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The varieties of minimal tomographically complete measurements

QBism Group, Physics Department, University of Massachusetts Boston, 100 Morrissey Boulevard, Boston MA 02125, USA

Stellenbosch Institute for Advanced Study (STIAS), Wallenberg Research Center at Stellenbosch University, Marais Street, Stellenbosch 7600, South Africa

E-mail Address: jdebrota@gmail.com

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Christopher A. Fuchs

QBism Group, Physics Department, University of Massachusetts Boston, 100 Morrissey Boulevard, Boston MA 02125, USA

Stellenbosch Institute for Advanced Study (STIAS), Wallenberg Research Center at Stellenbosch University, Marais Street, Stellenbosch 7600, South Africa

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Blake C. Stacey

QBism Group, Physics Department, University of Massachusetts Boston, 100 Morrissey Boulevard, Boston MA 02125, USA

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https://doi.org/10.1142/S0219749920400055Cited by:9 (Source: Crossref)

This article is part of the issue:

Special Issue on Quantum Tomography — Part I
Guest Editors: Jiangwei Shang and Yong Siah Teo

Abstract

Minimal Informationally Complete quantum measurements, or MICs, illuminate the structure of quantum theory and how it departs from the classical. Central to this capacity is their role as tomographically complete measurements with the fewest possible number of outcomes for a given finite dimension. Despite their advantages, little is known about them. We establish general properties of MICs, explore constructions of several classes of them, and make some developments to the theory of MIC Gram matrices. These Gram matrices turn out to be a rich subject of inquiry, relating linear algebra, number theory and probability. Among our results are some equivalent conditions for unbiased MICs, a characterization of rank-1 MICs through the Hadamard product, several ways in which immediate properties of MICs capture the abandonment of classical phase space intuitions, and a numerical study of MIC Gram matrix spectra. We also present, to our knowledge, the first example of an unbiased rank-1 MIC which is not group covariant. This work provides further context to the discovery that the symmetric informationally complete quantum measurements (SICs) are in many ways optimal among MICs. In a deep sense, the ideal measurements of quantum physics are not orthogonal bases.

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