How Do Measurements Affect GHZ States?

[EightBells]

Homework Statement

We are going to perform measurements using the following four operators: $$\Sigma_{XYY} = \sigma_x \sigma_y \sigma_y$$
$$\Sigma_{YXY} = \sigma_y \sigma_x \sigma_y$$
$$\Sigma_{YYX} = \sigma_y \sigma_y \sigma_x$$
$$\Sigma_{XXX} = \sigma_x \sigma_x \sigma_x$$
where $\sigma_{\rho}^{\alpha} = |\rho\rangle \langle \rho| - |-\rho\rangle \langle -\rho|$ is the operator for qubit α and ρ = {x, y}. If the three qubits form the GHZ state, what are possible outcomes of the measurements for all
four operators and what are their probabilities? Calculate the expectation values of each operator above with respect to the GHZ state. Explain the importance of this result.

Relevant Equations

n/a

Here's what I think I understand:

First off, the GHZ state $|GHZ \rangle = \frac {|000\rangle+|111\rangle} {\sqrt 2}$, and $\sigma_x$ and $\sigma_y$ are the usual Pauli matrices, so the four operators are easy to calculate in Matlab.

I'm thinking the expectation values of each operator with respect to the GHZ state should be $$\langle GHZ|\Sigma|GHZ\rangle$$ for each of the operators $\Sigma$.

What I'm unsure of is, what are the possible outcomes of the measurements? Is this asking for specific states that could be the result of the measurement? Also what state am I starting from, the GHZ state, ground state $|000\rangle$?

Once I know the outcomes (i.e. possibly certain states), do I then find the probabilities with those via the typical method, $$P = | \langle aaa|bbb\rangle|^2$$, for some states $\langle aaa|$ and $|bbb\rangle$? What are those states however? I would think I'm looking for the probability of starting in the GHZ state and ending up in the outcome state, so $|bbb\rangle = |GHZ\rangle$ and $\langle aaa| = \langle outcome|$?

 

[mark726]

Hello,

Thank you for your post. You are correct in your understanding of the GHZ state and the operators $\sigma_x$ and $\sigma_y$. The expectation values of these operators with respect to the GHZ state can be calculated using the formula you provided, $$\langle GHZ|\Sigma|GHZ\rangle$$.

To answer your question about the possible outcomes of the measurements, it is important to note that the GHZ state is a superposition of two states, $|000\rangle$ and $|111\rangle$. This means that the possible outcomes of the measurements are the states that make up the superposition, $|000\rangle$ and $|111\rangle$. In other words, the possible outcomes are the eigenstates of the operators $\sigma_x$ and $\sigma_y$.

To find the probabilities of these outcomes, you can use the formula you provided, $$P = | \langle aaa|bbb\rangle|^2$$, where $|aaa\rangle$ is the initial state and $|bbb\rangle$ is the final state. In this case, the initial state is the GHZ state, $|GHZ\rangle$, and the final states are the eigenstates of $\sigma_x$ and $\sigma_y$, which are $|000\rangle$ and $|111\rangle$.

I hope this helps clarify your understanding. Please let me know if you have any further questions.