Leggett-Garg Inequality Research Articles

Transition from quantum to classical dynamics in weak measurements and reconstruction of quantum correlation

The ability to follow the dynamics of a quantum system in a quantitative manner is of key importance for quantum technology. Despite its central role, justifiable deduction of the quantum dynamics of a single quantum system in terms of a macroscopical observable remains a challenge. Here we show that the relation between the readout signal of a single electron spin and the quantum dynamics of the single nuclear spin is given by a parameter related to the measurement strength. We determine this measurement strength in independent experiments and use this value to compare our analysis of the quantum dynamics with experimental results. We prove the validity of our approach by measuring violations of the Leggett-Garg inequality.

Leggett-Garg violations for continuous-variable systems with Gaussian states

Macrorealism (MR) is the worldview that certain quantities may take definite values at all times irrespective of past or future measurements and may be experimentally falsified via the Leggett-Garg (LG) inequalities. We put this worldview to the test for systems described by a continuous variable $x$ by seeking LG violations for measurements of a dichotomic variable $Q=\text{sign}(x)$, in the case of Gaussian initial states in a quantum harmonic oscillator. Extending our earlier analysis [C. Mawby and J. J. Halliwell, Phys. Rev. A 105, 022221 (2022)], we find analytic expressions for the temporal correlators. An exploration of parameter space reveals significant regimes in which the two-time LG inequalities are violated, and likewise at three and four times. To obtain a physical picture of the LG violations, we exploit the continuous nature of the underlying position variable and analyze the relevant quantum-mechanical currents, Bohm trajectories, and Wigner function. We also show that larger violations are possible using the Wigner LG inequalities. Further, we extend the analysis to LG tests using coherent state projectors, thermal coherent states, and squeezed states.

Leggett-Garg inequalities with deformed Pegg-Barnett phase observables

We investigate the Leggett-Garg inequalities (LGIs) for a boson system whose observables are given by deforming the Pegg-Barnett phase operators. We consider two observables and show that the quantum Fourier transform is useful in the realization of the required measurements. Deriving explicit forms for the LGIs using the coherent state $|\ensuremath{\alpha}\ensuremath{\rangle}$ as the initial state, we explore the regimes where they are violated when the time difference between observations of the phase operators is varied. We show that the system remains nonclassical in the large amplitude limit without dissipation; however, with dissipation, our violation diminishes rapidly.

Experimental violation of the Leggett-Garg inequality in a three-level trapped-ion system

Leggett-Garg inequality (LGI) studies the temporal correlation in the evolution of physical systems. Classical systems obey the LGI but quantum systems may violate it. The extent of the violation depends on the dimension of the quantum system and the state update rule. In this work, we experimentally test the LGI in a three-level trapped-ion system under the model of a large spin precessing in a magnetic field. The Von Neumann and L\"uders state update rules are employed in our system for direct comparative analysis. The maximum observed value of Leggett-Garg correlator under the Von Neumann state update rule is $K_3 = 1.739 \pm 0.014$, which demonstrates a violation of the L\"uders bound by 17 standard deviations and is by far the most significant violation in natural three-level systems.

Classical model of quantum interferometry tests of macrorealism

Macrorealism is a characteristic feature of many, but not all, classical systems. It is known, for example, that classical light can violate a Leggett–Garg inequality and, hence, reject a macrorealist interpretation. A recent experiment has used entangled light and negative measurements to demonstrate a loophole-free test of macrorealism [Joarder et al., PRX Quantum 3, 010307 (2022)]. This paper shows that such an experiment, while soundly rejecting macrorealism, may nevertheless be open to a classical interpretation. This is done by offering an explicit classical model of heralded photon detection in an optical interferometer with beam blockers. A numerical analysis of the model shows good agreement with experimental observations and consistency with both local realism and a rejection of macrorealism.

