Quantum entanglement and measurement operator

[superiority]

Homework Statement

A system in a state $\frac{1}{\sqrt{2}}(\left<\phi\right| + \left<\psi\right|)$ undergoes an interaction with a second system (which is initially in $\left<\alpha\right|$) and ands up in an entangled state $\frac{1}{\sqrt{2}}\left(\left\langle\phi\right| \otimes \left\langle\alpha\right| + \left\langle\psi\right|\otimes \left\langle\beta\right|\right)$. Find a unitary operator that will perform that interaction/measurement.

The respective states in each space are orthonormal, and φ and ψ form a complete basis.

Homework Equations

I assume the linearity of tensor products is relevant. Other than that, I'm really not sure what.

The Attempt at a Solution

I don't know nearly enough about entanglement or unitary operators here. An entangled state means one that isn't separable, I know. A unitary operator U satisfies U*U = UU* = I. From there... I don't know.

 

[mark726]

I'm not sure how to find the specific operator that would perform this interaction/measurement.

First, let's define the states in each space:

- System 1: φ, ψ (orthonormal)
- System 2: α, β (orthonormal)

Now, we can express the initial state of the system as:

|Ψ> = \frac{1}{\sqrt{2}}(\left<\phi\right| + \left<\psi\right|) \otimes \left<\alpha\right|

We can rewrite this as:

|Ψ> = \frac{1}{\sqrt{2}}(\left<\phi\right| \otimes \left<\alpha\right| + \left<\psi\right| \otimes \left<\alpha\right|)

We can see that this is similar to the entangled state given in the problem, except for the second term (\left<\psi\right| \otimes \left<\beta\right|). This suggests that the interaction/measurement will somehow involve the second state in system 2, β.

To find the unitary operator that will perform this interaction/measurement, we can start by considering the action of this operator on the basis states:

U|φ> = |φ> \otimes |α> + |ψ> \otimes |β>

U|ψ> = |φ> \otimes |α> - |ψ> \otimes |β>

From these equations, we can see that the operator U must perform a swap operation between the states in system 1 and system 2. This can be achieved using the Pauli X gate, which is a unitary operator that performs a bit-flip:

X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}

Applying this operator to the initial state, we get:

X|Ψ> = \frac{1}{\sqrt{2}}(\left<\phi\right| \otimes \left<\alpha\right| + \left<\psi\right| \otimes \left<\beta\right|)

This is the desired entangled state, so the unitary operator that will perform the interaction/measurement is the Pauli X gate.