AbstractIt is almost universally believed that in quantum theory the two following statements hold: (1) all transformations are achieved by a unitary interaction followed by a von-Neumann measurement; (2) all mixed states are marginals of pure entangled states. I name this doctrine the dogma of purification ontology. The source of the dogma is the original von Neumann axiomatisation of the theory, which largely relies on the Schrődinger equation as a postulate, which holds in a nonrelativistic context, and whose operator version holds only in free quantum field theory, but no longer in the interacting theory. In the present paper I prove that both ontologies of unitarity and state-purity are unfalsifiable, even in principle, and therefore axiomatically spurious. I propose instead a minimal four-postulate axiomatisation: (1) associate a Hilbert space $${\mathcal {H}}_\text{A}$$
H
A
to each system$$\text{A}$$
A
; (2) compose two systems by the tensor product rule $${\mathcal {H}}_{\text{A}\text{B}}={\mathcal {H}}_\text{A}\otimes {\mathcal {H}}_\text{B}$$
H
AB
=
H
A
⊗
H
B
; (3) associate a transformation from system $$\text{A}$$
A
to $$\text{B}$$
B
to a quantum operation, i.e. to a completely positive trace-non-increasing map between the trace-class operators of $$\text{A}$$
A
and $$\text{B}$$
B
; (4) (Born rule) evaluate all joint probabilities through that of a special type of quantum operation: the state preparation. I then conclude that quantum paradoxes—such as the Schroedinger-cat’s, and, most relevantly, the information paradox—are originated only by the dogma of purification ontology, and they are no longer paradoxes of the theory in the minimal formulation. For the same reason, most interpretations of the theory (e.g. many-world, relational, Darwinism, transactional, von Neumann–Wigner, time-symmetric,...) interpret the same dogma, not the strict theory stripped of the spurious postulates.
On weighted means and their inequalities
