Quantum Country exercise 9

This is one in a series of posts working through the exercises in the Quantum Country online introduction to quantum computing and related topics. The exercises in the original document are not numbered; I have added my own numbers for convenience in referring to them.

Exercise 9. Show that for any matrix M and vector \vert \psi \rangle the following identity holds: \Vert M \vert \psi \rangle \Vert^2 = \langle \psi \vert M^\dagger M \vert \psi \rangle.

Answer: By definition we have \Vert M \vert \psi \rangle \Vert^2 = \left( M \vert \psi \rangle \right)^\dagger M \vert \psi \rangle. It was shown in the text that \left( M \vert \psi \rangle \right)^\dagger = \vert \psi \rangle^\dagger M^\dagger. By definition we also have \langle \psi \vert = \vert \psi \rangle^\dagger.

We then have

\Vert M \vert \psi \rangle \Vert^2 = \left( M \vert \psi \rangle \right)^\dagger M \vert \psi \rangle = = \vert \psi \rangle^\dagger M^\dagger M \vert \psi \rangle = \langle \psi \vert M^\dagger M \vert \psi \rangle

So we have shown that \Vert M \vert \psi \rangle \Vert^2 = \langle \psi \vert M^\dagger M \vert \psi \rangle for any matrix M and vector \vert \psi \rangle.

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