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arXiv:2204.10498 · Physical Review A

Detecting quantumness in uniform precessions

Lin Htoo Zaw, Clive Cenxin Aw, Zakarya Lasmar, Valerio Scarani

Published Sep 29, 2022• 12 citations
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Abstract

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 $j=2$).

DOI
10.1103/physreva.106.032222

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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≥32j\ensuremath{\ge}\frac{3}{2}j\ensuremath≥23​ (excluding j=2j=2j=2).
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.
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.
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.
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.
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.
It is shown that a nonclassicality witness, whose expectation value can be measured for all quantum states, can be constructed from every nonclassicality filter. This finding leads to a set of universal witnesses, parameterized by only three real numbers, for the detection of nonclassicality of any quantum state of a harmonic oscillator. An explicit operator expression is given for such a universal witness. The application of the witnesses is demonstrated for a nontrivial example, and its experimental measurement is briefly considered.