Tighter Einstein-Podolsky-Rosen steering inequality based on the sum-uncertainty relation

We consider the uncertainty bound on the sum of variances of two incompatible observables in order to derive a corresponding steering inequality. Our steering criterion, when applied to discrete variables, yields the optimum steering range for two-qubit Werner states in the two-measurement and two-outcome scenario. We further employ the derived steering relation for several classes of continuous-variable systems. We show that non-Gaussian entangled states such as the photon-subtracted squeezed vacuum state and the two-dimensional harmonic-oscillator state furnish greater violation of the sum steering relation compared to the Reid criterion as well as the entropic steering criterion. The sum steering inequality provides a tighter steering condition to reveal the steerability of continuous-variable states.

Figure 1

Shown on the left is the entropic steering inequality for spin-$1/2$ observables on the Werner state. The dotted curve is given by Eq. (22), while the solid line denotes the lower bound of the entropic steering condition. In the right panel the dotted curve is given by the right-hand side of Eq. (21), whereas the solid line represents the lower bound of the sum-uncertainty relation that shows steerability for $p>\frac{1}{\sqrt{2}}$.

Figure 2

The left panel shows that the product of inferred uncertainties ${\left({\mathrm{\Delta }}_{\text{inf}}{X}_{{\theta }_1}\right)}^2{\left({\mathrm{\Delta }}_{\text{inf}}{X}_{{\theta }_2}\right)}^2$ for the two-mode squeezed vacuum state given by the dotted curve falls below the bound (solid line) obtained from the Reid inequality. The right panel shows that the sum of inferred uncertainties (dotted curve) falls below the bound (28) (solid curve) obtained using our sum steering relation.

Figure 3

Steering by the single-photon-subtracted squeezed vacuum state. Plotted on the left are the product of inferred uncertainties ${\left({\mathrm{\Delta }}_{\text{inf}}{X}_{{\theta }_1}\right)}^2{\left({\mathrm{\Delta }}_{\text{inf}}{X}_{{\theta }_2}\right)}^2$ (dotted curve) and the Reid bound (solid line) versus the squeezing parameter $r$. Depicted on the right are the sum of inferred uncertainties $({\mathrm{\Delta }}_{\text{inf}}{X}_{{\theta }_1}+{\mathrm{\Delta }}_{\text{inf}}{X}_{{\theta }_2})$ (dotted curve) and sum steering bound $\mathrm{\Delta }(X_{{\theta }_1}+X_{{\theta }_2})$ (solid curve). It is clear that the steering inequality based on the sum-uncertainty relation is able to show steering for all values of the squeezing parameter.

Figure 4

Sum-uncertainty bound (solid bar) and the sum of inferred uncertainties (crossed bar) plotted versus angular momentum $n$ for the LG beams. The figure shows clearly that the violation of the sum steering inequality increases with larger angular momentum.