I am new to quantum mechanics, and I'm trying to learn about Majorization in quantum states. These are the notes I am following http://michaelnielsen.org/blog/talks/2002/maj/book.ps by Michael Nielsen. I am stuck in an exercise that already has an answer, but I still can't understand it. It is the following:

Exercise 1.3.1: Let $\rho$ be an arbitrary state of a d-dimensional quantum system. Prove that there always exists an orthonormal basis $|e_k >$ such that the probabilities for a measurement in that basis are uniformly distributed. Given $\rho$ can you explicitly construct a basis $|e_k >$ such that this is true?

At the end of chapter 1, it says that the answer is: The basis $|e_k >$ to measure in is the Fourier transform of the eigenbasis of $\rho$.

But I can't see how this is proved. Or even what is the intuition of why this is true?

Thank you for any help you can give me!

I am new to quantum mechanics, and I'm trying to learn about Majorization in quantum states. These are the notes I am following http://michaelnielsen.org/blog/talks/2002/maj/book.ps by Michael Nielsen. I am stuck in an exercise that already has an answer, but I still can't understand it. It is the following:

Exercise 1.3.1: Let $\rho$ be an arbitrary state of a $d$-dimensional quantum system. Prove that there always exists an orthonormal basis $|e_k >$ such that the probabilities for a measurement in that basis are uniformly distributed. Given $\rho$ can you explicitly construct a basis $|e_k >$ such that this is true?

At the end of chapter 1, it says that the answer is: The basis $|e_k >$ to measure in is the Fourier transform of the eigenbasis of $\rho$.

But I can't see how this is proved. Or even what is the intuition of why this is true?