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Negativity Bounds for Weyl–Heisenberg Quasiprobability Representations

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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 (Phys Rev Lett 117 (12):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 (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.

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Notes

  1. A dual basis is one for which the bases considered together are biorthogonal, \(\text {Tr}F_iQ_j=\delta _{ij}.\)

  2. That is, it is impossible to represent quantum theory in a way which eliminates the appearance of negativity in both \(\mathfrak {p}\) and \(\mathfrak {r}\) in (5) for all quantum states and POVMs. It is possible, in general, to eliminate the negativity appearing in one or the other.

  3. Veitch et al. use the term sum negativity specifically for the sum of the negative elements of the discrete Wigner function for a quantum state [37] whereas we will be considering the equivalent notion with respect to any Q-rep.

  4. We have chosen to omit multiplication by d in our negativity definitions so that the negativity can more immediately be associated with the negative values in a quasiprobability vector. As the dual basis is calculationally easier to work with, a downside of our convention is that factors of 1 / d crop up more frequently.

  5. We won’t explore it further here, but the situation is more interesting. Although the sum negativity is insensitive to the value of the parameter t which defines the inequivalent SIC Q-reps, the quantum states whose quasiprobability representations achieve these sum negativity values do depend on the parameter. It turns out that the sum negativity for \(\{Q^-_j\}\) constructed from the Hesse SIC (\(t=0\) in [40]) is achieved by a complete set of mutually unbiased bases [41], that is, 12 [\({=}d(d+1)\)] vectors which form four orthogonal bases such that any vector from one basis has an equal overlap with any vector from another basis. For all the other SICs in dimension 3, the states which achieve the sum negativity of \(\{Q^-\}\) form a single basis instead. The complete set of mutually unbiased bases also turns out to be the set of states which minimize the Shannon entropy in the Hesse SIC representation [42, 43].

  6. And in dimension 8, it may be interesting to look at general \(\text {WH}\otimes \text {WH}\otimes \text {WH}\) covariant Q-reps.

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Acknowledgements

The authors would like to thank Marcus Appleby, Blake Stacey, and Huangjun Zhu for helpful discussions and suggestions. This research was supported by the Foundational Questions Institute Fund on the Physics of the Observer (Grant FQXi-RFP-1612), a Donor Advised Fund at the Silicon Valley Community Foundation.

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DeBrota, J.B., Fuchs, C.A. Negativity Bounds for Weyl–Heisenberg Quasiprobability Representations. Found Phys 47, 1009–1030 (2017). https://doi.org/10.1007/s10701-017-0098-z

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