# Worked Problems in Physics

## Help Needed with Nielsen and Chuang Exercise 2.73

Posted in Nielsen/Chuang by rpmuller on June 1, 2010

Exercise 2.73 Given density matrix ${\rho}$, the minimal ensemble ${\{p_k,\left|k\right>\}}$ contains a number of elements equal to the rank of ${\rho}$. Let ${\left|i\right>}$ be any state in the support of ${\rho}$, where the support is defined as being spanned by the eigenvectors of ${\rho}$ with nonzero eigenvalues. Show there exists a minimal ensemble for ${\rho}$ that contains ${\left|i\right>}$, and that ${\left|i\right>}$ appears with probability

$\displaystyle p_i = \frac{1}{\left}$

where ${\rho^{-1}}$ is the inverse that acts only on the support of ${\rho}$.

The density matrix ${\rho}$ has the eigendecomposition ${\rho=p_k\left|k\right>\left. We are given state ${\left|i\right>}$ in the support of ${\rho}$, meaning that it is spanned by the eigenvectors with nonzero eigenvalues. This means that we can expand

$\displaystyle \left|i\right> = \sum_{k'}\left\left|k'\right>= \sum_{k'}C_{k'i}\left|k'\right>$

where the primed summation is only over those vectors in the support of ${\rho}$. Among other things, this means that the rank of ${\left|i\right>}$ is no greater than ${\rho}$. Since we can expand ${\left|i\right>}$ in the eigenvectors of ${\rho}$, this also means that ${\left|i\right>}$ we can create a minimal ensemble for ${\rho}$ than contains ${\left|i\right>}$.

Given ${\rho}$ with rank ${m}$, consider the ${m\times m}$ matrix ${\sum_{k'}p_k\left|k\right>\left. We can write the projection of ${\rho}$ onto ${\left|i\right>}$ via

$\displaystyle \rho\left|i\right>=\sum_{kl'} C_{li}p_k\left|k\right>\left=\sum_{k'}C_{ki}p_k\left|k\right>.$

I need help here, if anyone has any suggestions. Just can’t push the equations home.

### 3 Responses

1. gf said, on January 17, 2011 at 5:30 pm

Basically we want to prove that =1.
Using theorem 2.6 we can rewrite this in terms of the eigenfunctions of ρ as
=Sum over j and j’ of (Uij’*Uij). It’s not that hard to show that = δj’j (so we’ve solved the problem in this special case where the pure states are eigenstates of ρ.)
We then have that
=Sum over j of (Uij*Uij)=1. The second equality follows since the U’s are unitary.

2. an3 said, on May 27, 2015 at 11:40 am