We investigate the violation of the Leggett-Garg inequalities for a harmonic oscillator in various quantum states and with various choices of a projection operator for a dichotomic variable. We focus on the two-time quasiprobability distribution function with a dichotomic variable constructed with the position or momentum operator of a harmonic oscillator. Our results are the generalization of the previous work by Mawby and Halliwell [Phys. Rev. A 107, 032216 (2023)], but the new points are the following. We first obtain the explicit expression for the two-time quasiprobability distribution function for the (thermal) squeezed coherent state. Second, we find the two-time quasiprobability distribution function with the dichotomic variable and the projection operator constructed in terms of the momentum operator. Third, we demonstrate that the violation of the Leggett-Garg inequalities can be boosted by adopting the dichotomic variable and the projection operator defined by a finite range of the position/momentum, in which a larger violation appears for the ground state and the squeezed state. We give an intuitive interpretation when the violation of the Leggett-Garg inequalities appears. We also present a mathematical formula to compute the quasiprobability distribution function by using the integral representation of the Heaviside function, which is useful to generalize it to a quantum field theory.
We investigate violations of Leggett-Garg inequalities (LGIs) for a harmonic oscillator and a
(1+1)-dimensional chiral scalar field with coherent-state projectors, which is equivalent to a heterodyne-type measurement scheme. For the harmonic oscillator, we found that the vacuum and thermal states violated the LGIs by evaluating the two-time quasiprobability distribution function. In particular, we demonstrate that the value of the two-time quasiprobability reaches $\ensuremath{-}0.123$ for a squeezed coherent-state projector, which is equivalent to
98 of the L\"uders bound corresponding to the maximal violation of the LGIs. We also find a violation of the LGIs for the local mode of a quantum chiral scalar field by constructing a coherent-state projector similar to the harmonic-oscillator case. In contrast with the harmonic oscillator, the periodicity in the time direction of the quasiprobability disappears, which is related to the existence of quantum entanglement between the local mode and its complementary degrees of freedom.
For multilevel spin systems, robustness of the quantum mechanical (QM) violation of macrorealism (MR) with respect to coarse-grained measurements is investigated using three different necessary conditions of MR, namely, the Leggett-Garg inequality (LGI), Wigner's form of the Leggett-Garg inequality (WLGI), and the condition of no-signaling in time (NSIT). It is shown that for dichotomic sharp measurements, in the asymptotic limit of spin, the algebraic maxima of the QM violations of all these three necessary conditions of MR are attained. Importantly, the QM violations of all these persist in that limit even for arbitrary unsharp measurements, i.e., for any nonzero value of the sharpness parameter characterizing the degree of fuzziness of the relevant measurements. We also find that, when different measurement outcomes are clubbed into two groups for the sake of dichotomizing the outcomes, the asymmetry or symmetry in the number of outcomes in the two groups, signifying the degree of coarse graining of measurements, has a crucial role in discerning quantum violation of MR. The results clearly demonstrate that classicality does not emerge in the asymptotic limit of spin, whatever be the unsharpness and degree of coarse graining of the measurements.
We overcome the long-standing noninvasive measurability (NIM) challenge in Leggett-Garg tests by exploiting the causal structure of quantum field theory (QFT). Our protocol uses three independent ensembles of the vacuum state, each measured by a different pair of observers at spacelike-separated events, yielding the three two-time correlators. By placing these events at positions
(0,0),
(τ,L), and
(2τ,2L) with
L>τ+2τ0, we rigorously ensure that no measurement can influence another. We investigate the vacuum state of a free massive scalar field in 1+1 dimensions, employing the dichotomic observable
Q(f)=sign(φ(f)) where
φ(f) is the smeared field. In the Heisenberg picture, the time evolution is absorbed into a translation of the time-window function, allowing us to derive the two-time correlation function
C(τ,L) and the Leggett-Garg parameter
K3=2C(τ,L)−C(2τ,2L). For non-overlapping time windows, we find that the correlation function decays exponentially with
τ for a massive field. For overlapping windows, our numerical computation for a rectangular time window yields
K3<1 across the entire mass range, firmly establishing that the vacuum does not violate the LGI. Thus, under strict noninvasive conditions, the vacuum shows no violation of macrorealism, in stark contrast to its well-known violation of spatial Bell inequalities. Our spacelike-separated protocol provides the first LGI test in QFT with rigorously satisfied NIM, setting a methodological benchmark for future studies and highlighting the fundamental distinction between spacelike entanglement and temporal macrorealism in relativistic quantum fields.
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.
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
t1 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
t1 and
t2, 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.