About the operatior-sum representation

[dream_chaser]

Homework Statement

Homework Equations

Suppose we have a single qubit principal system ,interacting with a single qubit environment through the transformation
U=P0$$\otimes$$I+P1$$\otimes$$X
where X is the usual Pauli matrix (acting on the enviornment)and P0=|0><0| ,P1=|1><1| are projectors (acting on the system ).Give the quantum operation for this process ,in the operator-sum representation ,assuming the environment starts in the state |0>

The Attempt at a Solution

I have two problem about this exercise :
1.how to represent the principal system
2.how to find an orthonormal basis for the environment so that the operatior-sum representation can be arrived

 

[anathema]

at.

Hello, thank you for your post. I would be happy to help you with your questions. To answer your first question, the principal system can be represented by a single qubit state, which can be written as a linear combination of the basis states |0> and |1>. As for your second question, finding an orthonormal basis for the environment can be done by using the Gram-Schmidt process. This process allows you to take a set of linearly independent vectors and create an orthonormal basis from them. Once you have an orthonormal basis, you can use it to construct the operator-sum representation for the quantum operation. I hope this helps! Let me know if you have any further questions.

 

[m2003]

at

I would like to offer some guidance on how to approach this problem. First, it is important to understand the concept of the operator-sum representation. This representation is used to describe quantum operations, which are transformations on quantum states. In this case, the quantum operation is the transformation of the principal system interacting with the environment.

To answer the first question, the principal system can be represented by a single qubit state, which can be written as a linear combination of the basis states |0> and |1>. For example, the principal system can be in the state |ψ> = α|0> + β|1>, where α and β are complex coefficients.

For the second question, the orthonormal basis for the environment can be found by considering the possible states of the environment. In this case, the environment starts in the state |0>, so the basis states can be written as |0> and |1>. These two states are orthogonal, meaning they are perpendicular to each other, and can form an orthonormal basis.

Now, to find the quantum operation in the operator-sum representation, we can use the transformation given in the homework statement: U = P0⊗I + P1⊗X. This can be written in matrix form as:

U = [1 0; 0 1] ⊗ [1 0; 0 1] + [0 1; 1 0] ⊗ [0 1; 1 0]

= [1 0 0 0; 0 1 0 0; 0 0 0 1; 0 0 1 0]

This is a 4x4 matrix, representing a transformation on a 2-qubit system (the principal system and the environment). The operation can be written in the operator-sum representation as:

U = ∑i Ei ⊗ Ui

where Ei are the basis operators for the environment (in this case, E0 = |0><0| and E1 = |1><1|) and Ui are the unitary operators on the principal system (in this case, U0 = I and U1 = X). This representation shows how the transformation on the system and the environment are connected.

I hope this helps to clarify the concept of the operator-sum representation and how it can be applied in this specific scenario.