Tsirelson's Inequality for the Precession Protocol is Maximally Violated by Quantum Theory

Abstract

The precession protocol involves measuring $P_3$, the probability that a uniformly precessing observable (like the position of a harmonic oscillator or a coordinate undergoing spatial rotation) is positive at one of three equally spaced times. Tsirelson's inequality, which states that $P_3 \leq 2/3$ in classical theory, is violated in quantum theory by certain states. In this Letter, we address some open questions about the inequality: What is the maximum violation of Tsirelson's inequality possible in quantum theory? Might other theories do better? By considering the precession protocol in a theory-independent manner for systems with finitely many outcomes, we derive a general bound for the maximum possible violation. This theory-independent bound must be satisfied by any theory whose expectation values are linear functions of observables -- which includes classical, quantum, and all general probabilistic theories -- and depends only on the minimum positive and negative measurement outcomes. Given any such two values, we prove by construction that quantum theory always saturates this bound. Some notable examples include the angular momentum of a spin-$3/2$ particle and a family of observables that outperform the quantum harmonic oscillator in the precession protocol. Finally, we also relate our findings to the recently introduced notion of constrained conditional probabilities.