Angle and angular momentum: Uncertainty relations, simultaneous measurement, and phase-space representation

Reaching ultimate performance of quantum technologies requires the use of detection at quantum limits and access to all resources of the underlying physical system. We establish a full quantum analogy between the pair of angular momentum and exponential angular variable, and the structure of canonically conjugate position and momentum. This includes the notion of optimal simultaneous measurement of the angular momentum and angular variable, the identification of Einstein-Podolsky-Rosen-like variables and states, and, finally, a phase -space representation of quantum states. Our construction is based on close interconnection of the three concepts and may serve as a template for the treatment of other observables. This theory also provides a test bed for implementation of quantum technologies combining discrete and continuous quantum variables.

Automated summary

This study establishes a full quantum analogy between angular momentum and exponential angular variable, as well as the structure of canonically conjugate position and momentum. It introduces the concept of optimal simultaneous measurement, Einstein-Podolsky-Rosen-like variables and states, and a phase-space representation of quantum states. The research is significant for the implementation of quantum technologies combining discrete and continuous quantum variables.