Measurement-device-independent and arbitrarily loss-tolerant verification of quantum steering

We propose a method to verify quantum steering for two-qubit states with an arbitrary amount of null results when both the steering and steered parties cannot be trusted. We converted the steering inequality proposed in a recent article [Phys. Rev. X 2, 031003 ( 2012)] to a corresponding measurement-device-independent steering criterion that depends on the heralding efficiency of the steering party, number of measurement settings, and imperfection of the state preparation. As a result, for a relative frequency of valid measurement outcomes eta(H), the steering can be verified using a number of different measurement settings larger than 1/eta(H) and maximally entangled states. Furthermore, steerability is guaranteed as long as the measurement efficiency of the steered party is nonzero. Our result is useful for loss-tolerant and measurement-device-independent steering tasks.