About the joint measurability of observables

This report deals with some aspects about the joint measurability of quantum

observables. Since W. Heisenberg reviewed the concepts of momentum and

position, it has been known that Quantum Mechanics furnishes the impossibility

of measuring them together, i.e., it is not possible to measure the one without

disturbing the other. This fact holds for many other sets of observables, and

it is commonly known by the name of complementarity.

Some of the examples taught regularly in introductory Quantum Mechanics courses

also deal with these constraints in the quantum measurement, however they are

rarely presented in its most generality. The mathematical model describing

observables in these courses has severe limitations concerning both theoretical

and experimental aspects. Even at the beginning of Quantum Mechanics there were

doubts whether or not

when  measuring  the position of some quantum particle the outcome or

probability distribution was really that of the model, or otherwise some other

fuzzy version of it. It is looking for a complete Theory of

Measurement that many aspects discussed in this report arose for the first

time. One of these new properties in the Theory of Measurement is the concept

of noise, which we will see plays an important role in questions of

measurement.

In this project we have put our focus on two outcome observables, i.e.,

physical measurable properties that may have as an outcome one given value or

another. We have developed geometrical methods (joint measurability

graphs) to visualize and characterize the joint measurability of any set of

two outcome observables, together with constraints for noise which must be

added to the system in order to make them jointly measurable.