I am working through R.de Wolf lecture notes on Quantum Computing and have difficulties figuring out a certain property of matrix norms.
Consider the operator norm (induced norm on matrices) : $\|A\| = \max_{\|v\| = 1} \|Av\| $ .
Define the distance between two matrices A and B as : $\|A-B\|$.
- Given a matrix (phase gate) : $$P_\phi=\begin{pmatrix}
1 & 0 \\
0 & e^{i\phi}
\end{pmatrix}.
$$
What is the distance between the 2 by 2 identity matrix and the phase gate?
With a simple calculation I ended up with :
\begin{equation}
\begin{pmatrix}
0 & 0 \\
0 & 1-e^{i\phi}
\end{pmatrix} \begin{pmatrix} v_1 & v_2 \end{pmatrix}.
\end{equation}
Deriving a final condition such that the distance is $\max \sqrt{|v_2 – v_2.e^{i\phi}|^2}.$
But I am not sure how to go from here (I assumed values for $\phi$ such as $\phi = \pi$)
But I am not sure how to "cover all the cases" and how to explicitly write the distance given the unknown $v_2$ term.
Best Answer
You can avoid maximization over the unit sphere by using the fact that the operator norm is equal to the largest singular value. We can write the singular value decomposition of the difference of the two matrices as
$$ I - P_\phi = \begin{bmatrix}0 & 0\\ 0 & 1-e^{i\phi}\end{bmatrix} = \begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}\begin{bmatrix}0 & 0\\ 0 & r\end{bmatrix}\begin{bmatrix}1 & 0\\ 0 & e^{i\theta}\end{bmatrix} $$
where $1-e^{i\phi}=re^{i\theta}$ and we have chosen to write singular values in the increasing$^1$ order along the diagonal of the middle matrix. Therefore,
$$ \|I-P_\phi\| = r = |1-e^{i\phi}|=\sqrt{\left(1-e^{-i\phi}\right)\left(1-e^{i\phi}\right)}=\sqrt{2-2\cos\phi}. $$
$^1$ Admittedly, this goes against the prevailing convention of writing singular values in decreasing order, but the unusual convention simplifies the formula by making all three matrices diagonal.