Violation Of Leggett-Garg Inequality Research Articles

Quantum violation of LGI under an energy constraint for different scenarios systems

In this paper, we consider a qubit in four scenarios: with drive, without drive, and in the presence of dissipation and dephasing, to investigate the quantum violation of the Leggett–Garg inequality (LGI) in an energy constraint. In the case of the energy constraint, we find that under the coarsening measurement in reference and final resolution, the quantum violation of the LGI for the pure qubit is the most robust; on the other hand, the quantum violation of the LGI for the dephasing qubit is the most vulnerable, and the quantum violation of the LGI for driven qubit lies between that of pure qubit and dissipation qubit. Under the coarsening of measurement temporal reference, the quantum violation of the LGI for the pure qubit is more robust than that of the qubit with driven. Moreover, in the case of a qubit that is subjected to driving and is in the presence of dissipation and dephasing, the robustness of quantum violations of the LGI for these scenario systems will become vulnerable, with the driven intensity and the rate of spontaneous emission increasing, respectively, for coarsening measurement both in reference and in final resolution. In addition, in the energy constraint and the projective measurement, the LGI can attain its maximum violation value, 1.5, for the coherent dynamics; while for drive, dissipative and dephasing qubits, the LGI cannot attain the value of 1.5. For systems in the presence of dissipation and dephasing, we find that in the energy constraint, the robustness of the coarsening measurement in final resolution exhibits more vulnerable than that of the coarsening measurement in reference. And for systems with drive and without drive, the robustness of the coarsening measurement in temporal reference is the most robust, and the robustness of the coarsening of measurement final measurement resolution is the most vulnerable.

Leggett–Garg inequality in Markovian quantum dynamics: role of temporal sequencing of coupling to bath

We study Leggett–Garg inequalities (LGIs) for a two level system (TLS) undergoing Markovian dynamics described by unital maps. We first find the analytic expression of LG parameter K 3 (simplest variant of LGIs) in terms of the parameters of two distinct unital maps representing time evolution for intervals: t 1 to t 2 and t 2 to t 3. We then show that the maximum violation of LGI for all possible unital maps can never exceed well known Lüders bound of over the full parameter space. We further show that if the map for the time interval t 1 to t 2 is non-unitary unital then irrespective of the choice of the map for interval t 2 to t 3 we can never reach Lüders bound. On the other hand, if the measurement operator eigenstates remain pure upon evolution from t 1 to t 2, then depending on the degree of decoherence induced by the unital map for the interval t 2 to t 3 we may or may not obtain Lüders bound. Specifically, we find that if the unital map for interval t 2 to t 3 leads to the shrinking of the Bloch vector beyond half of its unit length, then achieving the bound is not possible. Hence our findings not only establish a threshold for decoherence which will allow for , but also demonstrate the importance of temporal sequencing of the exposure of a TLS to Markovian baths in obtaining Lüders bound.

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.

Enhanced violation of Leggett–Garg inequality in three flavour neutrino oscillations via non-standard interactions

Neutrino oscillations occur due to non-zero masses and mixings and most importantly they are believed to maintain quantum coherence even over astrophysical length scales. In the present study, we explore the quantumness of three flavour neutrino oscillations by studying the extent of violation of Leggett–Garg inequalities (LGI) if non-standard interactions (NSI) are taken into account. We report an enhancement in violation of LGI with respect to the standard scenario for appropriate choice of NSI parameters.

Temporal correlation beyond quantum bounds in non-Hermitian PT- symmetric dynamics of a two level system

