Doubled Hilbert space in double-scaled SYK

We consider matter correlators in the double-scaled SYK (DSSYK) model. It turns out that matter correlators have a simple expression in terms of the doubled Hilbert space $\mathcal{H}\otimes\mathcal{H}$, where $\mathcal{H}$ is the Fock space of $q$-deformed oscillator (also known as the chord Hilbert space). In this formalism, we find that the operator which counts the intersection of chords should be conjugated by certain ``entangler'' and ``disentangler''. We explicitly demonstrate this structure for the two- and four-point functions of matter operators in DSSYK.