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.

 

[Entropia]

Hi there,

I am happy to provide a response to your question. First, let me clarify that the observable Z1 and Z2-X1 refer to the Pauli Z and X operators acting on the first and second qubit respectively. These operators are used to measure the spin states of the qubits along the z and x axes. So, when you are finding the expectation value of the product, you are essentially measuring the correlation between the two qubits.

To answer your first question, yes, finding the average of the system after these gates have acted on the system is the correct interpretation. The gates are used to manipulate the state of the qubits and then the expectation value is calculated to measure the outcome of the experiment.

As for your second question, it is possible to find the average of a two qubit system without using the density operator. This can be done by directly calculating the expectation value of the product of the two qubits. However, using the density operator can simplify the calculations and make it easier to analyze the system. There are various methods to calculate the average using the density operator, such as the trace rule or the inner product rule.

I hope this helps clarify your confusion. As a final point, I would like to mention that it is always important to fully understand the concepts and equations before moving on to applications. I recommend going back to the basics if you are still unsure about any aspect of the CHSH inequality or finding averages of observables. Best of luck with your studies!