Joint measurability, steering and entropic uncertainty

Abstract

The notion of incompatibility of measurements in quantum theory is in stark contrast with the corresponding classical perspective, where all physical observables are jointly measurable. It is of interest to examine if the results of two or more measurements in the quantum scenario can be perceived from a classical point of view or they still exhibit non-classical features. Clearly, commuting observables can be measured jointly using projective measurements and their statistical outcomes can be discerned classically. However, such simple minded association of compatibility of measurements with commutativity turns out to be limited in an extended framework, where the usual notion of sharp projective valued measurements of self adjoint observables gets broadened to include unsharp measurements of generalized observables constituting positive operator valued measures (POVM). There is a surge of research activity recently towards gaining new physical insights on the emergence of classical behavior via joint measurability of unsharp observables. Here, we explore the entropic uncertainty relation for a pair of discrete observables (of Alice's system) when an entangled quantum memory of Bob is restricted to record outcomes of jointly measurable POVMs only. Within the joint measurability regime, the sum of entropies associated with Alice's measurement outcomes - conditioned by the results registered at Bob's end - are constrained to obey an entropic steering inequality. In this case, Bob's non-steerability reflects itself as his inability in predicting the outcomes of Alice's pair of non-commuting observables with better precision, even when they share an entangled state. As a further consequence, the quantum advantage envisaged for the construction of security proofs in key distribution is lost, when Bob's measurements are restricted to the joint measurability regime.

arXiv comment5 pages, RevTeX, 1 pdf figure, Comments welcome

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Cited by in corpus (10)

• Tight steering inequalities from generalized entropic uncertainty relations
• Unbounded violation of quantum steering inequalities
• Quantum Correlations and Quantum Non-locality: a review and a few new ideas
• Unsharpness of generalized measurement and its effects in entropic uncertainty relations
• Joint Measurability and Temporal Steering
• Local measurement uncertainties impose a limit on non-local quantum correlations
• Measurement-device-independent and arbitrarily loss-tolerant verification of quantum steering
• Complementarity relations for design-structured POVMs in terms of generalized entropies of order $\alpha \in (0,2)$
• Local sum uncertainty relations for angular momentum operators of bipartite permutation symmetric systems
• Operational nonlocality