  In standard quantum mechanics (QM), a state vector $| \psi \rangle$ may
belong to infinitely many different orthogonal bases, as soon as the dimension
$N$ of the Hilbert space is at least three. On the other hand, a complete
physical observable $A$ (with no degeneracy left) is associated with a
$N$-dimensional orthogonal basis of eigenvectors. In an idealized case,
measuring $A$ again and again will give repeatedly the same result, with the
same eigenvalue. Let us call this repeatable result a modality $\mu$, and the
corresponding eigenstate $| \psi \rangle$. A question is then: does $| \psi
\rangle$ give a complete description of $\mu$ ?
  The answer is obviously no, since $| \psi \rangle$ does not specify the full
observable $A$ that allowed us to obtain $\mu$; hence the physical description
given by $| \psi \rangle$ is incomplete, as claimed by Einstein, Podolsky and
Rosen in their famous article in 1935. Here we want to spell out this
provocative statement, and in particular to answer the questions: if $| \psi
\rangle$ is an incomplete description of $\mu$, what does it describe ? is it
possible to obtain a complete description, maybe algebraic ? Our conclusion is
that the incompleteness of standard QM is due to its attempt to describe
systems without contexts, whereas both are always required, even if they can be
separated outside the measurement periods.
