Bohm's spin polarization modification of the paradox of Einstein, Podolsky, and Rosen has served as the basis for numerous analyses of the roles of hidden variables and Einstein locality in quantum mechanics. The present note points out how corresponding analyses can be carried out on the original formulation of Einstein, Podolsky, and Rosen (EPR) in terms of observations of position and momentum. Concrete conceptual experiments are proposed to furnish tangible illustrations of EPR tests of reality, and a position-momentum variant of Bell's inequality is formulated. Implications regarding joint measurements of position and momentum, delayed choice experiments, and superluminal communication are discussed briefly.
We propose delayed-choice experiments carried out with macroscopic qubits, realized as macroscopically distinct coherent states $|\ensuremath{\alpha}\ensuremath{\rangle}$ and $|\ensuremath{-}\ensuremath{\alpha}\ensuremath{\rangle}$. Quantum superpositions of $|\ensuremath{\alpha}\ensuremath{\rangle}$ and $|\ensuremath{-}\ensuremath{\alpha}\ensuremath{\rangle}$ are created via a unitary interaction $U(\ensuremath{\theta})$ based on a nonlinear Hamiltonian, in analogy with polarizing beam splitters used in photonic experiments. Macroscopic delayed-choice experiments give a compelling reason to develop interpretations not allowing macroscopic retrocausality: This would otherwise suggest changes to the macroscopic qubit value based on a future measurement setting $\ensuremath{\phi}$. We therefore consider weak macroscopic realism (wMR), which for the system at time
t specifies a hidden variable ${\ensuremath{\lambda}}_{\ensuremath{\theta}}$ to determine the macroscopic qubit value, independently of $\ensuremath{\phi}$. Using entangled cat states, we demonstrate a quantum eraser where the choice to measure a which-way or wave-type property is delayed. Consistency with wMR is possible, if we interpret the macroscopic qubit value to be determined by ${\ensuremath{\lambda}}_{\ensuremath{\theta}}$ without specification of the state at the level of $\ensuremath{\hbar}$. We then demonstrate violations of a delayed-choice Leggett-Garg inequality, and of the Wheeler-Chaves-Lemos-Pienaar dimension witness inequality, for the macroscopic qubits. This negates all two-dimensional nonretrocausal wMR models. However, one can interpret consistently with wMR, thus avoiding conclusions of macroscopic retrocausality, by noting extra dimensions, and that violations require further unitary dynamics
U after
t. The violations are then explained as failure of deterministic macroscopic realism, which specifies the validity of ${\ensuremath{\lambda}}_{\ensuremath{\theta}}$ prior to the dynamics $U(\ensuremath{\theta})$ determining the measurement setting $\ensuremath{\theta}$. Finally, although there is consistency with wMR for macroscopic observations, Einstein-Podolsky-Rosen--type paradoxes pointing to the incompleteness of quantum mechanics exist at a microscopic level, based on fringe distributions.