Violation of Continuous-Variable Einstein-Podolsky-Rosen Steering with Discrete Measurements

In this Letter, we derive an entropic Einstein-Podolsky-Rosen (EPR) steering inequality for continuous-variable systems using only experimentally measured discrete probability distributions and details of the measurement apparatus. We use this inequality to witness EPR steering between the positions and momenta of photon pairs generated in spontaneous parametric down-conversion. We examine the asymmetry between parties in this inequality, and show that this asymmetry can be used to reduce the technical requirements of experimental setups intended to demonstrate the EPR paradox. Furthermore, we develop a more stringent steering inequality that is symmetric between parties, and use it to show that the down-converted photon pairs also exhibit symmetric EPR steering.

Figure 1

Violation of our EPR steering inequality (2) for measurements of party $B$ conditioned on measurements of party $A$ as measured in standard deviations of the difference between the bound and the sum of conditional entropies. This plot was formed by independently downsampling the joint distributions for a $24\times 24$ data set. A positive value represents a violation. We see that below $4\times 4$ resolution for party $B$ as suggested by theory, the inequality is not violated for any resolution of DMD $A$, but violation is possible for large resolution of party $B$ and party $A$ having resolution below $4\times 4$.

Figure 2

Experimentally determined sums of conditional entropies in our EPR steering inequality (15) as a function of the product of resolutions $\frac{1}{\Delta x}$ and $\frac{1}{\Delta k}$. The different symbols designate different resolutions of experimental data which were then downsampled. For example, the triangles represent data from an experiment recording at $24\times 24$ resolution. Since the product of the window sizes is dimensionless, this can be expressed in any units we wish. The region below the black curve is the region where our EPR steering inequality (15) is violated; the sum is less than the bound.