Prove that the partial trace preserves density operators

Question

Let $H_A$ and $H_B$ be two finite dimensional Hilbert spaces and $rho_{AB}$ a density operator acting on $H_Aotimes H_B$. I am to show that $rho_{A} = operatorname{tr}_Brho_{AB}$ is also a density operator.
I've been able to show $operatorname{tr}rho_A = 1$, but am struggling to show $rho_A$ is a positive semi-definite operator. I feel like it should be simple, but just don't know where to take it. Here's an attempt:
The positive semi-definite-ness of $rho_{AB}$ shows us that
$$sum_{ijkell}overline{psi}_{ij}p_{ijkell}psi_{kell} geq 0$$
for all $|psirangle = sum_{ij}psi_{ij}|a_irangleotimes|b_jrangle$ and $p_{ijkell}$ are the matrix elements in the tensor product basis. That said I have no idea how to extrapolate any information from this in order to apply it to the elements of $rho_A$.

Jakob · Answer

Let $mathscr{H}equiv mathscr{H}_{mathrm{A}} otimes  mathscr{H}_mathrm{B}$. Then consider a density operator $rho$  on $mathscr{H}$ in its spectral decomposition:
$$rho = sumlimits_k lambda_k , |lambda_krangle langle lambda_k| quad , $$
with $ langle lambda_k|lambda_qrangle = delta_{kq}$, $lambda_k geq 0$ and $sumlimits_k lambda_k = 1$. We calculate the reduced density matrix of subsystem $mathrm{A}$ as
$$ rho_{mathrm{A}} equiv mathrm{Tr}_{mathrm{B}}(rho) = sumlimits_k lambda_k ,  mathrm{Tr}_{mathrm{B}}(|lambda_krangle langle lambda_k|) quad , $$
where the second equality follows from the linearity of the partial trace. We then note that we can express each rank-one projection $ |lambda_krangle langle lambda_k|$ in terms of the Schmidt decomposition of $|lambda_krangle in mathscr{H}$. By doing so, we find that
$$mathrm{Tr}_{mathrm{B}}(|lambda_krangle langle lambda_k|) = sumlimits_j |alpha^k_j|^2, |a^k_jrangle langle a^k_j| quad . $$
Here, ${ |a^k_jrangle }_j$ denotes a complete and orthonormal basis of $mathscr{H}_{mathrm{A}}$. It is easy to see that $mathrm{Tr}_{mathrm{B}}(|lambda_krangle langle lambda_k|)$ is positive semi-definite for all $k$ and since $lambda_k geq 0$  we find that $rho_{mathrm{A}} geq 0$. Additionally, the normalization of $rho_{mathrm{A}}$ follows from the normalization of the Schmidt coefficients.

Prove that the partial trace preserves density operators

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