Unbounded sequence of observers exhibiting Einstein-Podolsky-Rosen steering

A sequential steering scenario is investigated, where multiple Bobs aim at demonstrating steering using successively the same half of an entangled quantum state. With isotropic entangled states of local dimension $d$, the number of Bobs that can steer Alice is found to be $N_{\mathrm{Bob}}\sim d/logd$, thus leading to an arbitrary large number of successive instances of steering with independently chosen and unbiased inputs. This scaling is achieved when considering a general class of measurements along orthonormal bases, as well as complete sets of mutually unbiased bases. Moreover, we show that similar results can be obtained in an anonymous sequential scenario, where none of the Bobs know their position in the sequence. Finally, we briefly discuss the implication of our results for sequential tests of Bell nonlocality.

Figure 1

Sequential steering scenario: multiple Bobs aim at steering Alice. Note that each state ${\rho }_i$ is a shared state between Alice and the $i\mathrm{th}$ Bob $B_i$.

Figure 2

Graphical representation of the number of Bobs that are able to exhibit steering in the limit case in which each of them just saturates the steering criterion, here for the case of qubits $d=2$. In the shaded area, the noise is too strong for steering to be demonstrated. As the dashed line can only do five steps before falling in this area, we see that (at most) five Bobs can exhibit steering.

Figure 3

Similar curves as in Fig. 2, here for isotropic states in dimensions 2, 4, and 16 (from bottom to top). The noise thresholds $p_{\text{steer}}^{\text{iso}}\left(d\right)$ are approximately 0.50, 0.36, and 0.16 (respectively), and 5, 6, and 13 Bobs can demonstrate steering (respectively). These numbers of Bobs correspond to the last $i$ such that $p_i⩾p_{\text{steer}}^{\text{iso}}\left(d\right)$.

Figure 4

Behavior of the function $f_{\mathrm{ano}}$ for $d=2...16$. The horizontal axis has been rescaled to allow for a meaningful comparison of the curves.