Sanity check: Kitaev's quantum computing book

[Physics Monkey]

I was flipping through Kitaev's quantum computing book today and noticed something really strange. I thought it might be fun to post it here and figure out what's going on.

See http://books.google.com/books?id=Tr...AEwAA#v=onepage&q=measuring operators&f=false for the relevant page. Basically he's talking about what he calls measuring operators, but then he gives a really strange formula (the action of W on the state midway down the page) which looks like a typo to me. In particular, why isn't there a double sum, one from W and one from W^+?

For those in know, he's basically describing how to measure eigenvalues of unitary operators using interference. This is part of the buildup to the phase estimation algorithm and various other algorithms for abelian groups.

Any thoughts?

 

[martinbn]

It seems to me that you are right.

 

[arojo]

Hi,

There is no problem in the equation, the density matrix $\rho$ is a diagonal matrix (with the probabilities of each state in the diagonal), that is why you have just a sum over one index. Actually you could see it otherwise, an Observable with physical must be a real quantity, then what you are doing in the previous definition is to take the trace.

 

[Physics Monkey]

Hi,

There is no problem in the equation, the density matrix $\rho$ is a diagonal matrix (with the probabilities of each state in the diagonal), that is why you have just a sum over one index. Actually you could see it otherwise, an Observable with physical must be a real quantity, then what you are doing in the previous definition is to take the trace.

I agree that if the density matrix is diagonal then there is only one sum. However, I don't see where Kitaev has made that assumption.