Supersinglets can be self-tested with perfect quantum strategies

Supersinglets are states of spin-zero of d ge 3 particles of d levels. They are invariant under unitary transformations of the form U^{otimes d} and have applications in metrology, error protection, and communication. They also violate some specific Bell inequalities. However, none any of these applications {\em require} supersinglets nor do any of these Bell inequality violations capture the unique properties of the supersinglets. This leads to two questions. Question 1 is whether there exists a task that can be solved only with supersinglets. Question 2 is whether supersinglets can produce a unique d-partite, d-dimensional nonlocal signature. We answer both questions affirmatively by presenting a protocol that self-test all supersinglets by producing d-partite, d-dimensional {\em perfect} quantum strategies for any d ge 3.