Uniqueness of $\rho = AA^*$.

Question

If $\rho$ is self-adjoint and positive semi-definite so that $\rho =AA^*$, then is $A$ unique up to some transform? We may assume finite-dim Hilbert space if necessary.

Answer 1

I will mainly talk about the finite dimensional case.

So the question is asking when does $AA^* = BB^*$.

If $\rho$ is invertible (i.e. positive definite), then of course this is equivalent to $A^{-1}B$ being unitary, so $A$ and $B$ differ by a unitary transform.

If $\rho = 0$, then we have $A^*A = 0$ which implies $A = 0$ (because for any vector $x$, we have $\langle Ax, Ax\rangle = x^*A^*Ax = 0$, hence $Ax = 0$).

For general $\rho$, we may decompose it into the positive definite part and the zero part. This completely describes all the answers to $AA^* = \rho$.