Toolbox for reconstructing quantum theory from rules on information acquisition

Philipp Andres Höhn

Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5

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Updated version: The authors have uploaded version v9 of this work to the arXiv which may contain updates or corrections not contained in the published version v8. The authors left the following comment on the arXiv:
78 pages, many figures, graphs and references. Version accepted for publication in Quantum (completed missing part in the proof of reversibility of time evolution, combined previous sections 6 and 7 to a rewritten section 6, added clarifications and minor corrections throughout -- overall improved presentation, but results unaffected by revision)


We develop an operational approach for reconstructing the quantum theory of qubit systems from elementary rules on information acquisition. The focus lies on an observer $O$ interrogating a system $S$ with binary questions and $S$'s state is taken as $O$'s `catalogue of knowledge' about $S$. The mathematical tools of the framework are simple and we attempt to highlight all underlying assumptions. Four rules are imposed, asserting (1) a limit on the amount of information available to $O$; (2) the mere existence of complementary information; (3) $O$'s total amount of information to be preserved in-between interrogations; and, (4) $O$'s `catalogue of knowledge' to change continuously in time in-between interrogations and every consistent such evolution to be possible. This approach permits a {\it constructive} derivation of quantum theory, elucidating how the ensuing independence, complementarity and compatibility structure of $O$'s questions matches that of projective measurements in quantum theory, how entanglement and monogamy of entanglement, non-locality and, more generally, how the correlation structure of arbitrarily many qubits and rebits arises. The rules yield a reversible time evolution and a quadratic measure, quantifying $O$'s information about $S$. Finally, it is shown that the four rules admit two solutions for the simplest case of a single elementary system: the Bloch ball and disc as state spaces for a qubit and rebit, respectively, together with their symmetries as time evolution groups. The reconstruction for arbitrarily many qubits is completed in a companion paper [P. A. Höhn and C. S. P. Wever, Phys. Rev. A 95 (2017) 012102] where an additional rule eliminates the rebit case. This approach is inspired by (but does not rely on) the relational interpretation and yields a novel formulation of quantum theory in terms of questions.

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