Loss-tolerant Einstein-Podolsky-Rosen steering for arbitrary-dimensional states: Joint measurability and unbounded violations under losses

We show how to construct loss-tolerant linear steering inequalities using a generic set of von Neumann measurements that are violated by $d$-dimensional states, and that rely only upon a simple property of the set of measurements used (the maximal overlap between measurement directions). Using these inequalities we show that the critical detection efficiency above which $n$ von Neumann measurements can demonstrate steering is $1/n$. We show furthermore that using our construction and high-dimensional states allows for steering demonstrations which are also highly robust to depolarizing noise and produce unbounded violations in the presence of loss. Finally, our results provide an explicit means to certify the nonjoint measurability of any set of inefficient von Neuman measurements.

Figure 1

Region plot of $w$ against $\eta$ for demonstrating steering using the inequality (6) with $d+1$ MUB measurements and the $d$-dimensional isotropic state (10).

Figure 2

Critical amount of white noise $w$ that can be tolerated by the isotropic state (10) as a function of dimension for different steering and nonlocality tests. From bottom to top the curves refer to the following. Purple lowest curve: bound below which a LHS model for projective measurements exists [4, 20]; blue curve: critical $w$ derived using the steering inequality (6) with $d+1$ MUB measurements. Above this curve steering can be observed; orange curve: bound obtained using the entropic steering inequality for $d+1$ MUB measurements derived in [21]; red curve: bound derived from the inequality (6) using only two MUB measurements; green highest curve: bound obtained by the Collins-Gisin-Linden-Massar-Popescu Bell inequality [22], above which Bell nonlocality can be demonstrated. We have not plotted the bounds obtained from the inequalities derived in Ref. [5] for spin measurements as they are only violated for $d=2$ and 3.