Leggett-Garg Inequality Research Articles

Particle physics violating crypto-nonlocal realism

It has been well established that quantum mechanics (QM) violates Bell inequalities (BI), which are consequences of local realism (LR). Remarkably QM also violates Leggett inequalities (LI), which are consequences of a class of nonlocal realism called crypto-nonlocal realism (CNR). Both LR and CNR assume that measurement outcomes are determined by preexisting objective properties, as well as hidden variables (HV) not considered in QM. We extend CNR and LI to include the case that the measurement settings are not externally fixed, but determined by HV. We derive a new version of LI, which is then shown to be violated by entangled B_d mesons, if charge–conjugation–parity (CP) symmetry is indirectly violated, as indeed established. The experimental result is quantitatively estimated by using the indirect CP violation parameters, and the maximum of a suitably defined relative violation is about 2.7%. Our work implies that particle physics violates CNR. Our LI can also be tested in other systems such as photon polarizations.

Experimental violations of Leggett-Garg inequalities up to the algebraic maximum for a photonic qubit

Leggett and Garg defined macrorealism by positing that a macroscopic system will exist in a well-defined state at all times, and that this state can be measured without disturbing it. They also formulated a class of inequalities which hold under macrorealism but can be violated by quantum mechanics. The degree to which quantum systems can violate these inequalities is restricted by the temporal Tsirelson bounds. Recently variants of Leggett-Garg inequalities which can be violated to the algebraic maximum 3 for a sufficient large number of measurement times have been proposed. In this work we report an experimental test of quantum violation of the variants of Leggett-Garg inequalities beyond the temporal Tsirelson bounds, up to $2.248\ifmmode\pm\else\textpm\fi{}0.039$ in five-time measurement scenarios for a photonic qubit. Our results clearly show the advantages of the modified measurement scenarios.

Probing inequivalent forms of Leggett–Garg inequality in subatomic systems

We study various formulations of Leggett–Garg inequality (LGI), specifically, the Wigner and Clauser–Horne forms of LGI, in the context of subatomic systems, in particular, three flavor neutrino as well as meson systems. The optimal forms of various LGIs for either neutrinos or mesons are seen to depend on measurement settings. For the neutrinos, some of these inequalities can be written completely in terms of experimentally measurable probabilities. Hence, the Wigner and Clauser–Horne forms of LGI are found to be more suitable as compared to the standard LGI from the experimental point of view for the neutrino system. Further, these inequalities exhibit maximum quantum violation around the energies roughly corresponding to the maximum neutrino flux. The Leggett–Garg type inequality is seen to be more suited for the meson dynamics. The meson system being inherently a decaying system allows one to see the effect of decoherence on the extent of violation of various inequalities. Decoherence is observed to reduce the degree of violation, and hence the nonclassical nature of the system.

Experimental test of Leggett's inequalities with solid-state spins

Bell's theorem states that no local hidden variable model is compatible with quantum mechanics. Surprisingly, even if we release the locality constraint, certain nonlocal hidden variable models, such as the one proposed by Leggett, may still be at variance with the predictions of quantum physics. Here, we report an experimental test of Leggett's nonlocal model with solid-state spins in a diamond nitrogen-vacancy center. We entangle an electron spin with a surrounding weakly coupled $^{13}C$ nuclear spin and observe that the entangled states violate Leggett-type inequalities by more than four and seven standard deviations for six and eight measurement settings, respectively. Our experimental results are in full agreement with quantum predictions and violate Leggett's nonlocal hidden variable inequality with a high level of confidence.

