22aWA-5 Temporal and Spatial Bell's Inequalities

Tommaso Calarco

文明 森越

Peter Millington, Zong-Gang Mou, Paul M. Saffin +1 more

Abstract Inspired by Lefschetz thimble theory, we treat Quantum Field Theory as a statistical theory with a complex Probability Distribution Function (PDF). Such complex-valued PDFs permit the violation of Bell-type inequalities, which cannot be violated by a real-valued, non-negative PDF. In this paper, we consider the Classical-Statistical approximation in the context of Bell-type inequalities, viz. the familiar (spatial) Bell inequalities and the temporal Leggett-Garg inequalities. We show that the Classical-Statistical approximation does not violate temporal Bell-type inequalities, even though it is in some sense exact for a free theory, whereas the full quantum theory does. We explain the origin of this discrepancy, and point out the key difference between the spatial and temporal Bell-type inequalities. We comment on the import of this work for applications of the Classical-Statistical approximation.

Susana F. Huelga, Trevor W. Marshall, Emilio Santos

Aurindam Mondal

We investigate the role of the bipartite temporal Bell inequality, an analogue of the spatial Bell inequality, in probing the quantum imprints of primordial perturbations when the initially chosen Bunch-Davies vacuum is replaced by a coherent state. Although it is based on the same principles of locality and realism, its primary advantage lies in the fact that it does not require two distinct set of observable for its construction. Instead, measurements performed on a single component of the pseudo-spin operator at different times are sufficient. Consequently, it is particularly well suited for cosmological scenarios, where observational constraints typically allow access to only one component of the pseudo-spin operator. Assuming a coherent state as the initial condition, we derive an analytical expression for the expectation value of the bipartite temporal Bell operator and demonstrate the absence of temporal Bell violation in such a scenario. Interestingly, the results for squeezed coherent state is found to differ - albeit slightly - from those of squeezed vacuum state for large values of the squeezing parameter. This suggests that the ability to distinguish among different initial states of primordial perturbations does not rely on the violation of temporal Bell inequality. Furthermore, the dependence of the temporal Bell inequality on a purely imaginary phase factor of the wave function appears to be an unique feature, which is entirely absent in the context of spatial Bell inequalities.

Luigi Schiavulli