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Not sure if you've been following but there have been some really cool results about the amount of negativity in a discrete Wigner distribution (or variants) and efficient simulatability. This talk by @earltcampbell talks about some of these resultshttps://youtu.be/mvPMJ5ciYQ4
Only at the vaguest of levels - I knew this was the kind of thing people had been thinking about. Reminds me slightly of the note @quantum_aram and I wrote on efficient simulatability: https://arxiv.org/pdf/quant-ph/0301108.pdf …