Conditions for macrorealism for systems described by many-valued variables

Macrorealism (MR) is the view that a system evolving in time possesses definite properties independent of past or future measurements and is traditionally tested for systems described at each time by a single dichotomic variable $Q$. A number of necessary and sufficient conditions for macrorealism have been derived for a dichtomic variable using sets of Leggett-Garg (LG) inequalities, or the stronger no-signaling in time (NSIT) conditions, or a combination thereof. Here, we extend this framework by establishing necessary and sufficient conditions for macrorealism for measurements made at two and three times for systems described by variables taking three or more values at each time. Our results include a generalization of Fine's theorem to many-valued variables for measurements at three pairs of times and we derive the corresponding complete set of LG inequalities. We find that LG inequalities and NSIT conditions for many-valued variables do not enjoy the simple hierarchical relationship exhibited by the dichotomic case. This sheds light on some recent experiments on three-level systems which exhibit a LG inequality violation even though certain NSIT conditions are satisfied. Under measurements of dichotomic variables using the Luders projection rule the three-time LG inequalities cannot be violated beyond the Luders bound (which coincides numerically with the Tsirelson bound obeyed by correlators in Bell experiments), but this bound can be violated in LG tests using degeneracy-breaking (von Neumann) measurements. We identify precisely which MR conditions are violated under these circumstances.

Influence of equilibrium and nonequilibrium environments on macroscopic realism through the Leggett-Garg inequalities

We study the macroscopic realism (macrorealism) through the two- and three-time Leggett-Garg inequalities (LGIs) in a two interacting qubits system. The two qubits are coupled either with two bosonic (thermal or photonic) baths or fermionic (electronic) baths. We study both how the equilibrium and nonequilibrium environments influence the LGIs. One way to characterize the nonequilibrium condition is by the temperature difference (for the bosonic bath) or the chemical potential difference (for the fermionic bath). We also study the heat or particle current and the entropy production rate generated by the nonequilibrium environments. Analytical forms of LGIs and the maximal value of LGIs based on the quantum master equation beyond the secular approximation are derived. The LGI functions and the corresponding maximal value have separated contributions, the part describing the coherent evolution and the part describing the coupling between the system and environments. The environment-coupling part can be from the equilibrium environment or the nonequilibrium environment. The nonequilibrium dynamics is quantified by the Bloch-Redfield equation which is beyond the Lindblad form. We found that the nonequilibriumness quantified by the temperature difference or the chemical potential difference can lead to the LGIs violations or the increase of the maximal value of LGIs, restoring the quantum nature from certain equilibrium cases where LGIs are preserved. The corresponding nonequilibrium thermodynamic cost is quantified by the nonzero entropy production rate. Our finding of the nonequilibrium promoted LGIs violations suggests a new strategy for the design of quantum information processing and quantum computational devices to maintain the quantum nature and quantum correlations for long.

Temporal nonlocality of a two-level system interacting with a dephasing environment

Temporal nonlocality of a two-level system interacting with a thermal reservoir is studied in terms of the Leggett–Garg inequality and the quantum witness, where the dephasing coupling and the Ohmic-like spectral density are assumed for an system-reservoir interaction. When the whole system is initially in the thermal equilibrium state, which is a quantum-classical correlated state, violation of the Leggett–Garg inequality and the quantum witness are investigated in detail. It is shown how the time evolution of the temporal nonlocality depends on the system-reservoir coupling strength, the reservoir temperature and the Ohmicity parameter. The results are compared with those obtained for a non-correlated initial state, where the two-level system and the thermal reservoir are initially in their own thermal equilibrium states. The comparison reveals that ignoring the initial correlation yields an additional phase shift in two-time correlation functions of the two-level system. The effect of the system-reservoir initial correlation becomes more significant as the Ohmicity parameter is smaller.

