Marcus Appleby
University of Sydney
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Featured researches published by Marcus Appleby.
Foundations of Physics | 2017
Marcus Appleby; Gary McConnell; Jon Yard
We give an overview of some remarkable connections between symmetric informationally complete measurements (SIC-POVMs, or SICs) and algebraic number theory, in particular, a connection with Hilbert’s 12th problem. The paper is meant to be intelligible to a physicist who has no prior knowledge of either Galois theory or algebraic number theory.
European Physical Journal D | 2017
Marcus Appleby; Christopher A. Fuchs; Blake C. Stacey; Huangjun Zhu
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. Graphical abstract
Journal of Mathematical Physics | 2017
Marcus Appleby; Ingemar Bengtsson; Irina Dumitru
Algebraic number theory relates SIC-POVMs in dimension d > 3 to those in dimension d(d − 2). We define a SIC in dimension d(d − 2) to be aligned to a SIC in dimension d if and only if the squares of the overlap phases in dimension d appear as a subset of the overlap phases in dimension d(d − 2) in a specified way. We give 19 (mostly numerical) examples of aligned SICs. We conjecture that given any SIC in dimension d, there exists an aligned SIC in dimension d(d − 2). In all our examples, the aligned SIC has lower dimensional equiangular tight frames embedded in it. If d is odd so that a natural tensor product structure exists, we prove that the individual vectors in the aligned SIC have a very special entanglement structure, and the existence of the embedded tight frames follows as a theorem. If d − 2 is an odd prime number, we prove that a complete set of mutually unbiased bases can be obtained by reducing an aligned SIC to this dimension.
arXiv: Number Theory | 2016
Marcus Appleby; Gary McConnell; Jon Yard
Journal of Physics A | 2018
Marcus Appleby; Tuan-Yow Chien; Shayne Waldron
Bulletin of the American Physical Society | 2013
Matthew Graydon; Marcus Appleby
Bulletin of the American Physical Society | 2013
Hoan Bui Dang; Kate Blanchfield; Ingemar Bengtsson; Marcus Appleby
Bulletin of the American Physical Society | 2012
Hoan Dang; Marcus Appleby; Christopher A. Fuchs
Bulletin of the American Physical Society | 2012
Marcus Appleby; Hulya Yadsan-Appleby; Gerhard Zauner
Bulletin of the American Physical Society | 2011
Hoan Bui Dang; Marcus Appleby; Ingemar Bengtsson; Kate Blanchfield; Åsa Ericsson; Christopher A. Fuchs; Matthew Graydon; Gelo Noel Tabia