A Conjugate System for Twisted Araki-Woods von Neumann Algebras of finite dimensional spaces

We compute the conjugate system of twisted Araki-Woods von Neumann algebras ${\mathcal{L}}_T(H)$ for a compatible braided crossing symmetric twist $T$ on a finite dimensional Hilbert space $\mathcal{H}$ with norm $\mathrm{\parallel }T\mathrm{\parallel }<1$. This implies that those algebras have finite non-microstates free Fisher information and therefore are always factors of type ${\text{III}}_{\lambda }$ ($0<\lambda \le 1$) or ${\text{II}}_1$. Moreover, using the nontracial free monotone transport, we show that ${\mathcal{L}}_T(H)$ is isomorphic to the free Araki-Woods algebra ${\mathcal{L}}_0(H)$ when $\mathrm{\parallel }T\mathrm{\parallel }=q$ is small enough.