We study the dynamics of a non-Hermitian PT- symmetric Hamiltonian of a two level system (TLS) which has real eigenvalues. Within the framework of Hermitian quantum mechanics, it is known that maximal violation of Leggett–Garg inequality (LGI) is bounded by K 3 = 3/2 (Luder’s bound). We show that this absolute bound can be evaded when dynamics is governed by non-Hermitian PT- symmetric Hamiltonians with real eigenvalues. Moreover, the extent of violation can be optimized to asymptotically approach the algebraic maximum of , which is otherwise observed for Hermitian Hamiltonian with infinite dimensional Hilbert space. The extreme violation of LGI is shown to be directly related to the two basic ingredients: (i) the Bloch equation for the TLS having a non-linear terms which allow for accelerated dynamics of states on the Bloch sphere exceeding all known quantum speed limits of state evolution; and (ii) quantum trajectory of states lie on a great circle (geodesic path) on the Bloch sphere at all times. We demonstrate that such extreme temporal correlation of TLS can be simulated in realistic system by embedding the TLS into a higher dimensional Hilbert space such that the composite system obeys unitary dynamics. Specifically we show that a four dimensional embedding of non-Hermitian PT- symmetric TLS is enough to host K 3 → 3 limit. We also discuss the effect of random noise on our results. Finally we conclude with a comparative study of our results with existing experimental realization of embedding.

Effect of memory on the violation of Leggett–Garg inequality

The Leggett-Garg inequalities impose restrictions on the values taken by some combinations of the two-time correlation functions of observables in order to be explainable by a noninvasive-realist classical model. While in the unitary dynamics, it is straightforward to compute these correlation functions, open system effects bring in subtleties. Specifically, for non-Markovian dynamics, which involves setting up of system-bath correlations, the Leggett-Garg measurements disrupt these correlations, making a full system-bath Hamiltonian approach natural. However, here we point out that the problem can also be dealt with from a reduced dynamics perspective. The key point is that the noise superoperator acting on the system must be suitably updated after measurement interventions. Also considered is the effect of Markovian versus non-Markovian behavior as well as classically non-Markovian processes on the violation of Leggett-Garg inequalities.

Interference experiment, anomalous weak value, and Leggett-Garg test of macrorealism

Macrorealism is a classical world view asserting that the properties of macro-objects exist independently and irrespective of observation. One practical approach to test this view in quantum theory is to observe the quantum coherence for a macro-object in an interference experiment. An elegant and conceptually appealing approach for testing the notion of macrorealism in quantum theory is through the violation of Leggett-Garg inequality. However, a conclusive Leggett-Garg test hinges on how the noninvasive measurability criteria are guaranteed in an experiment and remains a debated issue to date. In this work, we connect the practical and the conceptual approaches for testing the macrorealism through the weak value. We argue that whenever a quantum effect is observed in an interference experiment there is an existence of anomalous weak value. Further, we demonstrate that whenever such weak value exists, one obtains the violation of a Leggett-Garg inequality in any interference experiment. Since in a path-only interference experiment effectively no prior measurement is performed, the Leggett-Garg test of macrorealism presented here is without assuming the noninvasive measurability. We provide a rigorous discussion about the assumptions involved in the Leggett-Garg scenario and how our scheme fits into it. Further, we provide a simple argument about the macrorealistic understanding of our results by using the recently developed approach involving quasiprobability.

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.

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.

Leggett-Garg tests of macrorealism for dynamical cat states evolving in a nonlinear medium

We show violations of Leggett-Garg inequalities to be possible for single-mode cat-states evolving dynamically in the presence of a nonlinear quantum interaction arising from, for instance, a Kerr medium. In order to prove the results, we derive a generalised version of the Leggett-Garg inequality involving different cat-states at different times. The violations demonstrate failure of the premise of macro-realism as defined by Leggett and Garg, provided extra assumptions associated with experimental tests are valid. With the additional assumption of stationarity, violations of the Leggett-Garg inequality are predicted for the multi-component cat-states observed in the Bose-Einstein condensate and superconducting circuit experiments of Greiner et al. (Nature 419, 51, (2002)) and Kirchmair et al. (Nature, 495, 205 (2013)). The violations demonstrate a mesoscopic quantum coherence, by negating that the system can be in a classical mixture of mesoscopically distinct coherent states. Higher orders of nonlinearity are also studied and shown to give strong violation of Leggett-Garg inequalities.

