In quantum mechanics, we can describe pure quantum states using an n-dimensional vector (also called a rank-1 tensor). This vector is written in Dirac notation as a ket and represents the complete information of the state: $$ \vert \Psi \rangle = \begin{pmatrix} \alpha_1 \\ \vdots \\ \alpha_n \end{pmatrix} $$
Each component is a complex number that encodes both amplitude and phase. Physically, the squared magnitude gives the probability of finding the system in the corresponding basis state when a measurement is performed.
From this ket we can construct the corresponding bra, which belongs to the dual space. It is obtained by transposing the ket and taking the complex conjugate of every component: $$ \langle \Psi \vert = \begin{pmatrix} \alpha_1^* & \cdots & \alpha_n^* \end{pmatrix} $$ At first instance we can define some products using bras and kets, the inner product, outer product and tensor product.
Inner product
The inner product (or scalar product) combines a bra with a ket and yields a single complex number: \begin{equation*} \langle \Psi \vert \Psi \rangle = \begin{pmatrix} \alpha_1^* & \cdots & \alpha_n^* \end{pmatrix} \begin{pmatrix} \alpha_1 \\ \vdots \\ \alpha_n \end{pmatrix} = \alpha_1^* \alpha_1 + \cdots \alpha_n^* \alpha_n = \vert \alpha_1 \vert ^2 + \cdots + \vert \alpha_n \vert ^2 \end{equation*}
This corresponds to the squared norm of the state. For physical states, this value must equal 1, reflecting the fact that the total probability across all possible outcomes is always conserved.
Outer product
The outer product combines a ket with its corresponding bra, producing a matrix instead of a number: \begin{align*} \vert \Psi \rangle \langle \Psi \vert &= \begin{pmatrix} \alpha_1 \\ \vdots \\ \alpha_n \end{pmatrix} \begin{pmatrix} \alpha_1^* & \cdots & \alpha_n^* \end{pmatrix} = \begin{pmatrix} \alpha_1 \alpha_1^* & \alpha_1 \alpha_2^* & \cdots & \alpha_1 \alpha_n^* \\ \alpha_2 \alpha_1^* & \alpha_2 \alpha_2^* & \cdots & \alpha_2 \alpha_n^* \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_n \alpha_1^* & \alpha_n \alpha_2^* & \cdots & \alpha_n \alpha_n^* \end{pmatrix} \\ \\ &= \begin{pmatrix} \vert \alpha_1 \vert ^2 & \alpha_1 \alpha_2^* & \cdots & \alpha_1 \alpha_n^* \\ \alpha_2 \alpha_1^* & \vert \alpha_2 \vert ^2 & \cdots & \alpha_2 \alpha_n^* \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_n \alpha_1^* & \alpha_n \alpha_2^* & \cdots & \vert \alpha_n \vert ^2 \end{pmatrix} \end{align*}
This operator is fundamental in quantum mechanics. For example, it is used to represent projectors onto states and to build density matrices, which describe both pure and mixed states in quantum theory.
Tensor product
Finally, the tensor product (or Kronecker product) is the operation that allows us to combine two independent quantum systems into a larger one: $ \vert \alpha \rangle \otimes \vert \beta \rangle \equiv \vert \alpha \rangle \vert \beta \rangle \equiv \vert \alpha , \beta \rangle \equiv \vert \alpha \beta \rangle $.
Supposing 2-dimensional kets $ \vert \alpha \rangle = \big(\begin{smallmatrix} \alpha_1 \\ \alpha_2 \end{smallmatrix}\big) $ and $\vert \beta \rangle = \big(\begin{smallmatrix} \beta_1 \\ \beta_2 \end{smallmatrix}\big) $ we can define the tensor product as:
\begin{align*} \vert \alpha \rangle \otimes \vert \beta \rangle &= \begin{pmatrix} \alpha_1 \\ \alpha_2 \end{pmatrix} \otimes \begin{pmatrix} \beta_1 \\ \beta_2 \end{pmatrix} = \begin{pmatrix} \alpha_1 \begin{pmatrix} \beta_1 \\ \beta_2 \end{pmatrix} \\ \alpha_2 \begin{pmatrix} \beta_1 \\ \beta_2 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} \alpha_1 \beta_1 \\ \alpha_1 \beta_2 \\ \alpha_2 \beta_1 \\ \alpha_2 \beta_2 \end{pmatrix} \end{align*}
The dimension of the resulting vector is the product of the dimensions of the original ones. This operation is the mathematical foundation for describing multi-particle systems and quantum entanglement.
