Exercise 2.63

[dandanua]

We actually need to prove the existence of unitaries $U_m$, not just verify the correctness of the formula $M_m = U_m\sqrt{E_m}$.

So, the correct solution is something like this:

Suppose $M_m^\dagger M_m = E_m$.
If $E_m > 0$ then denote $U_m = M_m(\sqrt{E_m})^{-1}$. We can verify that $U_m$ is indeed unitary, because we can deduce that $U_m^\dagger U_m = I$.
If $E_m \ge 0$ then use limiting argument similar to this https://en.wikipedia.org/wiki/Cholesky_decomposition#Proof_for_positive_semi-definite_matrices
I.e. consider $E_m^\prime = E_m + \frac{1}{k}I$ and so on...

[0moonlight]

The answer is a direct conclusion of polar decomposition.
https://en.wikipedia.org/wiki/Polar_decomposition
In short, any square complex matrix $M$ has a decomposition of the form $M=U P$, where $U$ is unitary and $P=\sqrt(M^\dagger M)$ is unique, so the unitary $U$ exists.

[goropikari]

If all measurement operators are square, I agree polar decomposition is sufficient.
But I'm not sure whether all measurement operators are square. I haven't found any information about squareness of measurement operators.
(I also think it is square through the context. But I wanted to prove this without squareness.)

[mikeevmm]

Doesn't the squareness depend on your basis? Unless I'm missing something you can always make the measurement operator have a square matrix representation by padding it with 0s where adequate.

P.S. Thank you for your work :)

[0moonlight]

Doesn't the squareness depend on your basis? Unless I'm missing something you can always make the measurement operator have a square matrix representation by padding it with 0s where adequate.

P.S. Thank you for your work :)

We can always choose an appropriate basis to transform a matrix into square matrix. A simple way is to pick up the support of input space and the range of output space