Leggett-Garg tests of macrorealism: Checks for noninvasiveness and generalizations to higher-order correlators

In the tests for macrorealism proposed by Leggett and Garg, the temporal correlation functions of a dichotomic variable Q must be measured in a non-invasive way to rule out alternative classical explanations of Leggett-Garg inequality violations. Ideal negative measurements, in which a null result is argued to be a non-invasive determination of the system's state, are often used. From a quantum-mechanical perspective, such a measurement collapses the wave function and will therefore typically be found to be invasive under any experimental check. Here, a simple modified ideal negative measurement protocol is described for measuring the correlation functions which is argued to be non-invasive from both classical and quantum perspectives and hence the non-invasiveness can then be checked experimentally, thereby permitting a quantitative measure of the degree of clumsiness of the measurement. It is also shown how this procedure may be extended to measure higher-order correlation functions and a number of higher-order conditions for macrorealism are derived.

Experimental observation of the Leggett-Garg inequality violation in classical light

Leggett-Garg (LG) inequalities have been derived to clarify the validity of generalizing quantum mechanics to macroscopic systems for more than thirty years. They have been used as tools to investigate the existence of quantum coherence in macroscopic systems. Here, we show that such inequality violation can also be observed in classical optical systems. Using the linear polarization degree of freedom as the dichotomic observables, in a classical optical experiment, we have observed the violation of LG inequality by implementing the non-invasive measurement. Furthermore, the coherence witness, which is analogous to the quantum witness, has also been observed in the classical optical experiment. Our results not only provide new physical insights into the macroscopic quantum correlation and quantum witness, but also show their potential applications in correlated information processes.

Violation of the Lüders bound of macrorealist and noncontextual inequalities

In a recent Letter [PRL, 113, 050401 (2014)], it is shown that the quantum violation of a three-time Leggett-Garg inequality (LGI) for a dichotomic qutrit system can exceed the L$\ddot{u}$ders bound. This is obtained by using a degeneracy breaking projective measurement rule which the authors termed as von Neumann rule. Such violation can even approach the algebraic maximum in the asymptotic limit of system size. In this paper, we question the implication of such violation of L$\ddot{u}$ders bound and its conceptual relevance in LG scenario. We note an important fact that the basis for implementing the proposed von Neumann rule for a degenerate observable is non-unique and show that the violation of L$\ddot{u}$ders bound is crucially dependent on the choice of basis. Further, we demonstrate the violation of L$\ddot{u}$ders bound of the simplest non-contextual inequality (NCI) which is in contrast to the reasoning provided in the aforementioned Letter. This result further raises the doubts regarding the validity of the proposed rule as a viable projective measurement. We discuss the relevance of such results with respect to the usual quantum violation of LGI and NCI.

Canonical Leggett-Garg inequality: Nonclassicality of temporal quantum correlations under energy constraint

Nonclassicality of temporal correlations pertaining to noncommutative sequential measurements is defined through the violation of macrorealistic inequalities, known as Leggett-Garg inequalities (LGI). We investigate the energy cost of the process associated with the Leggett-Garg test in the context of noiseless and Markovian noise for arbitrary initial states. We prove that in noiseless and in certain noisy scenarios, the maximal violations of LGI under the energy constraint occurs when the average energy of the process is equal to the negative of the energy of the initial state. Such a dependence of LGI on the choice of the initial state is not seen in the unconstrained case. Moreover, we find that in the presence of a moderate amount of Markovian noise, the amount of violation of LGI remains almost unaltered with a suitable choice of the evolution and dephasing operators in the neighborhood of the maximal violation line, thereby showing the robustness of temporal correlations under environmental effects.

Universal two-time correlations, out-of-time-ordered correlators, and Leggett-Garg inequality violation by edge Majorana fermion qubits

In the present work we propose that two-time correlations of Majorana edge localized fermions constitute a novel and versatile toolbox for assessing the topological phases of 1D open lattices. Using analytical and numerical calculations on the Kitaev model, we uncover universal relationships between the decay of the short-time correlations and a particular family of out-of-time-ordered correlators, which provide direct experimental alternatives to the quantitative analysis of the system regime, either normal or topological. Furthermore we show that the saturation of two-time correlations possesses features of an order parameter. Finally, we find that violations of Leggett-Garg inequalities can indicate the topological-normal phase transition by looking at different qubits formed by pairing local and non-local edge Majorana fermions.

