The Leggett-Garg inequality (LGI), based on the notions of realism and noninvasive measurability, is applied in the context of a linear harmonic oscillator. It is found that merely through observing at various instants which region of the potential well, the oscillating quantum object is in, the LGI can be violated without taking recourse to any ancillary quantum system. Strikingly, this violation reveals an unexplored nonclassicality of the state which is considered the most "classical-like" of all quantum states, namely the Schrödinger coherent state. In the macrolimit, the extent to which such nonclassicality persists for large values of mass and classical amplitudes of oscillation is quantitatively investigated. It is found that while 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, as the mass is sufficiently increased, actual observability of this violation becomes increasingly difficult. A feasible experimental setup for testing the predicted quantum mechanical violation of LGI is suggested using a trapped nano-object of $\sim 10^6-10^9$ amu mass.
The Leggett-Garg inequality (LGI), based on the notions of realism and noninvasive measurability, is applied in the context of a linear harmonic oscillator. It is found that merely through observing at various instants which region of the potential well, the oscillating quantum object is in, the LGI can be violated without taking recourse to any ancillary quantum system. Strikingly, this violation reveals an unexplored nonclassicality of the state which is considered the most "classical-like" of all quantum states, namely the Schrödinger coherent state. In the macrolimit, the extent to which such nonclassicality persists for large values of mass and classical amplitudes of oscillation is quantitatively investigated. It is found that while 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, as the mass is sufficiently increased, actual observability of this violation becomes increasingly difficult. A feasible experimental setup for testing the predicted quantum mechanical violation of LGI is suggested using a trapped nano-object of ∼106−109 amu mass.
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.
We show how a test of macroscopic realism based on Leggett-Garg inequalities (LGIs) can be performed in a macroscopic system. Using a continuous-variable approach, we consider quantum nondemolition (QND) measurements applied to atomic ensembles undergoing magnetically driven coherent oscillation. We identify measurement schemes requiring only Gaussian states as inputs and giving a significant LGI violation with realistic experimental parameters and imperfections. The predicted violation is shown to be due to true quantum effects rather than to a classical invasivity of the measurement. Using QND measurements to tighten the "clumsiness loophole" forces the stubborn macrorealist to recreate quantum backaction in his or her account of measurement.
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.
The descriptions of the quantum realm and the macroscopic classical world di�er signi
�cantly not only in their mathematical formulations, but also in their foundational concepts
and philosophical consequences. The assumptions of local realism, non-contextuality
and macroscopic realism (macro-realism) along with the quantum mechanical violations
of these assumptions are discussed.
A complete generalization is provided of Wigner's argument inferring quantum nonlocality,
originally restricted for the EPR-Bohm entangled state, based on assuming joint
probability distributions for all the relevant dichotomic observables. Such Generalized
Wigner Inequality (GWI) turns out to be the N-partite local realist inequality which is
violated by quantum mechanics for all pure entangled states for any N.
We then exploit the non-trivial manifestation of quantum non-locality via local contextuality
on three qubit system. Here, we propose an experimental scheme to realize this
feature by deriving a new local realist inequality. The way quantum entanglement can
be used for global contextual value assigning is discussed in a quantum pseudo-telepathy
game.
In the second part of this report, we formulate Leggett-Garg inequality (LGI) in terms
of joint probability and demonstrate its quantum mechanical violation for two state oscillation,
which is initial state dependent, unlike the case of standard LGI. It has been
shown that, the newly formulated LGI reveals robustness against unsharp measurement
for testing compatibility between macro-realism and quantum mechanics. Finally we
show that, the recently proposed No-signaling in time condition, contradicts with quantum
mechanical predictions for any degree of unsharpness in measurement.
Macroscopic Realism (MR) says that a macroscopic system is always determinately in one or other of the macroscopically distinguishable states available to it. The Leggett-Garg (LG) inequality was derived to allow experimental test of whether or not this doctrine is true; it is also often thought of as a temporal version of a Bell-inequality. Despite recent interest in the inequality, controversy remains regarding what would be shown by its violation. Here we resolve this controversy, which arises due to an insufficiently general and model-independent approach to the question so far. We argue that LG's initial characterisation of MR does not pick out a particularly natural realist position, so we articulate an operationally well-defined and well-motivated position in its place. We show that much weaker conditions than LG's are sufficient to derive the inequality: in the first instance, its violation only demonstrates that certain measurements fail to be non-disturbing at the operational level. We articulate three distinct species of MR-ist position, and argue that it is only the first of these which can be refuted by LG inequality violation. This first position is an attractive one, so ruling it out remains of interest, however. A crucial role is played in LG's argument by the assumption of noninvasive measurability. We show that this notion is ambiguous between the weaker notion of disturbance at the operational level, and the stronger notion of invasiveness at the ontic level of properties of the system. Ontic noninvasiveness would be required to rule out MR per se but this property is not entailed by MR, and its presence cannot be established in a model-independent way. It follows that despite the formal parallels, Bell's and LG's inequalities are not methodologically on a par. We close with some reflections on the implications of our analysis for the pedagogy of quantum superposition.
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.