Quantum State in a Nutshell

There are thousands of articles trying to explain what exactly a quantum state is. Many of them boiled down to “the state of a qubit is 0, 1, or 0 and 1 at the same time”. This statement leads to both confusion and misinterpretation. The explanation I found on Quantum computing for the very curious is by far the most elegant and simplest:

The state of a qubit is a vector in a two-dimensional vector space. This vector space is known as state space.

I will use many of great content from Quantum computing for the very curious to explain things.

Mapping qubits to classical bits

We’ve described what a qubit state is, but provided no link between a qubit state and a classical bit state. There are two possible states for a classical bit: 0 and 1. The corresponding states for a qubit is slightly fancier: \(|0\rangle \) and \(|1\rangle \).

The notation with \(|\) and \(\rangle\) is called a \(ket\) notation. With a number wrapped between them, \(0\) or \(1\) are called \(kets\). A \(ket\) is a fancy term for a vector. In fact, \(|0\rangle\) is really...