Metrological complementarity reveals the Einstein-Podolsky-Rosen paradox

Abstract

The Einstein-Podolsky-Rosen (EPR) paradox plays a fundamental role in our understanding of quantum mechanics, and is associated with the possibility of predicting the results of non-commuting measurements with a precision that seems to violate the uncertainty principle. This apparent contradiction to complementarity is made possible by nonclassical correlations stronger than entanglement, called steering. Quantum information recognises steering as an essential resource for a number of tasks but, contrary to entanglement, its role for metrology has so far remained unclear. Here, we formulate the EPR paradox in the framework of quantum metrology, showing that it enables the precise estimation of a local phase shift and of its generating observable. Employing a stricter formulation of quantum complementarity, we derive a criterion based on the quantum Fisher information that detects steering in a larger class of states than well-known uncertainty-based criteria. Our result identifies useful steering for quantum-enhanced precision measurements and allows one to uncover steering of non-Gaussian states in state-of-the-art experiments.

Entanglement of Local Hidden States

“…which describes a complementarity between Bob's phase sensitivity and the fluctuations for measurements of the generator H [18].…”

“…a|X , H]. Since Alice's results occur randomly, Bob's average sensitivity is given, after an optimization over Alice's setting X, by the quantum conditional Fisher information (QCFI) [18]…”

“…Similarly, using another measurement setting, Alice may remotely prepare conditional states for Bob that have small variances for measurements of the generator H and yield the quantum conditional variance (QCV) [3,18] Var…”

“…The observation of an EPR paradox [16], or more generally of steering, formally excludes the possibility to describe the observed data in terms of a local hidden state (LHS) model that assigns local quantum states to one of the subsystems [14]. Steering criteria rule out LHS descriptions by verifying a local complementarity principle in form of an uncertainty relation [17] or a metrological bound [18]. Uncertainty-based criteria formulated in terms of variances are most widely used in experiments and are particularly suited to reveal the steering of approximately Gaussian states [3].…”

“…Uncertainty-based criteria formulated in terms of variances are most widely used in experiments and are particularly suited to reveal the steering of approximately Gaussian states [3]. Metrological approaches [18,19], as well as entropic relations [20,21,22,23], have particular advantages for detecting steering of non-Gaussian states. So far, despite advances in multipartite steering [24,25,26,28,27,22,23,29], these criteria are mostly applied in bipartite settings.…”