# (Proof verification) Kaye Exercise 3.5.5, partial trace in larger system

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 $$\rho$$.

Yes, this is a perfectly reasonable answer. The only thing that I'd change is that I would make $$\{|k\rangle\}$$ the eigenbasis of $$\rho$$. Then, by definition, $$\{|k\rangle\}$$ gives you an orthonormal basis. This automatically takes care of the "we can always change our $$p_k$$" aspect.