An equality between entanglement and uncertainty

TL;DR

This paper establishes a precise equality linking quantum entanglement and measurement uncertainty, providing new insights into their fundamental relationship and practical criteria for entanglement detection.

Contribution

It introduces an exact equality between entanglement and uncertainty, offering a new operational perspective and tools for entanglement witnessing and monogamy relations.

Findings

Derived an equality between entanglement and uncertainty measures.

Provided a simple criterion for bipartite entanglement witnessing.

Established a novel entanglement monogamy equality.

Abstract

Heisenberg's uncertainty principle implies that if one party (Alice) prepares a system and randomly measures one of two incompatible observables, then another party (Bob) cannot perfectly predict the measurement outcomes. This implication assumes that Bob does not possess an additional system that is entangled to the measured one; indeed the seminal paper of Einstein, Podolsky and Rosen (EPR) showed that maximal entanglement allows Bob to perfectly win this guessing game. Although not in contradiction, the observations made by EPR and Heisenberg illustrate two extreme cases of the interplay between entanglement and uncertainty. On the one hand, no entanglement means that Bob's predictions must display some uncertainty. Yet on the other hand, maximal entanglement means that there is no more uncertainty at all. Here we follow an operational approach and give an exact relation - an equality - between the amount of uncertainty as measured by the guessing probability, and the amount of entanglement as measured by the recoverable entanglement fidelity. From this equality we deduce a simple criterion for witnessing bipartite entanglement and a novel entanglement monogamy equality.