Tight steering inequalities from generalized entropic uncertainty relations

We establish a general connection between entropic uncertainty relations, Einstein-Podolsky-Rosen steering, and joint measurability. Specifically, we construct steering inequalities from any entropic uncertainty relation, given that the latter satisfies two natural properties. We obtain steering inequalities based on Rényi entropies. These turn out to be tight in many scenarios using max- and min-entropies. Considering steering tests with two noisy measurements, our inequalities exactly recover the noise threshold for steerability. This is the case for any pair of qubit two-outcome measurements as well as for pairs of mutually unbiased bases in any dimension. This shows that easy-to-evaluate quantities, such as entropy, can optimally witness steering, despite the fact that they are coarse-grained representations of the underlying statistics.

Figure 1

Tolerable noise level for the detection of steering via our Rényi steering inequality (13) for two MUBs with white noise. Here, we consider the case of symmetric noise $\eta =\chi$ as a function of the dimension $d$. The optimal noise threshold is attained for a steering inequality based on the min- and max-entropies ($\alpha =1/2,\beta =\infty$) in which case our inequality is tight. For other cases, e.g., the Shannon entropy ($\alpha =\beta =1$) or the Rényi criteria with $\alpha >1/2$, the inequality is suboptimal. For comparison, the Tsallis entropy criteria derived in Ref. [14] is also depicted.

Figure 2

A section of the Bloch sphere designated by Alice's measurements ($Z_A^{\pm },X_A^{\pm }$). Optimal measurements of Bob ($Z_B^{\pm },X_B^{\pm }$) are shown for the case of Alice's measurements having symmetric noise ($\eta =\chi \in [0,1]$), no bias ($b_z=b_x=0$), and for Alice and Bob sharing a maximally entangled state $\frac{1}{\sqrt{2}}(|00\rangle +|11\rangle )$.

Figure 3

We depict sharpness, i.e., $1-\eta$, as a function of the parameter $t$ in $d=3$. For $t=0$, we are in the MUB case for which we have previously shown tightness of the Rényi criteria, hence the exact threshold and the detected threshold are the same. For $t=0.5$, Alice's measurements overlap, hence even for $\eta =0$, they are jointly measurable. Although the Rényi criteria are not tight, they outperform the Shannon criteria.