Negativity Bounds for Weyl-Heisenberg Quasiprobability Representations

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 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 (SICs). 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.

21 pages. v2: journal version, added references

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• Symmetric Informationally Complete Measurements Identify the Irreducible Difference between Classical and Quantum Systems
• Quasiprobability distributions for quantum-optical coherence and beyond
• Discrete Wigner Functions from Informationally Complete Quantum Measurements
• Lüders Channels and the Existence of Symmetric Informationally Complete Measurements
• The Poincaré half-plane for informationally complete POVMs
• Is the SIC Outcome There When Nobody Looks?
• Multiscale Structure of More-than-Binary Variables