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arXiv:2411.03132 · npj Quantum Information

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

Lin Htoo Zaw, Valerio Scarani

Published Sep 26, 2025• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1038/s41534-025-01098-7

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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.
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.
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.
We introduce an even-parity precession protocol that can detect the nonclassicality of some quantum states using only measurements of a uniformly precessing variable at different points in time. Depending on the system under study, the protocol may detect the Wigner negativity of a single quantum harmonic oscillator or of a single spin $j\ensuremath{\ge}2$; the non-Gaussian entanglement of two harmonic oscillators; or genuine multipartite entanglement of a spin ensemble, whose total spin is an integer. Unlike in other nonclassicality tests, simultaneous or sequential measurements are not required. Our protocol can also detect states that commute with the parity operator, which were missed by similar protocols built from Tsirelson's original precession protocol. This work also closes a long-standing gap by showing the possibility of detecting the Greenberger-Horne-Zeilinger entanglement of an even number of qubits using only collective spin measurements.
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).
The dimensionality of entanglement, quantified by the Schmidt number, is a core resource for quantum information processing. While its certification is well-established in finite-dimensional systems, many platforms harnessing high Schmidt numbers are natively continuous-variable (CV) systems, which are usually truncated to a finite subspace before canonical discrete-variable techniques are applied. For those instances, a core question is whether directly utilizing a genuinely CV toolbox leads to better and more robust characterisations. We answer this question by introducing Schmidt-number witnesses built directly from correlations and covariances of phase-space observables, accessible through characteristic- or Wigner-function measurements. They hold for arbitrary mixed states, including non-Gaussian ones, and only require correlations along a slice of phase space. We show that this direct estimation is more robust and versatile than first discretizing the system and using canonical discrete variable techniques. Moreover, we apply our method to certify arbitrarily-large Schmidt numbers of a bound-entangled Gaussian state. These results motivate genuine CV methods for characterizing high-dimensional entanglement across CV quantum technologies.
A dynamics-based test, originally proposed by Tsirelson for the harmonic oscillator, provides a method for certifying quantumness under the assumption of a known Hamiltonian. These tests, however, are typically proposed for isolated systems, an assumption that breaks down in experimental implementation. In this paper, we extend the protocol to the harmonic oscillator coupled to standard models of dissipation: thermal relaxation, pure dephasing, and the Caldeira--Leggett model. Using the Moyal-Wigner formalism of quantum mechanics in phase space, we show that the introduction of dissipation requires a dissipation-dependent shift of the classical bound, and provide the threshold under which the nonclassicality witness retains its validity.