Eur. Phys. J. D (2017) 71: 197
https://doi.org/10.1140/epjd/e2017-80024-y

Topical Review

Introducing the Qplex: a novel arena for quantum theory

¹ Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, Australia
² Physics Department, University of Massachusetts Boston, Boston, MA 02125, USA
³ Max Planck Institute for Quantum Optics, 85748 Garching, Germany
⁴ Department of Physics, University of Massachusetts Boston, Boston, MA 02125, USA
⁵ Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany

^a e-mail: qbism.fuchs@gmail.com

Received: 12 January 2017
Received in final form: 18 April 2017
Published online: 25 July 2017

Abstract

We reconstruct quantum theory starting from the premise that, as Asher Peres remarked, “Unperformed experiments have no results.” The tools of quantum information theory, and in particular the symmetric informationally complete (SIC) measurements, provide a concise expression of how exactly Peres’s dictum holds true. That expression is a constraint on how the probability distributions for outcomes of different, hypothetical and mutually exclusive experiments ought to mesh together, a type of constraint not foreseen in classical thinking. Taking this as our foundational principle, we show how to reconstruct the formalism of quantum theory in finite-dimensional Hilbert spaces. The central variety of mathematical entity in our reconstruction is the qplex, a very particular type of subset of a probability simplex. Along the way, by closely studying the symmetry properties of qplexes, we derive a condition for the existence of a d-dimensional SIC.