Question about orthonormal decompositions over unitary operators

Question

I'm teaching myself quantum information theory using Nielson and Chuang's "Quantum Computation and Quantum Information" and I'm at a point in the book where the formalism is starting to make my eyes bleed.

Exercise 2.56 Use the spectral decomposition to show that $K \equiv -i\log(U)$ is Hermitian for an unitary $U$, and thus $U=\exp(iK)$ for some Hermitian $K$.

Using the defition of Hermitian e.g. $[A^\dagger]^\dagger=A$, I plugged the product $-i\log(U)$ and am now stuck. The problem for me is that I've been using these definitions in the context of matrix computations, but now the book is no longer connecting the operators to matrices as explicitly, expecting me to find the connections for myself.

Answer 1

The most important thing that both unitary and Hermitian operators have in common is that they are both normal: that is, that you can diagonalize them, and consider functions such as $\exp$ and $\log$ (and similar matrix-operations) in terms of transforming the eigenvalues while keeping the same set of eigenvectors. This is what the exercise is teaching you to think about.

By way of analogy, for any normal matrix, you can describe $A^2$ both as the matrix obtained by multiplying $A$ by itself, or equally well as the matrix whose eigenvectors are the same as those of $A$, but where the respective eigenvalues have been squared. Similarly for any power $A^k$ of the matrix $A$; and so similarly for any function expressible by Taylor series, and their inverses.

(Note that in quantum information, there is rarely any harm in identifying linear operators on finite-dimensional Hilbert spaces with the matrices, as we usually describe them in terms of their action on the standard basis anyhow; only when dealing with continuous variables is it likely to be important to maintain the distinction quite clearly in mind.)

Answer 2

This one you just have to think about in the right way. Start from the fact that $U$ is unitary, $U^\dagger U = 1$.

That's actually a technique that is often useful when dealing with unitary operators: start from $U^\dagger U = 1$ and see what you can show by performing mathematical operations on that identity.

Answer 3

They gave you a hint: to use the spectral decomposition. Don't just try to use only the definitions of Hermitian or unitary, no wonder you got stuck. So go look up in the book where it tells you what the spectral decomposition of $U$ is, and then write it down and try to see what that tells you about $\log U$, etc. etc.