All three-angle variants of Tsirelson’s precession protocol, and improved bounds for wedge integrals of Wigner functions

Tsirelson’s precession protocol is a nonclassicality witness that can be defined for both discrete and continuous variable systems. Its original version involves measuring a precessing observable, like the quadrature of a harmonic oscillator or a component of angular momentum, along three equally-spaced angles. In this work, we characterise all three-angle variants of this protocol. For continuous variables, we show that the maximum score $${{\bf{P}}}_{3}^{\infty }$$ achievable by the quantum harmonic oscillator is the same for all such generalised protocols. We also derive markedly tighter bounds for $${{\bf{P}}}_{3}^{\infty }$$ , both rigorous and conjectured, which translate into improved bounds on the amount of negativity a Wigner function can have in certain wedge-shaped regions of phase space. For discrete variables, we show that changing the angles significantly improves the score for most spin systems. Like the original protocol, these generalised variants can detect non-Gaussian and genuine multipartite entanglement when applied on composite systems. Overall, this work broadens the scope of Tsirelson’s original protocol, making it capable to detect the nonclassicality and entanglement of many more states.