Fig. 1: Formulation of the EPR paradox as a metrological task. | Nature Communications

Fig. 1: Formulation of the EPR paradox as a metrological task.

From: Metrological complementarity reveals the Einstein-Podolsky-Rosen paradox

Fig. 1

a In the standard EPR scenario, Alice’s measurement setting X (Y), and result a (b), leave Bob in the conditional quantum states \({\rho }_{a| X}^{{\rm{B}}}\) (\({\rho }_{b| Y}^{{\rm{B}}}\)). Knowing Alice’s setting and result allows Bob to choose what measurement to perform on his state, and to make a prediction for the result. In an ideal scenario with strong quantum correlations, Alice’s measurement of X (Y) steers Bob into an eigenstate of his observable H (M), allowing him to predict the result with certainty. When H and M do not commute, this seems to contradict the complementarity principle. In practice, an EPR paradox is revealed whenever Bob’s predictions are precise enough to observe an apparent violation of Heisenberg’s uncertainty relation, see Eq. (1). b In our formulation of the EPR paradox as a metrological task, a local phase shift θ is generated by H on Bob’s state. Then, depending on Alice’s measurement setting and result, he decides whether to predict and measure H (as before), or to estimate θ from the measurement M. Here, Bob can choose the observable M as a function of Alice’s measurement result. The complementarity between θ and its generator H seems to be contradicted if the lower bound on their estimation errors, Eq. (3), is violated. This gives a metrological criterion for observing the EPR paradox. Since the metrological complementarity is sharper than the uncertainty-based notion, this approach leads to a tighter criterion to detect steering. Both results coincide in the special case when Bob estimates θ only from the observable M.

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