This follows from the fact that $\mathbb{E}[A^\dagger A]=d I$ (with $A^\dagger$ the conjugate transpose of $A$ and $I$ the $d\times d$ identity matrix). Hence $$E[\|C_n\|_F^2]=\operatorname{tr}\mathbb{E}[A_1A_2\cdots A_nA_n^\dagger\cdots A_2^\dagger A_1^\dagger]$$ $$=\operatorname{tr}\mathbb{E}[A_2\cdots A_nA_n^\dagger\cdots A_2^\dagger]\mathbb{E}[ A_1^\dagger A_1]$$ $$=d\operatorname{tr}\mathbb{E}[A_2\cdots A_nA_n^\dagger\cdots A_2^\dagger]$$ $$=d\operatorname{tr}\mathbb{E}[A_3\cdots A_nA_n^\dagger\cdots A_3^\dagger]\mathbb{E}[ A_2^\dagger A_2]$$ $$=d^2\operatorname{tr}\mathbb{E}[A_3\cdots A_nA_n^\dagger\cdots A_3^\dagger]$$ $$=d^{n-1}\operatorname{tr}\mathbb{E}[A_n^\dagger A]=d^n\operatorname{tr} I=d^{n+1}.$$

This follows from the fact that $\mathbb{E}[A^\dagger A]=d I$ (with $A^\dagger$ the conjugate transpose of $A$ and $I$ the $d\times d$ identity matrix). Hence $$\mathbb{E}[\|C_n\|_F^2]=\operatorname{tr}\mathbb{E}[A_1A_2\cdots A_nA_n^\dagger\cdots A_2^\dagger A_1^\dagger]$$ $$=\operatorname{tr}\mathbb{E}[A_2\cdots A_nA_n^\dagger\cdots A_2^\dagger]\mathbb{E}[ A_1^\dagger A_1]$$ $$=d\operatorname{tr}\mathbb{E}[A_2\cdots A_nA_n^\dagger\cdots A_2^\dagger]$$ $$=d\operatorname{tr}\mathbb{E}[A_3\cdots A_nA_n^\dagger\cdots A_3^\dagger]\mathbb{E}[ A_2^\dagger A_2]$$ $$=d^2\operatorname{tr}\mathbb{E}[A_3\cdots A_nA_n^\dagger\cdots A_3^\dagger]$$ $$=d^{n-1}\operatorname{tr}\mathbb{E}[A_n^\dagger A]=d^n\operatorname{tr} I=d^{n+1}.$$

This follows from the fact that $\mathbb{E}[A^\dagger A]=d I$ (with $A^\dagger$ the conjugate transpose of $A$ and $I$ the $d\times d$ identity matrix). Hence $$\operatorname{tr}\mathbb{E}[A_1A_2\cdots A_nA_n^\dagger\cdots A_2^\dagger A_1^\dagger]$$ $$=\operatorname{tr}\mathbb{E}[A_2\cdots A_nA_n^\dagger\cdots A_2^\dagger]\mathbb{E}[ A_1^\dagger A_1]$$ $$=\operatorname{tr}d\mathbb{E}[A_2\cdots A_nA_n^\dagger\cdots A_2^\dagger]$$ $$=\operatorname{tr}d\mathbb{E}[A_3\cdots A_nA_n^\dagger\cdots A_3^\dagger]\mathbb{E}[ A_2^\dagger A_2]$$ $$=\operatorname{tr}d^2\mathbb{E}[A_3\cdots A_nA_n^\dagger\cdots A_3^\dagger]$$ $$=\operatorname{tr}d^{n-1}\mathbb{E}[A_nA_n^\dagger]=\operatorname{tr}d^n I=d^{n+1}.$$

This follows from the fact that $\mathbb{E}[A^\dagger A]=d I$ (with $A^\dagger$ the conjugate transpose of $A$ and $I$ the $d\times d$ identity matrix). Hence $$E[\|C_n\|_F^2]=\operatorname{tr}\mathbb{E}[A_1A_2\cdots A_nA_n^\dagger\cdots A_2^\dagger A_1^\dagger]$$ $$=\operatorname{tr}\mathbb{E}[A_2\cdots A_nA_n^\dagger\cdots A_2^\dagger]\mathbb{E}[ A_1^\dagger A_1]$$ $$=d\operatorname{tr}\mathbb{E}[A_2\cdots A_nA_n^\dagger\cdots A_2^\dagger]$$ $$=d\operatorname{tr}\mathbb{E}[A_3\cdots A_nA_n^\dagger\cdots A_3^\dagger]\mathbb{E}[ A_2^\dagger A_2]$$ $$=d^2\operatorname{tr}\mathbb{E}[A_3\cdots A_nA_n^\dagger\cdots A_3^\dagger]$$ $$=d^{n-1}\operatorname{tr}\mathbb{E}[A_n^\dagger A]=d^n\operatorname{tr} I=d^{n+1}.$$