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

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

• Hi Jimarious :). I guess the question, as it is described now, is not connected with the tags: physical-realization and physical-qubit. Is there any connection that I don't see? May 12, 2020 at 14:50
• No, there is no underlying connection, I just cannot create tags, but was forced to add five tags so I could post this question. You can edit this if you want. May 12, 2020 at 14:51
• Ok :). how I remember one can use less than 5 tags. May 12, 2020 at 15:05

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

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

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

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

How to calculate the outer product of two vectors:

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

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

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

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:

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