# What is the correct name of this quantum gate? Possibly state control gate

Let $$\vec v \in \mathbb{C}^2$$ be the following quantum state: $$\vec v = \frac{1}{\sqrt{2}}\begin{bmatrix} v_{1} \\ v_{2} \\ \end{bmatrix},\space \lvert v_1 \rvert = 1, \space \lvert v_2 \rvert = 1$$

Let $$\mathbb{A}$$ be the following quantum gate:

$$\mathbb{A} = cos (\phi)\cdot\sigma_1+sin(\phi)\cdot\sigma_2,$$ where, $$\sigma_1,\sigma_2$$ are Pauli matrices and the angle $$\phi$$ is the solution to the equation: $$\frac{v_2}{v_1} = e^{i\phi}$$

With these definitions the matrix $$\mathbb{A}$$ has the property: $$\mathbb{A}\vec v = \vec v$$

I am a mathematical physicist not well versed in quantum computing terminology, does this gate have a specific name in the field? I would call it a state control gate. I would believe it must be an important notion in the field, I would welcome any further explanation of the importance of this gate in the field of quantum computing.

Note: $$\mathbb{A}$$ and $$\vec{v}$$ can be generalized to $$n$$-dimensions. The general form of the matrix is the following: $$\mathbb{A} = \begin{bmatrix} 0 & \frac{v_2}{v_1} & \cdots & \frac{v_n}{v_1}\\ \frac{v_1}{v_2} & 0 & \cdots & \frac{v_n}{v_2}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{v_1}{v_n} & \frac{v_2}{v_n} & \cdots & 0\\ \end{bmatrix},$$ where, $$\mathbb{A}\vec{v} = (n-1)\vec{v}$$

In dimension 3 the matrix is a combination of the Gell-mann matrices.