Suppose that $A_1, \dots , A_n, B_1, \dots , B_n \in \Bbb C^{d \times d}$ and that, for every $x \in \mathbb{C}^n$, the following holds $$\left( \sum_i x_i A_i \right) \left( \sum_i x_i A_i \right)^{\dagger} = \left( \sum_i x_i B_i \right) \left( \sum_i x_i B_i \right)^{\dagger} $$

where $\dagger$ denotes the adjoint (i.e., transpose conjugate) of a matrix. Is it true that there exists a unitary matrix $\Sigma \in {\rm U}(d, \mathbb{C})$ such that $A = \Sigma B$?

Suppose that $A_1, \dots , A_n, B_1, \dots , B_n \in \Bbb C^{d \times d}$ and that, for every $x \in \mathbb{C}^n$, the following holds $$\left( \sum_i x_i A_i \right) \left( \sum_i x_i A_i \right)^{\dagger} = \left( \sum_i x_i B_i \right) \left( \sum_i x_i B_i \right)^{\dagger} $$

where $\dagger$ denotes the adjoint (i.e., transpose conjugate) of a matrix. Is it true that there exists a unitary matrix $\Sigma \in {\rm U}(d, \mathbb{C})$ such that $A_i = \Sigma B_i$ for all $i$?

Suppose that $A_1, ... , A_n$ and $B_1, ... , B_n$ are complex $d \times d$ matrices. Suppose that for every $x \in \mathbb{C}^n$ it holds that: $$\left( \sum_i A_i x_i \right) \left( \sum_i A_i x_i \right)^{\dagger} = \left( \sum_i B_i x_i \right) \left( \sum_i B_i x_i \right)^{\dagger} $$ where $\dagger$ denotes the adjoint (i.e., transpose conjugate) of a matrix.

  Is it true that there exists a unitary matrix $\Sigma \in {\rm U}(d, \mathbb{C})$ such that $A = \Sigma B$?

Thank you in advance for your help.

Suppose that $A_1, \dots , A_n, B_1, \dots , B_n \in \Bbb C^{d \times d}$ and that, for every $x \in \mathbb{C}^n$, the following holds $$\left( \sum_i x_i A_i \right) \left( \sum_i x_i A_i \right)^{\dagger} = \left( \sum_i x_i B_i \right) \left( \sum_i x_i B_i \right)^{\dagger} $$

where $\dagger$ denotes the adjoint (i.e., transpose conjugate) of a matrix. Is it true that there exists a unitary matrix $\Sigma \in {\rm U}(d, \mathbb{C})$ such that $A = \Sigma B$?