The Role of Scalar Product and Wigner Distribution in Optical and Quantum Mechanical Measurements

Antoine Royer

Robert F. O’Connell, A. K. Rajagopal

A new interpretation of the scalar product in Hilbert space was presented recently by Aharonov, Albert, and Au. Here the essential equivalence between such a view-point and Wigner's exact reformulation of nonrelativistic quantum mechanics using distribution functions is pointed out.

Wolfgang P. Schleich, Daniel F. Walls, JOHN ARCHIBALD WHEELER

In the semiclassical approximation, the quantum-mechanical scalar product between two quantum states is governed by (1) the areas of overlap in phase space of these states and (2) interference between the probability amplitudes contributed by these areas. We compare and contrast this principle with Wigner's concept of pseudoprobabilities in phase space and illustrate the essential points of both treatments by the oscillations in the photon distribution of a highly squeezed state.

Martin J. Bastiaans

Hartmut O. Bartelt, Karl‐Heinz Brenner, Adolf W. Lohmann

B. M. Rodríguez-Lara, Elohim Becerra

Abstract Optical scalar beams provide a platform to explore classical analogies to quantum mechanics. We introduce a Lie group-theoretic su ( 1 , 1 ) framework to construct optical beam analogs of continuous-variable (CV) quantum states, including one- and two-mode squeezing. Using Fourier methods and the optical Wigner distribution, we investigate how squeezing reshapes the spatial and spectral structure of these beams, and find that the achievable diffraction-limited spatial compression is bounded by the numerical aperture of the optical system. We construct an optical analog of the Duan–Simon inseparability criterion and use marginal optical Wigner distributions, which encode partial phase-space information, as witnesses of CV classical entanglement. Our framework provides a unified approach to quantum-inspired beam engineering with potential applications in optical imaging, metrology, and communication.