Nonclassicality of the Harmonic-Oscillator Coherent State Persisting up to the Macroscopic Domain.

Can the most "classical-like" of all quantum states, namely the Schrödinger coherent state of a harmonic oscillator, exhibit nonclassical behavior? We find that for an oscillating object initially in a coherent state, merely by observing at various instants which spatial region the object is in, the Leggett-Garg inequality (LGI) can be violated through a genuine negative result measurement, thereby repudiating the everyday notion of macrorealism. This violation thus reveals an unnoticed nonclassicality of the very state which epitomizes classicality within the quantum description. It is found that for any given mass and oscillator frequency, a significant quantum violation of LGI can be obtained by suitably choosing the initial peak momentum of the coherent state wave packet. It thus opens up potentially the simplest way (without coupling with any ancillary quantum system or using nonlinearity) for testing whether various recently engineered and sought after macroscopic oscillators, such as feedback cooled thermal trapped nanocrystals of ∼10^{6}-10^{9} amu mass, are indeed bona fide nonclassical objects.

Leggett-Garg Inequality for a Two-Level System under Decoherence

We consider a macroscopic quantum system subjected to an asymmetric double-well potential in a harmonic environment. By using a time-dependent approach, we calculate tunnelling probabilities for the system which contain oscillation effects. To show how one can decide between quantum mechanics and the implications of macrorealism assumptions, a given form of Leggett-Garg inequality is considered. The violation of this inequality occurs for a broader range of the system-environment interactions, compared to previous results obtained for two-level systems. Assuming that the coupling strength between the system and the environment can be controlled with time, one can see the violation even for strong decoherence effects. We also investigate the variation of the tilt/tunnelling parameters on the violation of Leggett-Garg inequality.

Constraints on macroscopic realism without assuming non-invasive measurability

Macroscopic realism is the thesis that macroscopically observable properties must always have definite values. The idea was introduced by Leggett and Garg (1985), who wished to show a conflict with the predictions of quantum theory, by using it to derive an inequality that quantum theory violates. However, Leggett and Garg's analysis required not just the assumption of macroscopic realism per se, but also that the observable properties could be measured non-invasively. In recent years there has been increasing interest in experimental tests of the violation of the Leggett-Garg inequality, but it has remained a matter of controversy whether this second assumption is a reasonable requirement for a macroscopic realist view of quantum theory. In a recent critical assessment Maroney and Timpson (2014) identified three different categories of macroscopic realism, and argued that only the simplest category could be ruled out by Leggett-Garg inequality violations. Allen, Maroney, and Gogioso (2016) then showed that the second of these approaches was also incompatible with quantum theory in Hilbert spaces of dimension 4 or higher. However, we show that the distinction introduced by Maroney and Timpson between the second and third approaches is not noise tolerant, so unfortunately Allen's result, as given, is not directly empirically testable. In this paper we replace Maroney and Timpson's three categories with a parameterization of macroscopic realist models, which can be related to experimental observations in a noise tolerant way, and recover the original definitions in the noise-free limit. We show how this parameterization can be used to experimentally rule out classes of macroscopic realism in Hilbert spaces of dimension 3 or higher, without any use of the non-invasive measurability assumption. Even for relatively low precision experiments, this will rule out the original category of macroscopic realism, that is tested by the Leggett-Garg inequality, while as the precision of the experiments increases, all cases of the second category and many cases of the third category, will become experimentally ruled out.

Enhanced violations of Leggett-Garg inequalities in an experimental three-level system.

Leggett-Garg inequalities are tests of macroscopic realism that can be violated by quantum mechanics. In this letter, we realise photonic Leggett-Garg tests on a three-level system and implement measurements that admit three distinct measurement outcomes, rather than the usual two. In this way we obtain violations of three- and four-time Leggett-Garg inequalities that are significantly in excess of those obtainable in standard Leggett-Garg tests. We also report violations the quantum-witness equality up to the maximum permitted for a three-outcome measurement. Our results highlight differences between spatial and temporal correlations in quantum mechanics.