More on doubled Hilbert space in double-scaled SYK

We continue our study of the doubled Hilbert space formalism of double-scaled SYK model initiated in [arXiv:2401.07403]. We show that the 1-particle Hilbert space introduced by Lin and Stanford in [arXiv:2307.15725] is related to the doubled Hilbert space $\mathcal{H}_0\otimes \mathcal{H}_0$ of the 0-particle Hilbert space $\mathcal{H}_0$ by some linear isomorphism $\mathcal{C}$. It turns out that the entangler and disentangler appeared in our previous study are given by the dual (or adjoint) of $\mathcal{C}^{-1}$ and $\mathcal{C}$.