Joint measurability, steering, and entropic uncertainty

Abstract

There has been a surge of research activity recently on the role of joint measurability of unsharp observables in nonlocal features, viz., violation of Bell inequality and EPR steering. Here, we investigate the entropic uncertainty relation for a pair of noncommuting observables (of Alice's system) when an entangled quantum memory of Bob is restricted to record outcomes of jointly measurable positive operator valued measures. We show that with this imposed constraint of joint measurability at Bob's end, the entropic uncertainties associated with Alice's measurement outcomes—conditioned by the results registered at Bob's end—obey an entropic steering inequality. Thus, Bob's nonsteerability is intrinsically linked to his inability to predict the outcomes of Alice's pair of noncommuting observables with better precision, even when they share an entangled state. As a further consequence, we prove that in the joint measurability regime, the quantum advantage envisaged for the construction of security proofs in quantum key distribution is lost.