Toolbox for reconstructing quantum theory from rules on information  acquisition

Philipp Andres Höhn

doi:10.22331/q-2017-12-14-38

Toolbox for reconstructing quantum theory from rules on information acquisition

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

Published:	2017-12-14, volume 1, page 38
Eprint:	arXiv:1412.8323v8
Doi:	https://doi.org/10.22331/q-2017-12-14-38
Citation:	Quantum 1, 38 (2017).

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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)

Abstract

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.

Featured image: Lattice representation of the reconstructed compatibility, complementarity and correlation structure of an informationally complete binary question set for two qubits. Every vertex corresponds to one of the 15 pairwise independent questions. If two questions are connected by an edge they are maximally compatible, otherwise maximally complementary. Every triangle represents a context. Red/green triangles denote odd/even correlation; i.e., a question in a red/green triangle is equal to the XOR/XNOR connective of the other two questions in the triangle.

► BibTeX data

@article{Hohn2017toolbox,
  doi = {10.22331/q-2017-12-14-38},
  url = {https://doi.org/10.22331/q-2017-12-14-38},
  title = {Toolbox for reconstructing quantum theory from rules on information  acquisition},
  author = {H{\"{o}}hn, Philipp Andres},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {1},
  pages = {38},
  month = dec,
  year = {2017}
}

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