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Probing quantum entanglement from magnetic-sublevels populations: beyond spin squeezing inequalities

Guillem M\\"uller-RigatMaciej LewensteinIr\\'en\\'ee Fr\\'erot
摘要
Spin squeezing inequalities (SSI) represent a major tool to probe quantumentanglement among a collection of few-level atoms, and are based on collectivespin measurements and their fluctuations. Yet, for atomic ensembles of spin-$j$atoms and ultracold spinor gases, many experiments can image the populations inall Zeeman sublevels $s=-j, -j+1, \dots, j$, potentially revealing finerfeatures of quantum entanglement not captured by SSI. Here we present asystematic approach which exploits Zeeman-sublevel population measurements inorder to construct novel entanglement criteria, and illustrate our approach onground states of spin-1 and spin-2 Bose-Einstein condensates. Beyond thesespecific examples, our approach allows one to infer, in a systematic manner,the optimal permutationally-invariant entanglement witness for any given set ofcollective measurements in an ensemble of $d$-level quantum systems.
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