Quantification of quantumness in neutrino oscillations

Neutrino oscillation is an important physical phenomenon in elementary particle physics, and its nonclassical features can be revealed by the Leggett–Garg inequality. It shows that its quantum coherence can be sustained over astrophysical length scales. In this work, we investigate the measure of quantumness in experimentally observed neutrino oscillations via the nonlocal advantage of quantum coherence (NAQC), quantum steering, and Bell nonlocality. From various neutrino sources, ensembles of reactor and accelerator neutrinos are analyzed at distinct energies, such as Daya Bay (0.5 km and 1.6 km) and MINOS (735 km) collaborations. The NAQC of two-flavor neutrino oscillation is characterized experimentally compared to the theoretical prediction. It exhibits non-monotonously evolutive phenomenon with the increase of energy. Furthermore, it is found that the NAQC is a stronger quantum correlation than quantum steering and Bell nonlocality even in the order of km. Hence, for an arbitrary bipartite neutrino-flavor state with achieving a NAQC, it must be also a steerable and Bell nonlocal state. The results might offer an insight into the neutrino oscillation for the further applications on quantum information processing.

Theory of Quantum Path Entanglement and Interference with Multiplane Diffraction of Classical Light Sources.

Quantum history states were recently formulated by extending the consistent histories approach of Griffiths to the entangled superposition of evolution paths and were then experimented with Greenberger–Horne–Zeilinger states. Tensor product structure of history-dependent correlations was also recently exploited as a quantum computing resource in simple linear optical setups performing multiplane diffraction (MPD) of fermionic and bosonic particles with remarkable promises. This significantly motivates the definition of quantum histories of MPD as entanglement resources with the inherent capability of generating an exponentially increasing number of Feynman paths through diffraction planes in a scalable manner and experimental low complexity combining the utilization of coherent light sources and photon-counting detection. In this article, quantum temporal correlation and interference among MPD paths are denoted with quantum path entanglement (QPE) and interference (QPI), respectively, as novel quantum resources. Operator theory modeling of QPE and counterintuitive properties of QPI are presented by combining history-based formulations with Feynman’s path integral approach. Leggett–Garg inequality as temporal analog of Bell’s inequality is violated for MPD with all signaling constraints in the ambiguous form recently formulated by Emary. The proposed theory for MPD-based histories is highly promising for exploiting QPE and QPI as important resources for quantum computation and communications in future architectures.

Comparing Leggett–Garg inequality for work moments with Leggett–Garg inequality and NSIT

In this paper, we investigate non-violations of the work Leggett–Garg (WLG) inequality, the Leggett–Garg (LG) inequality and the no-signaling-in-time (NSIT) condition in a two-level system. We consider two kinds of initial states: the thermal state and the state with coherence. We find that the non-violation condition of WLG inequality for the first work moment is similar to the NSIT conditions. The WLG inequality of the first work moment cannot be violated in the high temperature limit or for a specific driving intensity for the initial thermal state, while for the initial state with coherence, the non-violation of it strongly depends on the relative phase in the quantum coherence. For the initial thermal state and state with coherence, the NSIT conditions cannot be violated for a specific driving intensity when the projective measurement operators at different measurement times are the same, while when the projective measurement operators at different measurement times are different, we cannot find any circumstance to simultaneously make all the NSIT conditions non-violated. We find that Gaussian measurements have no effect on the non-violation conditions of the WLG inequality for the first work moment and the NSIT conditions. We also find that the non-violation condition of WLG inequality for the second work moment is similar to the LG inequality. The non-violation condition of the WLG inequality for the second work moment is the same as the LG inequality under projective measurements, while it cannot be violated for a wider parameter regime than the LG inequality under Gaussian measurements.

Quantumness and memory of one qubit in a dissipative cavity under classical control

Hybrid quantum–classical systems constitute a promising architecture for useful control strategies of quantum systems by means of a classical device. Here we provide a comprehensive study of the dynamics of various manifestations of quantumness with memory effects, identified by non-Markovianity, for a qubit controlled by a classical field and embedded in a leaky cavity. We consider both Leggett–Garg inequality and quantum witness as experimentally-friendly indicators of quantumness, also studying the geometric phase of the evolved (noisy) quantum state. We show that, under resonant qubit-classical field interaction, a stronger coupling to the classical control leads to enhancement of quantumness despite a disappearance of non-Markovianity. Differently, increasing the qubit-field detuning (out-of-resonance) reduces the nonclassical behavior of the qubit while recovering non-Markovian features. We then find that the qubit geometric phase can be remarkably preserved irrespective of the cavity spectral width via strong coupling to the classical field. The controllable interaction with the classical field inhibits the effective time-dependent decay rate of the open qubit. These results supply practical insights towards a classical harnessing of quantum properties in a quantum information scenario.

