Probing quantumness of superpositions of Gaussian states via Tsirelson probability

Abstract: Superpositions of Gaussian states, including circular states and generalized circular states, exhibit a rich variety of nonclassical features such as Wigner negativity and sub-Planck phase-space structures. The Tsirelson probability, central to the Tsirelson precession protocol, is defined as the average probability that a precessing quadrature yields a positive outcome when measured at equally spaced times, with the classical bound given by $1/2 \pm 1/(2d).$ In this work, we compute this probability for superpositions of Gaussian states. For circular states (superpositions of $d$ coherent states), we analytically calculate their Tsirelson probability and find violations of the classical bounds for $d=3,5,7$. For generalized circular states, which incorporate squeezing, we derive a general expression and identify parameter regimes that enhance the violation. Our results provide a comprehensive map of the dynamical nonclassicality of superpositions of Gaussian states and establish them as versatile platforms for testing nonclassicality witnesses.