This is the exercise as stated in Kaye's book, Introduction An Introduction to Quantum Computing:
Show that for any density operator $\rho$ on a system $A$, there exists a pure state $|\psi\rangle$ on some larger system $A \otimes B$ such that $\rho = \text{Tr}_B|\psi\rangle \langle \psi|$ and $\dim(A)\geq \dim(B).$
Here's my solution: Suppose $\rho = \sum_{k=1}^m p_k |k\rangle \langle k|$. Wlog, we can assume that $\{|k\rangle\}$ are linearly independent. Let $B=\text{span}_{k=1,...m}\{|k\rangle\}$ and select $\{|\varphi_i\rangle\}_{i=1}^m$ an o.n.b for $B$. Then $\dim(A)\geq \dim(B)$ and for the state $|\psi\rangle = \sum_k \sqrt{p_k}\,|k\rangle_A|\varphi_k\rangle_B$ we get $\rho = \text{Tr}_B |\psi\rangle \langle \psi|$.
Is this solution correct? I am specifically concerned about the part where I assumed that $\{|k\rangle\}$ are linearly independent. My reasoning is that even if $\rho$ can be written in a way that contains linearly dependent states we can always change our $p_k$ accordingly and get rid of them one by one leaving us with the same $\rho$.