Trade-off relations of quantum resource theory in neutrino oscillations

The violation of the classical bounds imposed by Leggett-Garg inequalities has tested the quantumness of neutrino oscillations (NOs) over a long distance during the propagation. The measure of quantumness in experimentally observed NOs is studied via quantum resource theory (QRT). Here, we focus on the trade-off relations of QRT in the three-flavor NOs, based on Bell-type violations, first-order coherence and intrinsic concurrence, and the relative entropy of coherence. For the electron and muon antineutrino oscillations, the analytical trade-off relations obeyed by the Bell-CHSH inequality of pairwise flavor states in this three-flavor neutrino system are obtained; the sum of the maximal violation of the CHSH tests for three pairwise flavor states is less than or equal to 12. Moreover, there exists an equality relation concerning first-order coherence and intrinsic concurrence in NOs, showing how much quantum resources flow between first-order coherence and intrinsic concurrence during the neutrino propagation. In addition, it is found that the tripartite coherence of three-flavor system is equal to or larger than the sum of the coherence of reduced bipartite flavor states. The trade-off relations of QRT provide a method for studying how the quantum resources convert and distribute in NOs, which might inspire the future applications in quantum information processing using neutrinos.

Nonlocality under uncertainty-disturbance relations and self-duality

We exploit the interplay between uncertainty, disturbance, nonlocality, and self-duality within the framework of generalized probabilistic theories (GPTs). We first introduce an operational uncertainty-disturbance relation that reveals the discrepancies of GPTs in uncertainty and disturbance exhibited by one single measurement. We then apply the relation to Bell's scenarios and derive upper bounds for the basic spatial Clauser-Horne-Shimony-Holt and the temporal Leggett-Garg inequalities with and without the assumption of strong self-duality, respectively. It turns out that self-duality, though being thought of as one pillar of the axiomatization of quantum theory and implying additional constraints, is insufficient to explain the quantum violations. These insights may serve as a seed for a universal understanding of these basic properties in GPTs.

Quantum nonlinear spectroscopy of single nuclear spins

Conventional nonlinear spectroscopy, which use classical probes, can only access a limited set of correlations in a quantum system. Here we demonstrate that quantum nonlinear spectroscopy, in which a quantum sensor and a quantum object are first entangled and the sensor is measured along a chosen basis, can extract arbitrary types and orders of correlations in a quantum system. We measured fourth-order correlations of single nuclear spins that cannot be measured in conventional nonlinear spectroscopy, using sequential weak measurement via a nitrogen-vacancy center in diamond. The quantum nonlinear spectroscopy provides fingerprint features to identify different types of objects, such as Gaussian noises, random-phased AC fields, and quantum spins, which would be indistinguishable in second-order correlations. This work constitutes an initial step toward the application of higher-order correlations to quantum sensing, to examining the quantum foundation (by, e.g., higher-order Leggett-Garg inequality), and to studying quantum many-body physics.

Geuine tripartite entanglement in three-flavor neutrino oscillations

The violation of Leggett–Garg inequalities tested the quantumness of neutrino oscillations (NOs) across macroscopic distances. The quantumness can be quantified by using the tools of the quantum resource theories. Recently, a new genuine tripartite entanglement measure (Xie et al. in Phys Rev Lett 127:040403, 2021), concurrence fill, is defined as the square root of the area of the concurrence triangle satisfying all genuine multipartite entanglement conditions. It has several advantages compared to other existing tripartite measures. Here, we focus on using concurrence fill to quantify the tripartite entanglement in three-flavor NOs. Concurrence fill can reach its maximum 0.89 for the experimentally-observed electron antineutrino oscillations, but it cannot for the muon antineutrino oscillations. In both cases, we compare its performance with other three tripartite entanglement measures, including the generalized geometric measure (GGM), the three-pi entanglement, and the genuinely multipartite concurrence (GMC), in the neutrino propagation, and accordingly show that concurrence fill contains the most quantum resource. Furthermore, concurrence fill and the three-pi entanglement are always smooth, while GGM and GMC measures have several sharp peaks. The genuine tripartite quantification of the quantumness of three-flavor NOs represents the first step towards the further potential application of neutrinos on quantum information processing.

