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arXiv:2405.17966 · Physical Review A

Even-parity precession protocol for detecting nonclassicality and entanglement

Jinyan Chen, Jackson Tiong, Lin Htoo Zaw, Valerio Scarani

Published Dec 4, 2024• 2 citations
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Abstract

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.

DOI
10.1103/physreva.110.062408

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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≥2j\ensuremath{\ge}2j\ensuremath≥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.
Of existing entanglement witnesses that utilize only collective measurements of a spin ensemble, not all can detect genuine multipartite entanglement (GME), and none can detect Greenberger-Horne-Zeilinger (GHZ) states beyond the tripartite case. We fill this gap by introducing an entanglement witness that detects GME of spin ensembles, whose total spin is half-integer, using only collective spin measurements. Our witness is based on a nonclassicality test introduced by Tsirelson, and solely requires the measurement of total angular momentum along different directions. States detected by our witness are close to a family of GHZ-like states, which includes GHZ states of an odd number of spin-half particles. We also study the robustness of our witness under depolarizing noise, and derive exact noise bounds for detecting noisy GHZ 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).
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.
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 entanglement witnesses for spin ensembles which detect genuine multipartite entanglement using only measurements of the total angular momentum. States that are missed by most other angular-momentum-based witnesses for spin ensembles, which include Greenberger-Horne-Zeilinger states and certain superpositions of Dicke states, can be effectively detected by our witness. The protocol involves estimating the probability that the total angular momentum is positive along equally spaced directions on a plane. Alternatively, one could measure along a single direction at different times, under the assumption that the total spins undergo a uniform precession. Genuine multipartite entanglement is detected when the observed score exceeds a separable bound. Exact analytical expressions for the separable bound are obtained for spin ensembles ${j}_{1}\ensuremath{\bigotimes}{j}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{j}_{N}$ such that the total spin is a half-integer and numerical results are reported for the other cases. Finally, we conjecture an expression for the separable bound when the total spin is not known, which is well supported by the numerical results.
We introduce a family of entanglement witnesses for continuous variable systems, which rely on the sole assumption that their dynamics is that of coupled harmonic oscillators at the time of the test. Entanglement is inferred from the Tsirelson nonclassicality test on one of the normal modes, without any knowledge about the state of the other mode. In each round, the protocol requires measuring only the sign of one coordinate (e.g., position) at one among several times. This dynamic-based entanglement witness is more akin to a Bell inequality than to an uncertainty relation: in particular, it does not admit false positives from classical theory. Our criterion detects non-Gaussian states, some of which are missed by other criteria.