A quantum information theoretic quantity sensitive to the neutrino mass-hierarchy

In this work, we derive a quantum information theoretic quantity similar to the Leggett-Garg inequality, which can be defined in terms of neutrino transition probabilities. For the case of νμ→νe/ν¯μ→ν¯e transitions, this quantity is sensitive to CP violating effects as well as the neutrino mass-hierarchy, namely which neutrino mass eigenstate is heavier than the other ones. The violation of the inequality for this quantity shows an interesting dependence on mass-hierarchy. For normal (inverted) mass-hierarchy, it is significant for νμ→νe (ν¯μ→ν¯e) transitions. This is applied to the two ongoing accelerator experiments T2K and NOνA as well as the future experiment DUNE.

Wigner inequalities for testing the hypothesis of realism and concepts of macroscopic and local realism

We propose a new Wigner inequality suitable for test of the hypothesis of realism. We show that this inequality is not identical neither to the well-known Wigner inequality nor to the Leggett-Garg inequality in Wigner form. The obtained inequality is suitable for test of realism not only in quantum mechanical systems, but also in quantum field systems. Also we propose a mathematically consistent derivation of the Leggett--Garg inequality in Wigner form, which was recently presented in the literature, for three and $n$ distinct moments of time. Contrary to these works, our rigor derivation uses Kolmogorov axiomatics of probability theory. We pay special attention to the construction and studies of the spaces of elementary outcomes. Basing on the the Leggett--Garg inequality in Wigner form for $n$ distinct moments of time we prove that any unitary evolution of a quantum system contradicts the concept of macroscopic realism. We show that application of the concept of macroscopic realism to any quantum system leads to ``freezing'' of the system in the initial state. It is shown that for a particle with an infinite number of observables the probability to find a pair of the observables in some defined state is zero, even if the operators of these observables commute. This fact might serve as an additional logical argument for the contradiction between quantum theory and classical realism.

Enhancing violations of Leggett-Garg inequalities in nonequilibrium correlated many-body systems by interactions and decoherence

We identify different schemes to enhance the violation of Leggett-Garg inequalities in open many-body systems. Considering a nonequilibrium archetypical setup of quantum transport, we show that particle interactions control the direction and amplitude of maximal violation, and that in the strongly-interacting and strongly-driven regime bulk dephasing enhances the violation. Through an analytical study of a minimal model we unravel the basic ingredients to explain this decoherence-enhanced quantumness, illustrating that such an effect emerges in a wide variety of systems.

Exploration of an augmented set of Leggett-Garg inequalities using a noninvasive continuous-in-time velocity measurement

Macroscopic realism (MR) is the view that a system may possess definite properties at any time independent of past or future measurements, and may be tested experimentally using the Leggett-Garg inequalities (LGIs). In this work we advance the study of LGIs in two ways using experiments carried out on a nuclear magnetic resonance spectrometer. Firstly, we addresses the fact that the LGIs are only necessary conditions for MR but not sufficient ones. We implement a recently-proposed test of necessary and sufficient conditions for MR which consists of a combination of the original four three-time LGIs augmented with a set of twelve two-time LGIs. We explore different regimes in which the two- and three-time LGIs may each be satisfied or violated. Secondly, we implement a recent proposal for a measurement protocol which determines the temporal correlation functions in an approximately non-invasive manner. It employs a measurement of the velocity of a dichotomic variable $Q$, continuous in time, from which a possible sign change of $Q$ may be determined in a single measurement of an ancilla coupled to the velocity. This protocol involves a significantly different set of assumptions to the traditional ideal negative measurement protocol and a comparison with the latter is carried out.

