Asymptotic infinitesimal freeness with amalgamation for Haar quantum unitary random matrices

We consider the limiting distribution of $U_N{A}_N{U}_N^{\ast }$ and $B_N$ (and more general expressions), where $A_N$ and $B_N$ are $N\times N$ matrices with entries in a unital C${}^{\ast }$-algebra $\mathcal{B}$ which have limiting $\mathcal B$-valueddistributionsas$N \to \infty$,and$U_N$isa$N \times N$ Haar distributed quantum unitary random matrix with entries independent from $\mathcal{B}$. Under a boundedness assumption, we show that $U_N{A}_N{U}_N^{\ast }$ and $B_N$ are asymptotically free with amalgamation over $\mathcal{B}$. Moreover, this also holds in the stronger infinitesimal sense of Belinschi-Shlyakhtenko. We provide an example which demonstrates that this example may fail for classical Haar unitary random matrices when the algebra $\mathcal{B}$ is infinite-dimensional.