Leggett-Garg inequalities for testing quantumness of gravity

In this study, we determine a violation of the Leggett-Garg inequalities due to gravitational interaction in a hybrid system consisting of a harmonic oscillator and a spatially localized superposed particle. The violation of the Leggett-Garg inequalities is discussed using the two-time quasiprobability in connection with the entanglement negativity generated by gravitational interaction. It is demonstrated that the entanglement suppresses the violation of the Leggett-Garg inequalities when one of the two times of the quasiprobability ${t}_{1}$ is chosen as the initial time. Further, it is shown that the Leggett-Garg inequalities are generally violated due to gravitational interaction by properly choosing the configuration of the parameters, including ${t}_{1}$ and ${t}_{2}$, which are the times of the two-time quasiprobability. The feasibility of detecting violations of the Leggett-Garg inequalities in hybrid systems is also discussed.

Macroscopic delayed choice and retrocausality: Quantum eraser, Leggett-Garg, and dimension witness tests with cat states

We propose delayed choice experiments carried out with macroscopic qubits, realised as macroscopically-distinct coherent states $|\alpha\rangle$ and $|-\alpha\rangle$. Quantum superpositions of $|\alpha\rangle$ and $|-\alpha\rangle$ are created via a unitary interaction $U(\theta)$ based on a nonlinear Hamiltonian. Macroscopic delayed-choice experiments give a compelling reason to develop interpretations not allowing macroscopic retrocausality (MrC). We therefore consider weak macroscopic realism (wMR), which specifies a hidden variable $\lambda_{\theta}$ to determine the macroscopic qubit value (analogous to 'which-way' information), independent of any future measurement setting $\phi$. Using entangled states, we demonstrate a quantum eraser where the choice to measure a which-way or wave-type property is delayed. Consistency with wMR is possible, if we interpret the macroscopic qubit value to be determined by $\lambda_{\theta}$ without specification of the state at the level of $\hbar$, where fringes manifest. We then demonstrate violations of a delayed-choice Leggett-Garg inequality, and of the dimension witness inequality applied to the Wheeler-Chaves-Lemos-Pienaar experiment, where measurements need only distinguish the macroscopic qubit states. This negates all two-dimensional non-retrocausal models, thereby suggesting MrC. However, one can interpret consistently with wMR, thus avoiding MrC, by noting extra dimensions, and by noting that the violations require further unitary dynamics $U$ for each system. The violations are then explained as failure of deterministic macroscopic realism (dMR), which specifies validity of $\lambda_{\theta}$ prior to the dynamics $U(\theta)$ determining the measurement setting $\theta$. Finally, although there is consistency with wMR for macroscopic observations, we demonstrate EPR paradoxes at a microscopic level.

Gravitational effects on quantum coherence in neutrino oscillation

In this paper, we investigate the quantum coherence for two flavor neutrinos propagating in a Schwarzschild metric. In fact, this issue is explored both qualitatively via calculating the parameter ${K}_{3}$ in Leggett-Garg inequality and also quantitatively by evaluating the ${l}_{1}$-norm, $\mathcal{C}(\ensuremath{\rho})$. Using the weak field approximations, we show that the gravitational effects decrease the maximum value of ${K}_{3}$ for some intervals of energy in such a way that there is no violation, while it leaves the maximum amount of the quantum coherence, $\mathcal{C}(\ensuremath{\rho})$, unchanged.

Bipartite Leggett-Garg and macroscopic Bell-inequality violations using cat states: Distinguishing weak and deterministic macroscopic realism

