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Triply Positive Matrices and Quantum Measurements Motivated by QBism

DeBrota, J.B.Fuchs, C.A.Stacey, B.C. QBism Group,
Physics Department,
University of Massachusetts Boston,

Stellenbosch Institute for Advanced Study (STIAS),
Wallenberg Research Center at Stellenbosch University,
South Africa
2018 Physics

We study a class of quantum measurements that furnish probabilistic representations of finite- dimensional quantum theory. The Gram matrices associated with these Minimal Informationally Com- plete quantum measurements (MICs) exhibit a rich structure. They are “positive” matrices in three different senses, and conditions expressed in terms of them have shown that the Symmetric Informa- tionally Complete measurements (SICs) are in some ways optimal among MICs.

Here, we explore MICs more widely than before, comparing and contrasting SICs with other classes of MICs, and using Gram matrices to begin the process of mapping the territory of all MICs. Moreover, the Gram matrices of MICs turn out to be key tools for relating the probabilistic representations of quantum theory furnished by MICs to quasi-probabilistic representations, like Wigner functions, which have proven relevant for quantum computation. Finally, we pose a number of conjectures, leaving them open for future work.

The article was pulished in: arXiv preprint arXiv:1812.08762.

Full article

This work was supported (in part) by the Fetzer Franklin Fund of the John E. Fetzer Memorial Trust.