Fine's theorem for Leggett-Garg tests with an arbitrary number of measurement times

If the time evolution of a quantum system can be understood classically, then there must exist an underlying probability distribution for the variables describing the system at a sequence of times. It is well known that for systems described by a single time-evolving dichotomic variable $Q$ and for which a given set of temporal correlation functions are specified, a necessary set of conditions for the existence of such a probability are provided by the Leggett-Garg (LG) inequalities. Fine's theorem in this context is the nontrivial result that a suitably augmented set of LG inequalities are both necessary and sufficient conditions for the existence of an underlying probability. We present a proof of Fine's theorem for the case of measurements on a dichotomic variable at an arbitrary number of times, thereby generalizing the familiar proofs for three and four times. We demonstrate how the LG framework and Fine's theorem can be extended to the case in which all possible two-time correlation functions are measured (instead of the partial set of two-time correlators normally studied). We examine the limit of a large number of measurements for both of the above cases.

Necessary and sufficient conditions for macrorealism using two- and three-time Leggett-Garg inequalities

The Leggett-Garg (LG) inequalities were introduced, as a temporal parallel of the Bell inequalities, to test macroscopic realism – the view that a macroscopic system evolving in time possesses definite properties which can be determined without disturbing the future or past state. The original LG inequalities are only a necessary condition for macrorealism, and are therefore not a decisive test. We argue, for the case of measurements of a single dichotomic variable Q, that when the original four three-time LG inequalities are augmented with a set of twelve two-time inequalities also of the LG form, Fine’s theorem applies and these augmented conditions are then both necessary and sufficient. A comparison is carried out with the alternative necessary and sufficient conditions for macrorealism based on no-signaling in time conditions which ensure that all probabilities for Q at one and two times are independent of whether earlier or intermediate measurements are made. We argue that the two tests differ in their implementation of the key requirement of non-invasive measurability so are testing different notions of macrorealism, and these notions are elucidated.

Leggett-Garg Inequality and Quantumness Under the Influence of Random Telegraph Noise

The violation of the Leggett-Garg inequality and the quantum witness of a two-level system are investigated under the influence of stochastic dephasing which is described by a non-stationary and unbalanced random telegraph noise. An extended version of the characteristic function of the noise is derived by making use of the Lie algebra method. It is found how the maximum violation time of the Leggett-Garg inequality depends on the ratio of dephasing time of the two-level system and correlation time of the noise. It is also shown that to observe the violation, unsharpness of the measurement must be smaller as the correlation time of the stochastic process is shorter. Furthermore decay of the quantum witness of the two-level system is investigated. It is shown that the quantum witness is equivalent to fidelity difference in the quantum teleportation that uses two different entanglement resources under the influence of the common noise.

Paths, negative \u201cprobabilities\u201d, and the Leggett-Garg inequalities

We present a path analysis of the condition under which the outcomes of previous observation affect the results of the measurements yet to be made. It is shown that this effect, also known as “signalling in time”, occurs whenever the earlier measurements are set to destroy interference between two or more virtual paths. We also demonstrate that Feynman’s negative “probabilities” provide for a more reliable witness of “signalling in time”, than the Leggett-Garg inequalities, while both methods are frequently subject to failure.

Leggett-Garg inequality in the context of three flavor neutrino oscillation

The present work is devoted to the characterization of the Leggett-Garg inequality for three-flavoured neutrino oscillations in presence of both matter and Charge-Conjugation and Parity violating (CP) effects. This study complements and completes the recent one put forward in [17] by relaxing the stationary condition. At variance with the latter case, the LGI contains interference terms which cannot be expressed in terms of experimentally measurable quantities, thus drawing a clear-cut distinction between the two scenarios, as well as highlighting the role of the stationary assumption on such systems. We find that the additional terms are small for high energy neutrino beam compared to the maximum value attained by the Leggett-Garg parameter.