Finding averages of observables of Bell states

[poonintoon]

Hi I am looking at Quantum Computation by Neilsen and Chuang at the CHSH inequality.

Looking at the spin singlet state they make measurements of for example the observable Z₁
and Z₂-X₁ and then find the expectation value of the product.

I am slightly confused here because
a) Z and X are gates does it just mean to find the average of the system after these have acted on the system?
b)Secondly can I find the average of a two qubit system without using the density operator. If I need to use the density operator is there a simple way?


Thanks

(Sorry its hard to make it clear without waffling on for ages but since its a popular book hopefully someone will understand me).

 

[azntoon]

a) Yes, finding the expectation value of the product is equivalent to measuring the average of the system after the gates have been applied. b) To calculate the expectation value of a two qubit system, you need to use the density operator. The density operator encodes the state of the system, so it can be used to calculate the expectation value of any observable. To calculate the expectation value, you need to take the trace of the product of the density operator and the observable.