Can we stochastically distil quantum steering and measurement incompatibility?

Quantum steering and measurement incompatibility are two central resources in device-independent quantum information tasks. Here, we thoroughly study their stochastic distillation, that is, the possibility and impossibility to enhance them by filter operations on the trusted party. First, we prove a necessary and sufficient condition for steering conversion under local filter operations. We characterise the conversion by the steering-equivalent observable (SEO), which was first introduced to show that steering and measurement incompatibility are mathematically equivalent. We further provide an efficient method to compute the optimal steerability that can be stochastically distilled by local filters and show that it coincides with the SEO’s incompatibility. Hence, steering can be stochastically distilled, and measurement incompatibility sets a fundamental limitation for it. Finally, rather surprisingly, we show that measurement incompatibility cannot be stochastically distilled by combining measurements with a device that post-selects the incoming states on a fixed outcome of a stochastic transformation. Consequently, despite their mathematical equivalence, steering and measurement incompatibility can be very different operationally.