We consider tests of Leggett-Garg's macrorealism and of macroscopic local realism, where for spacelike separated measurements the assumption of macroscopic noninvasive measurability is justified by that of macroscopic Bell locality. We give a mapping between the Bell and Leggett-Garg experiments for microscopic qubits based on spin-$1/2$ eigenstates and gedanken experiments for macroscopic qubits based on two macroscopically distinct coherent states $|\ensuremath{\alpha}\ensuremath{\rangle}$ and $|\ensuremath{-}\ensuremath{\alpha}\ensuremath{\rangle}$ (as $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty}$). In this mapping, the unitary rotation ${U}_{\ensuremath{\theta}}$ of the Stern-Gerlach analyzer or polarizing beam splitter is realized by a local interaction $H=\mathrm{\ensuremath{\Omega}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}^{4}$ where $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}$ is the number of quanta. By adjusting the time of interaction, one alters the measurement setting $\ensuremath{\theta}$. We thus predict violations of Leggett-Garg and Bell inequalities in a macroscopic regime where coarse-grained measurements $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ need only discriminate between two macroscopically distinct coherent states. To interpret the violations, we distinguish between different definitions of macroscopic realism. Deterministic macroscopic local realism (dMR) assumes the system is in a state with a definite outcome ${\ensuremath{\lambda}}_{\ensuremath{\theta}}$ for the measurement $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ prior to the unitary rotation ${U}_{\ensuremath{\theta}}$, and is negated by the violations. Weak macroscopic realism (wMR) assumes a definite outcome for systems prepared in a superposition ${\ensuremath{\psi}}_{\text{pointer}}$ of two macroscopically distinct eigenstates of $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$, after the unitary rotation ${U}_{\ensuremath{\theta}}$. We find that wMR can be viewed consistently with the violations and with the predictions of quantum mechanics for two tests of wMR. A model is presented in which wMR holds and the macroscopic violations emerge over the course of the unitary dynamics occurring at both sites. Implications for the realism of the micro-macro state generated during a quantum measurement are discussed. Finally, we point out an EPR-type paradox, which demonstrates inconsistency between wMR and the completeness of quantum mechanics at a microscopic level.

Leggett–Garg inequality, Wigner form of Leggett–Garg inequality and no-signaling-in-time condition under coarsening measurement

Leggett–Garg inequality, Wigner form of Leggett–Garg inequality and no-signaling-in-time condition under coarsening measurement

Experimental violation of the Leggett-Garg inequality with a single-spin system

Investigation the boundary between quantum mechanical description and classical realistic view is of fundamental importance. The Leggett-Garg inequality provides a criterion to distinguish between quantum systems and classical systems, and can be used to prove the macroscopic superposition state. A larger upper bound of the LG function can be obtained in a multi-level system. Here, we present an experimental violation of the Leggett-Garg inequality in a three-level system using nitrogen-vacancy center in diamond by ideal negative result measurement. The experimental maximum value of Leggett-Garg function is $K_{3}^{exp}=1.625\pm0.022$ which exceeds the L\"uders bound with a $5\sigma$ level of confidence.

Experimental violations of Leggett-Garg inequalities on a quantum computer

Leggett-Garg's inequalities predict sharp bounds for some classical correlation functions that address the quantum or classical nature of real-time evolutions. We experimentally observe the violations of these bounds on single- and multiqubit systems, in different settings, exploiting the IBM Quantum platform. In the multiqubit case, we introduce the Leggett-Garg-Bell's inequalities as an alternative to the previous ones. Measuring these correlation functions, we find quantum error mitigation to be essential to spot inequalities violations. Accessing only two qubit readouts, we assess Leggett-Garg-Bell's inequalities to emerge as the most efficient quantum coherence witnesses to be used for investigating quantum hardware, among those introduced. Our analysis highlights the limits of current quantum platforms, showing that the above-mentioned correlation functions deviate from theoretical prediction as the number of qubits and the depth of the circuit grow.

Quantum violations of Lders bound Leggett–Garg inequalities for non-unitary quantum channel

Leggett Garg inequalities (LGIs) provides an elegant way for probing the incompatibility between the notion of macrorealism and quantum mechanics. For unitary dynamics the optimal quantum violation of a LGI is constrained by the Lders bound. In this paper, we have studied two formulations of LGIs in three-time LG scenario, viz, the standard LGIs and third-order LGIs both for unbiased and biased measurement settings. We show that if system evolves under non-unitary quantum channel between two measurements, the quantum violations of both forms of LGIs exceed their respective Lders bounds and can even reach their algebraic maximum in sharp measurement settings. We found that when the measurement is unsharp the quantum violations of both standard and third-order LGIs for non-unitary quantum channel can be obtained for lower value of the unsharpness parameter compared to the unitary dynamics. We critically examined the violation of Lders bound of LGIs and its relation to the violation of various no-signaling in time conditions, another formalism for testing macrorealism. It is shown that mere violations of no-signaling conditions are not enough to warrant the violation of standard LGIs, an interplay between the violations of various NSIT condition along with a threshold value play an important role. On the other hand violation of third-order LGI is obtained when the degree of violation of a specific no-signaling in time condition reaches a different threshold value.

