Postulate 4: Kronecker Product of Two Qubits

Motivation

Imagine we have a photon (a particle of light) which can be in state $\ket{H}$ (horizontal polarization), $\ket{V}$ (vertical polarization), or some superposition of the two.

When we have a second photon, or a third, or a fourth, etc. we need a mathematical way to describe their state together. The Kronecker product ($\otimes$) is our tool for this.

A key feature of nearly all quantum computing algorithms is the interaction and interference between many different qubits. Thus the Kronecker product is a very important mathematical concept to understand if one wishes to understand what is happening before, during, and after a quantum computation.

Mathematical Definition

Let $\ket{\psi} = \alpha\ket{0} + \beta\ket{1} = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}$ and $\ket{\phi} = \gamma\ket{0}+\delta\ket{1} = \begin{bmatrix} \gamma \\ \delta \end{bmatrix}$ be two quantum (pure) states.

The Kronecker product is defined as:

$$\begin{align*} \ket{\psi} \otimes \ket{\phi} &=  \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \otimes \begin{bmatrix} \gamma \\ \delta \end{bmatrix} \\ & = \begin{bmatrix} \alpha \gamma \\ \alpha \delta \\ \beta \gamma \\ \beta \delta \end{bmatrix}. \end{align*}$$

Note that the resulting vector has dimension $4$.

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