From spin squeezing to fast state discrimination

There is great interest in generating and controlling entanglement in Bose-Einstein condensates and similar ensembles for use in quantum computation, simulation, and sensing. One class of entangled states useful for enhanced metrology are spin-squeezed states of $N$ two-level atoms. After preparing a spin coherent state of width $1/\sqrt{N}$ centered at coordinates $( \theta, \phi) $ on the Bloch sphere, atomic interactions generate a nonlinear evolution that shears the state's probability density, stretching it to an ellipse and causing squeezing in a direction perpendicular to the major axis. Here we consider the same setup but in the $N \rightarrow \infty $ limit . This shrinks the initial coherent state to zero area. Large $N$ also suppresses two-particle entanglement and squeezing, as required by a monogamy bound. The torsion (1-axis twist) is still present, however, and the center of the large $N$ coherent state evolves as a qubit governed by a two-state Gross-Pitaevskii equation. The resulting nonlinearity is known to be a powerful resource in quantum computation. It can be used to implement single-input quantum state discrimination, an impossibility within linear one-particle quantum mechanics. We obtain a solution to the discrimination problem in terms of a Viviani curve on the Bloch sphere. We also consider an open-system variant containing both Bloch sphere torsion and dissipation. In this case it should be possible to generate two basins of attraction within the Bloch ball, having a shared boundary that can be used for a type of autonomous state discrimination. We explore these and other connections between spin squeezing in the large $N$ limit and nonlinear quantum gates, and argue that a two-component condensate is a promising platform for realizing a nonlinear qubit.