The connection between coarse-graining of measurement and emergence of classicality has been investigated
for some time, if not well understood. Recently in (PRL 112, 010402, (2014)) it was pointed out that coarsegraining
measurements can lead to non-violation of Bell-type inequalities by a state which would violate
it under sharp measurements. We study here the effects of coarse-grained measurements on bipartite cat
states. We show that while it is true that coarse-graining does indeed lead to non-violation of a Bell-type
inequality, this is not re
ected at the state level. Under such measurements the post-measurement states can
be non-classical (in the quantum optical sense) and in certain cases coarse-graning can lead to an increase
in this non-classicality with respect to the coarse-graining parameter. While there is no universal way to
quantify non-classicality, we do so using well understood notions in quantum optics such as the negativity
of the Wigner function and the singular nature of the Gluaber-Sudharshan P distribution.
The connection between coarse-graining of measurement and emergence of classicality has been investigated
for some time, if not well understood. Recently in (PRL 112, 010402, (2014)) it was pointed out that coarsegraining
measurements can lead to non-violation of Bell-type inequalities by a state which would violate
it under sharp measurements. We study here the effects of coarse-grained measurements on bipartite cat
states. We show that while it is true that coarse-graining does indeed lead to non-violation of a Bell-type
inequality, this is not re
ected at the state level. Under such measurements the post-measurement states can
be non-classical (in the quantum optical sense) and in certain cases coarse-graning can lead to an increase
in this non-classicality with respect to the coarse-graining parameter. While there is no universal way to
quantify non-classicality, we do so using well understood notions in quantum optics such as the negativity
of the Wigner function and the singular nature of the Gluaber-Sudharshan P distribution.
The connection between coarse-graining of measurement and emergence of classicality has been investigated for some time, if not well understood. Recently in (PRL 112, 010402, (2014)) it was pointed out that coarse-graining measurements can lead to non-violation of Bell-type inequalities by a state which would violate it under sharp measurements. We study here the effects of coarse-grained measurements on bipartite cat states. We show that while it is true that coarse-graining does indeed lead to non-violation of a Bell-type inequality, this is not reflected at the state level. Under such measurements the post-measurement states can be non-classical (in the quantum optical sense) and in certain cases coarse-graning can lead to an increase in this non-classicality with respect to the coarse-graining parameter. While there is no universal way to quantify non-classicality, we do so using well understood notions in quantum optics such as the negativity of the Wigner function and the singular nature of the Gluaber-Sudharshan P distribution.
The departure of the quantum world from the classical regime is captured through the observation of nonclassical correlations manifested in the behaviors of subatomic systems. However, once the dimension of the system becomes substantially large, the quantum behavior begins to decline, and subsequently it starts following the predictions of classical physics. The macroscopic limit at which such quantum-to-classical transition occurs remains one of the long-standing questions in the foundations of quantum theory. There are evidences that the macroscopic limit to which the quantumness of a system persists depends on the degree of interaction due to the measurement processes. For instance, with a system having a considerably large Hilbert-space dimension, if the measurement is performed in such a way that the outcome of the measurement only reveals a coarse-grained version of the information about the individual level of the concerned system then the disturbance due to the measurement process can be considered to be infinitesimally small. Based on such coarse-grained measurement the dependence of Bell inequality violation on the degree of coarsening has already been investigated [H. Jeong, Y. Lim, and M. S. Kim, Phys. Rev. Lett. 112, 010402 (2014)]. In this paper, we first capture the fact that, when local realism is taken to be the defining notion of classicality, the effect of the degree of coarsening on the downfall of quantumness of a macroscopic entangled state can be compensated by testing a Bell inequality of a higher number of settings from a family of symmetric Bell inequalities if the number of settings is odd. However, in contrast, we show that such compensation cannot be seen when we witness such quantum-to-classical transition using symmetric Bell inequalities having an even number of settings. Finally, complementing the above result, we show that, when unsteerability is taken as the classicality, for both odd and even numbers of settings the degree of coarsening at which the quantum-to-classical transition occurs can be consistently pushed ahead by testing a linear steering inequality of a higher number of settings and observing its violation. We further extend our treatment for mixed macroscopic entangled states.
It is usually believed that, in the macroscopic limit, coarse-graining of measurements leads to classical behavior and, in particular, that coarse-graining of quantum correlations leads to local realistic correlations, a principle known as macroscopic locality. We show that, in the most general case, the Hilbert space structure of quantum theory can be preserved in the macroscopic limit. This leads to a violation of a Bell inequality for coarse-grained collective measurements, thus breaking the principle of macroscopic locality.
Why do we not experience a violation of macroscopic realism in everyday life. Normally, no violation can be seen either because of decoherence or the restriction of coarse-grained measurements, transforming the time evolution of any quantum state into a classical time evolution of a statistical mixture. We find the sufficient condition for these classical evolutions for spin systems under coarse-grained measurements. However, there exist "nonclassical" Hamiltonians whose time evolution cannot be understood classically, although at every instant of time the quantum state appears as a classical mixture. We suggest that such Hamiltonians are unlikely to be realized in nature because of their high computational complexity.
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 show that failure of local realism can be revealed to observers for whom only extremely-coarse-grained measurements are available. In our instances, Bell's inequality is violated even up to the maximum limit while both the local measurements and the initial local states under scrutiny approach the classical limit. Furthermore, we can observe failure of local realism when an inequality enforced by nonlocal realistic theories is satisfied. This suggests that locality alone may be violated while realism cannot be excluded for specific observables and states. Small-scale experimental demonstration of our examples may be possible in the foreseeable future.