Negativity Bounds for Weyl–Heisenberg Quasiprobability Representations

Foundations of Physics 47 (8):1009-1030 (2017)
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Abstract

The appearance of negative terms in quasiprobability representations of quantum theory is known to be inevitable, and, due to its equivalence with the onset of contextuality, of central interest in quantum computation and information. Until recently, however, nothing has been known about how much negativity is necessary in a quasiprobability representation. Zhu :120404, 2016) proved that the upper and lower bounds with respect to one type of negativity measure are saturated by quasiprobability representations which are in one-to-one correspondence with the elusive symmetric informationally complete quantum measurements. We define a family of negativity measures which includes Zhu’s as a special case and consider another member of the family which we call “sum negativity.” We prove a sufficient condition for local maxima in sum negativity and find exact global maxima in dimensions 3 and 4. Notably, we find that Zhu’s result on the SICs does not generally extend to sum negativity, although the analogous result does hold in dimension 4. Finally, the Hoggar lines in dimension 8 make an appearance in a conjecture on sum negativity.

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Christopher A. Fuchs
University of Massachusetts, Boston

References found in this work

Negative probability.Richard P. Feynman - 1987 - In Basil J. Hiley & D. Peat (eds.), Quantum Implications: Essays in Honour of David Bohm. Methuen. pp. 235--248.
Towards Better Understanding QBism.Andrei Khrennikov - 2018 - Foundations of Science 23 (1):181-195.
The Number Behind the Simplest SIC–POVM.Ingemar Bengtsson - 2017 - Foundations of Physics 47 (8):1031-1041.
SICs and Algebraic Number Theory.Marcus Appleby, Steven Flammia, Gary McConnell & Jon Yard - 2017 - Foundations of Physics 47 (8):1042-1059.

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