  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
new testbed for implementation of quantum technologies combining discrete and
continuous quantum variables.
