Einstein-Podolsky-Rosen steering inequalities from entropic uncertainty relations

We use entropic uncertainty relations to formulate inequalities that witness Einstein-Podolsky-Rosen (EPR)-steering correlations in diverse quantum systems. We then use these inequalities to formulate symmetric EPR-steering inequalities using the mutual information. We explore the differing natures of the correlations captured by one-way and symmetric steering inequalities and examine the possibility of exclusive one-way steerability in two-qubit states. Furthermore, we show that steering inequalities can be extended to generalized positive operator-valued measures, and we also derive hybrid steering inequalities between alternate degrees of freedom.

Figure 1

Scatterplots of the violation of the conditional and symmetric steering inequalities which use all mutually unbiased bases. Each point is a random two-qubit state. The plot in (b) is color coded according to purity, $P$, as measured by the von Neumann entropy, scaled, and inverted so that 0 is maximally mixed and 1 is pure: $P=1-\frac{S\left(\rho \right)}{2}$. The well-defined diagonal line through the origin indicates that regardless of the orientation of the mutually unbiased bases, the symmetric violation never exceeds the conditional violation. The plots thin out to the upper right since maximally entangled states are rare when uniformly sampling over pure states and rarer still when uniformly sampling over all states.

Figure 2

Scatterplots of the violation in number of bits of the conditional steering inequalities when using an optimal set of mutually unbiased bases. The violation of the inequality conditioned on Alice's measurements is plotted against the violation of the inequality conditioned on Bob's measurements. (a, b): Each point is a random two-qubit state whose set of measurement bases have been selected for maximum violation. For higher violation, the scatterplots approach a diagonal line, where more entangled states tend to be more symmetric between parties. (c): Examination of the violation of a candidate two-qubit state in many different independently random sets of measurement bases. States in the upper left quadrants are ones where Alice's uncertainty is less than Bob's uncertainty. In the lower right quadrants, Bob's uncertainty is less than Alice's. The color coding in (b) is according to purity as defined in Fig. 1b.