Random ensembles of symplectic and unitary states are indistinguishable

A unitary state $t$-design is an ensemble of pure quantum states whose moments match up to the $t$-th order those of states uniformly sampled from a $d$-dimensional Hilbert space. Typically, unitary state $t$-designs are obtained by evolving some reference pure state with unitaries from an ensemble that forms a design over the unitary group $\mathbb{U}(d)$, as unitary designs induce state designs. However, in this work we study whether Haar random symplectic states -- i.e., states obtained by evolving some reference state with unitaries sampled according to the Haar measure over $\mathbb{SP}(d/2)$ -- form unitary state $t$-designs. Importantly, we recall that random symplectic unitaries fail to be unitary designs for $t>1$, and that, while it is known that symplectic unitaries are universal, this does not imply that their Haar measure leads to a state design. Notably, our main result states that Haar random symplectic states form unitary $t$-designs for all $t$, meaning that their distribution is unconditionally indistinguishable from that of unitary Haar random states, even with tests that use infinite copies of each state. As such, our work showcases the intriguing possibility of creating state $t$-designs using ensembles of unitaries which do not constitute designs over $\mathbb{U}(d)$ themselves, such as ensembles that form $t$-designs over $\mathbb{SP}(d/2)$.