Closed-form characterization of maximally violating Tsirelson states

Determine a simple closed-form expression for the quantum harmonic-oscillator states that achieve the maximum Tsirelson probability in the three-angle precession protocol.

Background

The paper reviews the Tsirelson precession protocol, in which the sign of a harmonic oscillator’s quadrature is measured at equally spaced times. For three measurement angles, the quantum Tsirelson probability has a nontrivial maximum exceeding the classical upper bound.

The maximally violating states are known to possess rotational symmetry and therefore have Fock-space expansions supported only on photon-number states whose occupations are multiples of three. Existing results obtain these states numerically by diagonalizing the Tsirelson operator in a truncated Fock space, but the paper states that a simple closed-form representation remains unavailable. This unresolved issue provides context for the paper’s investigation of whether structured superpositions of Gaussian states can approach the maximal violation.