A composite system AB, its initial state is a product state of A and B given by $(I_A\otimes \rho_B)$ (A in a completely mixed state). The composite system undergoes a unitary operation $U_{AB}$. My calculation seems to show that we have

$Tr_{B}(U_{AB}(I_A\otimes \rho_B)U_{AB}^{+})=I_A$

since if the dimensions of A and B are m and n respectively, then we have:

$I_A\otimes \rho_B = \left[ {\begin{array}{cccc} \rho_B & & & \\ & \rho_B & & \\ & & ... & \\ & & & \rho_B \end{array} } \right]$

$U_{AB} = \left[ {\begin{array}{ccccc} u_{11} & u_{12} & . & . & u_{1m} \\ u_{21} & u_{22} & . & . & u_{2m} \\ . & . & . & . & . \\ u_{m1} & u_{m2} & . & . & u_{mm} \\ \end{array} } \right]$

where $u_{ij}$ is a $n \times n$ matrix

Then $O_{AB}=U_{AB}(I_A\otimes \rho_B)U_{AB}^{+}$ is given by

$O_{AB} = \left[ {\begin{array}{ccccc} o_{11} & o_{12} & . & . & o_{1m} \\ o_{21} & o_{22} & . & . & o_{2m} \\ . & . & . & . & . \\ o_{m1} & o_{m2} & . & . & o_{mm} \\ \end{array} } \right]$

where $o_{ij}=\sum_{k}u_{ik}\rho_B u_{jk}^{+}$

And the $(i,j)$ item of $Tr_{B}(U_{AB}(I_A\otimes \rho_B)U_{AB}^{+})$ is given by $Tr(o_{ij})$.

And due to the fact that $U_{AB}$ is unitary, we can easily get

$Tr_{B}(U_{AB}(I_A\otimes \rho_B)U_{AB}^{+})=I_A$

This means that starting from a completely mixed state $I_{A}$, subsystem A will stay in a completely mixed state no matter how it's jointly evolved with another system B if A and B are in an initial product state.

Is my calculation correct? Or I made a mistake somewhere?