No Purification Ontology, No Quantum Paradoxes
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No Purification Ontology, No Quantum Paradoxes Giacomo Mauro D’Ariano1,2 Received: 8 October 2020 / Accepted: 31 October 2020 © The Author(s) 2020
Abstract It 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 vonNeumann 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 HA to each system A ; (2) compose two systems by the tensor product rule HAB = HA ⊗ HB ; (3) associate a transformation from system A to B to a quantum operation, i.e. to a completely positive trace-non-increasing map between the traceclass operators of A and 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. Keywords Conservation of information · Information paradox · Intrinsic irreversibility of quantum theory · Optionality of unitarity and purity
* Giacomo Mauro D’Ariano [email protected] 1
Dipartimento di Fisica, dell’Università di Pavia, via Bassi 6, 27100 Pavia, Italy
2
Istituto Lombardo Accademia di Scienze e Lettere, INFN, Gruppo IV, Sezione di Pavia of Giacomo, Pavia, Italy
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Foundations of Physics
1 Introduction We all have become accustomed to a set of rules that we call Quantum Theory (QT), which we believe must hold for the whole physical domain at the fundamental level, hence also in a theory of gravity. I emphasise the naming Quantum ”Theory”— instead of Quantum ”Mechanics”—to strip the rules from their mechanical instantiation in particle and matter physics. Such rules (concerning systems, states, observables, evolutions, and measurements) are now common ground for all physicists. The advent of Quantum Information has further stressed such backbone structure of the theory, with conceptual focus on composition rules of systems and transformations, hence on the underlying graph structure of QT. In quantum gravity the Hawki
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