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We consider the limiting distribution of $U_NA_NU_N^*$ and $B_N$ (and moregeneral expressions), where $A_N$ and $B_N$ are $N \times N$ matrices withentries in a unital C$^*$-algebra $\mathcal B$ which have limiting $\mathcalB$-valued distributions as $N \to \infty$, and $U_N$ is a $N \times N$ Haardistributed quantum unitary random matrix with entries independent from$\mathcal B$. Under a boundedness assumption, we show that $U_NA_NU_N^*$ 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 forclassical Haar unitary random matrices when the algebra $\mathcal B$ isinfinite-dimensional.

Asymptotic infinitesimal freeness with amalgamation for Haar quantum unitary random matrices