Lüders Bounds of Leggett–Garg Inequalities, PT$\mathcal {PT}$‐ Symmetric Evolution and Arrow‐of‐Time

AbstractLeggett–Garg inequalities (LGIs) test the incompatibility between the notion of macrorealism and quantum theory. For unitary dynamics, the optimal quantum violation of a Leggett–Garg inequality is constrained by the Lüders bound. However, the LGIs do not provide the necessary and sufficient condition of macrorealism. A suitably formulated set of no‐signaling in time conditions along with the arrow‐of‐time condition provides the same. In this paper, two formulations in the three‐time Leggett–Garg scenario, are studied namely, the standard LGIs and the recently formulated variant of LGIs when the system evolves under PT‐symmetric Hamiltonian. It is first demonstrated that the quantum violations of both forms of LGIs exceed their respective Lüders bounds and can even reach their algebraic maximum. It is further shown that for the case of standard Leggett–Garg inequality, the violation of Lüders bound can be obtained when both no‐signaling in time and arrow‐of‐time conditions are violated. Interestingly, for the case of a variant of LGIs, for suitable choices of relevant parameters, the quantum violation can even be obtained when only the arrow‐of‐time is violated. However, all no‐signaling in time conditions are satisfied. Such a feature is hitherto unexplored.

Leggett-Garg tests for macrorealism in the quantum harmonic oscillator and more general bound systems

The Leggett-Garg (LG) inequalities were introduced to test for the possible presence of macroscopic quantum coherence. Since such effects may be found in various types of macroscopic oscillators, we consider the application of the LG approach to the one-dimensional quantum harmonic oscillator and also to more general bound systems, using a single dichotomic variable $Q$ given by the sign of the oscillator position. We present a simple method to calculate the temporal correlation functions appearing in the LG inequalities for any bound system for which the eigenspectrum is (exactly or numerically) known. We apply this result to the quantum harmonic oscillator for a variety of experimentally accessible states, namely energy eigenstates, and superpositions thereof. For the subspace of states spanned by only the ground state and first excited state, we readily find substantial regions of parameter space in which the LG inequalities at two, three and four times can each be independently violated or satisfied. We also find that the violations persist, although are reduced, when the sign function defining $Q$ is smeared to reflect experimental imprecision. For higher energy eigenstates, we find that LG violations diminish, showing the expected classicalization. With a $Q$ defined using a more general type of position coarse graining, we find two-time LG violations even in the ground state. We also show that two-time LG violations in a gaussian state are readily found if the dichotomic variable at one of the times is taken to be the parity operator. To demonstrate the versatility of the approach, we repeat much of the LG analysis for the Morse potential, finding qualitatively similar physical results.

New test of neutrino oscillation coherence with Leggett\u2013Garg inequality

Leggett–Garg inequality (LGI) is a time analogue of Bell’s inequality that concerns measurements performed on a system at different times. Violation to LGI indicates quantum coherence. We present a Leggett–Garg-type inequality compatible with more general neutrino oscillation frameworks, allowing the effects of decoherence to be taken into consideration. The inequality is applied to test coherence for data from Daya Bay, MINOS, and KamLAND experiments, and their results are compared to theoretical predictions to investigate decoherence. Both Daya Bay and MINOS data exhibit clear violations of over 10sigma , and of over 90% of theoretical predictions, while the KamLAND data exhibit violation of 1.9sigma , being of 58% of the theoretical prediction. The present work is the first to have considered the energy uncertainties in neutrino coherence tests.