Entanglement witnesses based on first and second moments exist in the form of spin squeezing criteria for the detection of particle entanglement from collective measurements, and in the form of modified uncertainty relations for the detection of mode entanglement or steering. By revealing a correspondence between them, we show that metrologically useful spin squeezing reveals multimode entanglement in symmetric spin states that are distributed into addressable modes. We further derive tight state-independent multipartite entanglement bounds on the spin squeezing coefficient and point out their connection to widely used entanglement criteria that depend on the state's polarization. Our results are relevant for state-of-the-art experiments where symmetric entangled states are distributed into a number of addressable modes, such as split spin squeezed Bose-Einstein condensates.

• Received 7 April 2020
• Accepted 15 June 2020

DOI:https://doi.org/10.1103/PhysRevA.102.012412