Macroscopic mechanical correlations using single-photon spatial compass state and operational Wigner distribution

We propose a measurement scheme for observing quantum correlations and entanglement in the spatial properties of two macroscopic mirrors. Two spatial versions of compass states are generated by using a single Gaussian mode of single photons in a simple interferometer. The chessboard pattern of spatial compass states reflects the spatial properties of a mirror composed of N quantum mirrors. The displacement and tilt correlations of the two mirrors are manifested by single photons and projection measurements through a measuring device which measures the propensities of the compass states. The technique can extract mechanical correlations of the two mirrors and lock them into the Einstein-Podolsky-Rosen (EPR) correlation. The criteria for EPR entanglement of these mirrors are then verified by sub-Planck structures in the propensity. We formulate the discrete-like property of the propensity, a demonstration of quantum jumps of EPR correlations in phase space, which, hence, enables discrete phase-space quantum computing and information processing.