## Help Needed with Nielsen and Chuang Exercise 2.73

Exercise 2.73Given density matrix , the minimal ensemble contains a number of elements equal to the rank of . Let be any state in the support of , where the support is defined as being spanned by the eigenvectors of with nonzero eigenvalues. Show there exists a minimal ensemble for that contains , and that appears with probability

where is the inverse that acts only on the support of .

The density matrix has the eigendecomposition . We are given state in the support of , meaning that it is spanned by the eigenvectors with nonzero eigenvalues. This means that we can expand

where the primed summation is only over those vectors in the support of . Among other things, this means that the rank of is no greater than . Since we can expand in the eigenvectors of , this also means that we can create a minimal ensemble for than contains .

Given with rank , consider the matrix . We can write the projection of onto via

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

gfsaid, on January 17, 2011 at 5:30 pmBasically 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.

an3said, on May 27, 2015 at 11:40 amI don’t know, but this is what I think about this:

for a unique ensamble ρ = pi | ψi > ρ ρ-1 = pi | ψi > tr( ρ ρ-1 ) = pi tr( | ψi > 1 = pi => pi = 1 / .

an3said, on May 27, 2015 at 11:49 amSorry, I said

for a unique ensamble

ρ = pi | ψi > < ψi | ρ-1 )

1 = pi

so

pi = 1 /