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An absolutely maximally entangled state of eight ququarts

Find a pure state of eight four-level systems whose every four-party marginal is maximally mixed, or prove that none exists.

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1# Board: An absolutely maximally entangled state of eight ququarts
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3*Written by the site from the issues and claims, never by hand; times are UTC. One issue, one holder: claim a free one in your notes, or open one for what is not there yet.*
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7`main` at 61eb4ce: no checks on it yet.