Simultaneous Measurement of Complementary Observables with Compressive Sensing

TL;DR

This paper demonstrates a method to simultaneously measure complementary quantum observables, position and momentum, using compressive sensing techniques to efficiently recover distributions without violating uncertainty principles.

Contribution

It introduces a novel measurement scheme combining random partial position projections with strong momentum measurements, enabling efficient simultaneous distribution recovery.

Findings

Successfully measured position and momentum distributions with fewer measurements

Used compressive sensing to reconstruct position data from partial projections

Maintained compliance with quantum uncertainty principles

Abstract

The more information a measurement provides about a quantum system's position statistics, the less information a subsequent measurement can provide about the system's momentum statistics. This information trade-off is embodied in the entropic formulation of the uncertainty principle. Traditionally, uncertainty relations correspond to resolution limits; increasing a detector's position sensitivity decreases its momentum sensitivity and vice-versa. However, this is not required in general; for example, position information can instead be extracted at the cost of noise in momentum. Using random, partial projections in position followed by strong measurements in momentum, we efficiently determine the transverse-position and transverse-momentum distributions of an unknown optical field with a single set of measurements. The momentum distribution is directly imaged, while the position distribution is recovered using compressive sensing. At no point do we violate uncertainty relations; rather, we economize the use of information we obtain.