I have been hanging on this problem for a long time without a solution

[dream_chaser]

I have been hanging on this problem for a long time without a solution!

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>

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

 

[lippyka]

at?The answer to the first question is that the principal system can be represented by a Hilbert space called the system space. This space contains all possible states of the system, which can be represented using a basis of orthonormal vectors. The answer to the second question is that we can find an orthonormal basis for the environment using the standard basis of the Pauli matrices. For example, the basis can be chosen as \{|0\rangle, X|0\rangle, Y|0\rangle, Z|0\rangle\}.With these two answers, we can now calculate the operator-sum representation of the quantum operation. The quantum operation is given by \begin{equation}\mathcal{E}(\rho) = P_0\rho P_0 + P_1 \rho P_1 \otimes \langle 0 |X| 0 \rangle \end{equation}where $\rho$ is the density matrix for the system.

 

[m2003]

at.

Thank you for sharing your problem with me. I understand the frustration of being stuck on a problem without a solution for a long time. However, I believe that with persistence and a systematic approach, we can find a solution to this problem.

To address your first question, we can represent the principal system as a single qubit, which can be described by a state vector in a two-dimensional Hilbert space. This state vector can be written as |ψ⟩ = a|0⟩ + b|1⟩, where a and b are complex numbers and |0⟩ and |1⟩ are the basis states of the qubit.

Now, to find the quantum operation for the given process, we can use the operator-sum representation, also known as the Kraus representation. This representation describes a quantum operation as a sum of operators acting on the initial state of the system. In this case, the initial state of the environment is given as |0⟩.

To find the appropriate operators for this representation, we need to first find an orthonormal basis for the environment. This basis can be found by considering the eigenvectors of the Pauli matrix X, which are |+⟩ = (|0⟩ + |1⟩)/√2 and |−⟩ = (|0⟩ - |1⟩)/√2. These two vectors form an orthonormal basis for the environment.

Now, we can express the quantum operation as U = P0⊗I + P1⊗X = |0⟩⟨0|⊗I + |1⟩⟨1|⊗X = |0⟩⟨0|⊗I + |1⟩⟨1|⊗(|+⟩⟨+| - |−⟩⟨−|). This can be further simplified as U = (|0⟩⟨0| + |1⟩⟨1|)⊗I + |1⟩⟨1|⊗|+⟩⟨+| - |1⟩⟨1|⊗|−⟩⟨−|. Therefore, the corresponding operators for the operator-sum representation are given as E0 = |0⟩⟨0| + |1⟩⟨1| and E1 = |1⟩⟨1|⊗|+⟩⟨+| - |1⟩⟨1