Argument: Quantum Theory $=$ Classical Probability Theory with $2$-norm and continuity axiom (complex numbers).

1.

Why $2$-norm and why not $p$-norm?

Because if there are any linear transformations other than these trivial ones that preserve the $p$-norm, then either $p = 1$ or $p=2$.

If $p=1$ we get classical probability theory, while if $p =2$ we get quantum mechanics.

Also all transformations must be norm preserving. So what kind of matrix or linear transformation preserves the $2$-norm? Unitary Matrices $(UU^\dagger = I)$.

2.

Why complex numbers and not real numbers?

Axiom of continuity: There exists a continuous re-versible transformation on a system between anytwo pure states of that system.

Thus, we need our field associated with vector spaces to be algebraically closed. Hence, we need complex numbers.

If you want every unitary operation to have a square root, then you have to go to the complex numbers.

3.

Why do transformations need to be linear?

If quantum mechanics were nonlinear, then one could build a computer to solve NP-complete problems in polynomial time. (Abrams and Lloyd).