What is the intuition of the outer product of two states?

Question

If it is possible give me an intuition both with vectors on the plane and on the Bloch-Sphere.

Answer 1

The outer product of two states is a matrix. Here are some often used outer products:

\begin{equation} |0\rangle \langle 0 | = \begin{pmatrix} 1&0\\0&0 \end{pmatrix} \qquad |0\rangle \langle 1 | = \begin{pmatrix} 0&1\\0&0 \end{pmatrix} \\ |1\rangle \langle 0 | = \begin{pmatrix} 0&0\\1&0 \end{pmatrix} \qquad |1\rangle \langle 1 | = \begin{pmatrix} 0&0\\0&1 \end{pmatrix} \end{equation}

For any $2$ by $2$ matrix we can write:

\begin{equation} M = \begin{pmatrix} a&b\\c&d \end{pmatrix} = a|0\rangle \langle 0 | + b|0\rangle \langle 1 | +c |1\rangle \langle 0 | +d|1\rangle \langle 1 | \end{equation}

How to calculate the outer product of two vectors:

\begin{equation} |v\rangle \langle w | = \begin{pmatrix} v_1\\v_2 \end{pmatrix} \begin{pmatrix} w_1^{\dagger}&w_2^{\dagger}\end{pmatrix} = \begin{pmatrix} v_1 w_1^{\dagger} & v_1 w_2^{\dagger} \\ v_2 w_1^{\dagger} & v_2 w_2^{\dagger}\end{pmatrix} \end{equation}

The outer product of a state from the Bloch sphere $|\psi\rangle = \alpha |0\rangle + \beta |1\rangle$ with itself:

\begin{equation} |\psi\rangle \langle \psi | = \begin{pmatrix} |\alpha|^2 & \alpha \beta^{\dagger} \\ \beta \alpha^{\dagger} & |\beta|^2 \end{pmatrix} = \rho \end{equation}

It is equal to the density matrix $\rho$ that also describes the state of the system like the wavefunction $|\psi\rangle$ and has correspondence with the same point on the Bloch sphere as the $|\psi\rangle$ state.

Another useful thing to know about the outer products of two states. Assume we have orthonormal eigenbasis $|v_s \rangle$ and we have this matrix $| w \rangle \langle v_1 |$ (that is not a unitary matrix and $|w\rangle$ is some normalized state). What will be after the multiplication of this matrix with a geneal $|\psi \rangle = \sum_s \alpha_s |v_s\rangle$ state vector:

$$| w \rangle \langle v_1 | \sum_s \alpha_s |v_s\rangle = \alpha_1 | w \rangle$$

This kind of outer products are often used in QC. For example, for $X$ operator we can write:

$$ X = | 0\rangle \langle 1| + | 1\rangle \langle 0| $$