recoverable entanglement fidelity. From this equality, we deduce a simple criterion for witnessing bipartite entanglement and an entanglement monogamy equality.

Figure 1

Upper bounds (UB) and lower bounds (LB) on $P_{\text{guess}}^{\text{pg}}\left(n\right)$ as a function of $F^{\text{pg}}\left(A\right|B)$ for $d_A=5$, for various values of $n$. For $n=1,2,3,4,5$, the (trivial) lower bound is given by the solid black line. The upper bounds for these values of $n$ are, respectively, the black, purple, blue, green, and orange dashed lines. For $n=6$, the upper and lower bounds coincide, i.e., the allowed values are confined to live on the red dashed line. The regions where $H_2\left(A\right|B)\ge 0$ and $H_2\left(A\right|B)<0$ are labeled “Heisenberg-limited” and “enhanced,” respectively.