Formalisms (in fidelity calculation for quantum teleportation)

[ssampak]

Hi, I am trying to read a paper about quantum teleportation and got stuck with calculating the fidelity of the mixed(noisy) channel.

Fidelity F := < $$\Phi^{(-)}_{12} | \rho_{1} \otimes \rho_{23} | \Phi^{(-)}_{12}$$ >

where $$\rho_{1} = | \phi_{1} > < \phi_{1} |$$
and $$\rho_{23} = t | \Phi^{(+)}_{23}><\Phi^{(+)}_{23}| + (1-t) | \Phi^{(-)}_{23}><\Phi^{(-)}_{23}|$$

$$|\phi_{1}> = a|0_{1}> + b|1_{1}>$$
$$|\Phi^{(\pm)}_{23}> = 1/sqrt{2} (|00_{23}> \pm |11_{23}>)$$

Am I doing wrong if terms like < 0_{1} | 1_{3} > appear?
Then, if right what do I have to do with those? If wrong please give me the right way.

 

[azntoon]

No, you are not doing anything wrong. The terms < 0_{1} | 1_{3} > may appear in the calculation of the fidelity. In this case, since the two states being compared are orthogonal, the value of such a term will be zero, so you can simply ignore it and carry on with the calculation.

 

[denian]

I can understand how confusing and overwhelming the calculations for fidelity in quantum teleportation can be. First of all, it is important to note that the fidelity calculation is used to measure the accuracy of the teleportation process, by comparing the input state (|\phi_{1}>) with the output state after teleportation (|0_{3}> or |1_{3}>). It is a way of quantifying how well the information of the input state is preserved during the teleportation process.

Now, in order to calculate the fidelity, we need to consider the entangled state (|\Phi^{(-)}_{12}>) shared between the sender (1) and receiver (2) of the quantum information, and the noisy channel (|\rho_{23}>) used for the teleportation process.

The first step is to express the noisy channel |\rho_{23}> in terms of the Bell states (|\Phi^{(\pm)}_{23}>). This is done using the parameter t, which represents the noise level in the channel. The Bell states are used because they are the optimal states for teleportation.

Next, we need to plug in the expressions for |\rho_{1}> and |\rho_{23}> into the fidelity equation. It is important to pay attention to the indices, as they represent the different qubits involved in the process.

Regarding your question about terms like <0_{1}|1_{3}>, they are not wrong. These terms represent the overlap between the input state and the output state after teleportation. In other words, they tell us how much of the input state's information was successfully transmitted to the output state.

Lastly, to get the final result for the fidelity, we need to perform the necessary calculations and simplify the expression. I would recommend breaking down the equation into smaller parts and using the properties of inner products to simplify the calculations.

I hope this helps clarify the steps for calculating fidelity in quantum teleportation. It is a complex process, so don't hesitate to seek further guidance or clarification if needed. Keep persevering and good luck with your research!