# Negativity Bounds for Weyl-Heisenberg Quasiprobability Representations

## TL;DR

This paper investigates the necessary amount of negativity in quasiprobability representations of quantum theory, introducing new negativity measures, analyzing their bounds, and exploring their relation to SICs and Hoggar lines across different dimensions.

## Contribution

It defines a family of negativity measures, establishes bounds and maxima for sum negativity, and examines their relation to SICs and Hoggar lines in various dimensions.

## Key findings

- Exact global maxima of sum negativity in dimensions 3 and 4.
- Zhu's bounds do not generally extend to sum negativity.
- Hoggar lines relate to sum negativity conjectures in dimension 8.

## 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.