Calculate probability of a state after depolarization

Let’s say I have a particle in the quantum state $|+rangle = frac{1}{sqrt{2}}(|0rangle + |1rangle)$, represented as a density operator (1st matrix) that went through a depolarizing chanel (2nd matrix). Let’s call the depolarized matrix $D_p$.

$$
begin{bmatrix}
.5 & .5
.5 & .5
end{bmatrix}
rightarrow
begin{bmatrix}
.5 & .43
.43 & .5
end{bmatrix}
$$
Now, I have two of these $D_p$, and their resulting product state is:
$$
D_p^{otimes 2} = begin{bmatrix}
.5 & .43
.43 & .5
end{bmatrix}
otimes
begin{bmatrix}
.5 & .43
.43 & .5
end{bmatrix}.
$$
Now, if I want want to calculate the probability of finding some state $|psirangle = |0rangle otimes |0rangle$ in the above mentioned product system, then this is what I do:
$$
p(|psirangle | D_p^{otimes 2}) = trace(psiranglelangle psi | D_p^{otimes 2}).
$$
As you can see, calculating this trace is a $O(N^3)$ complexity operation and becomes very slow for even a small number of particles, i.e. for $D_p^{otimes 10}$ or higher. Is there a principled way to calculate these probabilities? Without using any matrix multiplication?

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Cross-posted on Physics.SE