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arXiv:2401.16147 · Physical Review Letters

Tsirelson’s Inequality for the Precession Protocol Is Maximally Violated by Quantum Theory

Lin Htoo Zaw, Mirjam Weilenmann, Valerio Scarani

Published May 12, 2025• 3 citations
arXivHTMLPDF

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}≤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.

DOI
10.1103/physrevlett.134.190201

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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}≤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.
Macrorealism posits that a system possesses definite properties at all times and that we can discover these properties, in principle, without disturbing the system's subsequent behavior. The Leggett-Garg inequalities are derived under these assumptions and are readily violated by standard quantum mechanics, thereby providing a scheme to test whether demonstrably macroscopic systems can exhibit quantum coherence. Unfortunately, Leggett-Garg tests suffer from the difficult to avoid clumsiness loophole: the difficulty of proving that sequential measurements have not inadvertently disturbed the system. The recently uncovered Tsirelson inequality is derived from the simple dynamical assumption of uniform precession, obeyed by many classical systems, and requires only single-time measurements. However, Tsirelson inequality violations could be explained by a macrorealistic system that merely breaks the dynamical assumption, rather than genuine quantum behavior. By carrying out a quantum-mechanical analysis of the Tsirelson inequality in the harmonic oscillator, we develop a protocol to rule out this possibility by assessing generalized conditions of uniform precession. We show that various measures of uniform precession, some of which are related to Leggett-Garg quantities, are satisfied well enough that the presence of quantum-mechanical interference terms must be implied. We derive several incidental mathematical results relating to violating states of Tsirelson's inequality, concerning dwell time, crossing number, and probability currents, and also consider a group-theoretic analysis of the Tsirelson operator.
Quantifying nonclassicality in continuous-variable systems remains a fundamental problem in quantum information science. 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 KKK equally spaced times, with the classical bound given by 1/2±1/(2K).1/2 \pm 1/(2K).1/2±1/(2K). In this work, we adopt this probability as a symmetry-sensitive probe to assess the quantumness of superpositions of Gaussian states in harmonic oscillators. We derive analytical constraints on Tsirelson probability arising from rotational and parity symmetries, and show that parity symmetry establishes a universal relation between the maximal and minimal Tsirelson probabilities within parity-related state families. Furthermore, we prove that even-fold rotational circular states cannot exhibit Tsirelson violation due to their definite parity. These results reveal how geometric symmetries of Gaussian-state superpositions determine their nonclassical behavior and provide a symmetry-based framework for probing quantumness in continuous-variable systems.
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 P3∞{{\bf{P}}}_{3}^{\infty }P3∞​ achievable by the quantum harmonic oscillator is the same for all such generalised protocols. We also derive markedly tighter bounds for P3∞{{\bf{P}}}_{3}^{\infty }P3∞​ , 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.
Building on work by Tsirelson, we present a family of protocols that detect the nonclassicality of suitable states of a single quantum system, under the sole assumption that the measured dynamical observable undergoes a uniform precession. The case of the harmonic oscillator was anticipated in the work by Tsirelson, which we extend. We then apply the protocols to finite-dimensional spins that undergo uniform precession in real space and find a gap between the classical and the quantum expectations for every $j\ensuremath{\ge}\frac{3}{2}$ (excluding j=2j=2j=2).
Quantumness refers to the peculiar and counterintuitive characteristics exhibited by quantum systems. Tsirelson inequalities have emerged as a powerful tool in quantum theory to detect quantumness and entanglement of harmonic oscillators, spins undergoing uniform precession, and anharmonic systems. In this paper we harness the versatility of Tsirelson inequalities to address two distinct problems: detecting cheating in classic shell games and probing quantumness in spatially separated systems and harmonic oscillators. By adopting a black-box approach and a geometric characterization of the space of conditional probabilities, we demonstrate that Tsirelson inequalities can be used in both scenarios, enabling us to uncover quantum signatures and identify cheaters in a single unified framework. This connection provides an intuitive different perspective on quantumness of mechanical systems.
Quantum theory imposes a strict limit on the strength of non-local correlations. It only allows for a violation of the CHSH inequality up to the value 2 sqrt(2), known as Tsirelson's bound. In this note, we consider generalized CHSH inequalities based on many measurement settings with two possible measurement outcomes each. We demonstrate how to prove Tsirelson bounds for any such generalized CHSH inequality using semidefinite programming. As an example, we show that for any shared entangled state and observables X_1,...,X_n and Y_1,...,Y_n with eigenvalues +/- 1 we have |<X_1 Y_1> + <X_2 Y_1> + <X_2 Y_2> + <X_3 Y_2> + ... + <X_n Y_n> - <X_1 Y_n>| <= 2 n cos(pi/(2n)). It is well known that there exist observables such that equality can be achieved. However, we show that these are indeed optimal. Our approach can easily be generalized to other inequalities for such observables.