Physics of information

The exploration of what physics has to say about randomness and reality invokes information as a central concept.

Quantum physics broadened the notions of randomness and reality in physics suggesting a two-way connection between physics and information:
> Physics is informational and that information is expressed in terms of physical randomness.
But what about predictability: does God play dice?
> Information is physical and that information defines physical reality.
But what about objectivity: is the moon still there if you don’t look at it?

Observables

Observables are (represented by) hermitian operators.
Two observables are incompatible if and only if their commutator is nonzero i.e. incompatible observables do not commute.

A hermitian operator is a linear operator.
The corresponding (orthogonal) eigenvectors and (real) eigenvalues represent possible system states and corresponding possible outcomes for that observable.

Indeterminacy

No simultaneous sharp values can be attributed to a set of two incompatible observables. A simultaneous observation of two observables Ô1 and Ô2 is represented by their product Ô1Ô2 which is not necessarily an observable:

the product of 2 hermitian operators (observables) is an hermitian operator (observable) if and only if the commutator is zero i.e. for incompatible observables no simultaneous sharp values can be determined.

From an epistemological point of view one could speak about quantum uncertainty (Heisenberg), as from an ontological point of view this is called quantum complementarity (Bohr).

Hereby, remember:
- an observable is (represented by) a hermitian operator.
- two incompatible observables have a nonzero commutator.

Superposition

The superposition principle lies at the very heart of quantum theory.

The superposition principle is closely related to the notion of incompatible observables:

an eigenstate of an observable Ô1 can be written as a superposition of eigenstates of another observable Ô2 if and only if Ô1 and Ô2 are incompatible observables i.e. their commutator is nonzero.

The actual system state is an eigenstate belonging to an observable and is therefore an expansion (i.e. superposition) of eigenfunctions of the corresponding incompatible observable.

Subconsequently, the superposition principle is also related to quantum indeterminacy.

In conclusion, the superposition principle, incompatible observables and quantum indeterminacy are closely related notions.

Remember:
- observables are (represented by) hermitian operators. The eigenvectors represent possible system states and the corresponding real eigenvalues are the possible outcomes.