Leggett-Garg (LG) tests for macrorealism were originally designed to explore quantum coherence on the macroscopic scale. Interference experiments and systems modeled by harmonic oscillators provide useful examples of situations in which macroscopicity has been approached experimentally and are readily turned into LG tests for a single dichotomic variable
Q. Applying this approach to the double-slit experiment in which a noninvasive measurement at the slits is included, we exhibit LG violations. We find that these violations are always accompanied by destructive interference. The converse is not true in general and we find that there are nontrivial regimes in which there is destructive interference but the two-time LG inequalities are satisfied, which implies that it is in fact often possible to assign (indirectly determined) probabilities for the interferometer paths. Similar features have been observed in recent work involving a LG analysis of a Mach-Zehnder interferometer and we compare with those results. We extend the analysis to the triple-slit experiment again finding LG violations, and we also exhibit examples of some surprising relationships between LG inequalities and NSIT conditions that do not exist for dichotomic variables. For the simple harmonic oscillator, we find an analytically tractable example showing a two-time LG violation with a gaussian initial state, echoing in simpler form recent results of Bose et al. [Phys. Rev. Lett. 120, 210402 (2018)].
In contrast to the spatial Bell's inequalities which probe entanglement between spatially separated systems, the Leggett–Garg inequalities test the correlations of a single system measured at different times. Violation of a genuine Leggett–Garg test implies either the absence of a realistic description of the system or the impossibility of measuring the system without disturbing it. Quantum mechanics violates the inequalities on both accounts and the original motivation for these inequalities was as a test for quantum coherence in macroscopic systems. The last few years has seen a number of experimental tests and violations of these inequalities in a variety of microscopic systems such as superconducting qubits, nuclear spins, and photons. In this article, we provide an introduction to the Leggett–Garg inequalities and review these latest experimental developments. We discuss important topics such as the significance of the non-invasive measurability assumption, the clumsiness loophole, and the role of weak measurements. Also covered are some recent theoretical proposals for the application of Leggett–Garg inequalities in quantum transport, quantum biology and nano-mechanical systems.
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 generalized version of the Leggett-Garg inequality involving different cat states at different times. The violations demonstrate failure of the premise of macrorealism 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 multicomponent cat states observed in the Bose-Einstein condensate and superconducting circuit experiments of Greiner et al. [Nature (London) 419, 51 (2002)] and Kirchmair et al. [Nature (London) 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. Loopholes for testing macrorealism are discussed.
We consider two different types of conditions which were proposed to test macrorealism in the context of a system described by a single dichotomic variable
Q. This is the view that a macroscopic system evolving in time possesses definite properties which can be determined without disturbing the future or past state. The Leggett-Garg (LG) inequalities, the most commonly studied test, are only necessary conditions for macrorealism, but, building on earlier work J. J. Halliwell, Phys. Rev. A 93, 022123 (2016), it is shown that when the four three-time LG inequalities are augmented with a certain set of 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 a very different set of necessary and sufficient conditions for macrorealism, namely the no-signaling in time (NSIT) conditions proposed by J. Kofler and C. Brukner, Phys. Rev. A 87, 052115 (2013) and L. Clemente and J. Kofler, Phys. Rev. A 91, 062103 (2015), which ensure that all probabilities for
Q at one and two times are independent of whether earlier or intermediate measurements are made in a given run, and do not require (but imply) the LG inequalities. We argue that tests based on the LG inequalities have the form of very weak classicality conditions and can be satisfied in the face of moderate interference effects, but those based on NSIT conditions have the form of much stronger coherence witness conditions, satisfied only for zero interference. The two tests differ in their implementation of noninvasive measurability and so are testing different notions of macrorealism: the augmented LG tests are indirect, entailing a combination of the results of different experiments with only compatible quantities measured in each experimental run, in close analogy with Bell tests, and are primarily tests for macrorealism per se; in contrast, the NSIT tests entail sequential measurements of incompatible quantities and are primarily tests for noninvasiveness.
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.
Abstract 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.