A steerable quantum state allows the uncharacterized party (say, Alice) to remotely steer the other characterized party (say, Bob). To certify the steerability of the underlying unknown quantum state, one can use a steering witness. In some situations, however, one would like to certify the steerability even when no party is characterized. Such a scheme, i.e., neither the underlying shared quantum state nor the local quantum measurements are characterized, is called a device-independent (DI) scenario. The usual certication of steerability in a DI scenario is based on the violation of a Bell inequality, therefore a Bell inequality is sometimes called a "DI steering witness". However, not all steerable states can lead to a Bell inequality violation. In the work [Phys. Rev. A 87, 032306 (2013)], Cavalcanti, Hall, and Wiseman considered a generalized DI scenario, called a measurement-device-independent (MDI) scenario, showing that all steerable states provide advantages than all nonsteerable states in a set of generalized Bell-type nonlocal games. In such a scenario, the cost is that Bob has to prepare for himself a tomographically complete set as the inputs of the measurements. In this work, we show how to systematically construct a set of MDI steering witnesses for all steerable states. Besides, we introduce an MDI measure of steerability and prove that it is equivalent to the steering robustness proposed by Piani and Watrous [Phys. Rev. Lett. 114, 060404 (2015)]. Therefore, our proposed MDI steering measure not only obeys the requirement of a measure of steerability, i.e., the so-called one-way local operation assisted by classical communication monotone (1W-LOCC monotone), but also corresponds the usefulness a quantum information processing task, i.e., the subchannel discrimination problem.
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