- #1

blueinfinity

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- Homework Statement
- Alice and Bob share the Bell state

\begin{align*}

|\psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle+|11\rangle).

\end{align*}

Consider the pair of observables

\begin{align*}

\mathcal{O}_A =

\begin{pmatrix}

1 & 0 \\ 0 & \frac{1}{2}

\end{pmatrix}

, \qquad \mathcal{O}_B =

\begin{pmatrix}

1 & 0 \\ 0 & \frac{1}{3}

\end{pmatrix}

.

\end{align*}

Show the mutual information between Alice and Bob is larger than $(\langle\psi | \mathcal{O}_A \otimes \mathcal{O}_B| \psi\rangle - \langle\psi |\mathcal{O}_A|\psi \rangle \langle\psi |\mathcal{O}_B|\psi \rangle)^2 $

- Relevant Equations
- \begin{align*}

|\psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle+|11\rangle).

\end{align*}

Consider the pair of observables

\begin{align*}

\mathcal{O}_A =

\begin{pmatrix}

1 & 0 \\ 0 & \frac{1}{2}

\end{pmatrix}

, \qquad \mathcal{O}_B =

\begin{pmatrix}

1 & 0 \\ 0 & \frac{1}{3}

\end{pmatrix}

.

\end{align*}

I've make progress in obtaining the values for the mutual information using the following:

$I(\rho_A:\rho_B) = S(\rho_A) +S(\rho_B) - S(\rho_{AB}) = 1 + 1 - 0 = 2.$

I would like to compute the expectation but I'm facing a problem in the case of $\langle\psi |\mathcal{O}_A|\psi \rangle$ since the size of matrices in this multiplication do not match. namely, $\langle\psi$ is of size $1\times 4$ and $|\psi \rangle$ is of size $4\times 1$ and the matrix $\mathcal{O}_A$ is $2 \times 2$.

I'm very new to the subject and I would greatly appreciate if I could have some guidance on how the computations for this expectation would be carried out.

additionally I have computed the $\langle\psi | \mathcal{O}_A \otimes \mathcal{O}_B| \psi\rangle$ by first computing the tensor product of the two matrices $A,B$ and then taken the multiplication with the Bra and Ket of the state respectively deducing

$$\langle\psi | \mathcal{O}_A \otimes \mathcal{O}_B| \psi\rangle = \frac{7}{12}$$.

I would appreciate any insight on this.

$I(\rho_A:\rho_B) = S(\rho_A) +S(\rho_B) - S(\rho_{AB}) = 1 + 1 - 0 = 2.$

I would like to compute the expectation but I'm facing a problem in the case of $\langle\psi |\mathcal{O}_A|\psi \rangle$ since the size of matrices in this multiplication do not match. namely, $\langle\psi$ is of size $1\times 4$ and $|\psi \rangle$ is of size $4\times 1$ and the matrix $\mathcal{O}_A$ is $2 \times 2$.

I'm very new to the subject and I would greatly appreciate if I could have some guidance on how the computations for this expectation would be carried out.

additionally I have computed the $\langle\psi | \mathcal{O}_A \otimes \mathcal{O}_B| \psi\rangle$ by first computing the tensor product of the two matrices $A,B$ and then taken the multiplication with the Bra and Ket of the state respectively deducing

$$\langle\psi | \mathcal{O}_A \otimes \mathcal{O}_B| \psi\rangle = \frac{7}{12}$$.

I would appreciate any insight on this.