Quantum States

Single qubit states

Classical bits, represented by low and high voltages in a transistor, are represented using two numbers, 0 and 1. Quantum states are different. Instead of numbers, quantum states are represented by vectors. As you would recall, vectors have both magnitude and direction in space associated with them. More specifically, a quantum state is a vector in two-dimensional vector space. We will refer to this two-dimensional vector space as the state-space of a single qubit.

Bra-kets

In an earlier lesson, we introduced a shorthand notation for representing quantum states. Remember those fancy $|0\rangle$ and $|1\rangle$? Here they are again.

Here’s how you define $|0\rangle$ $=$ $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and here’s how you define $|1\rangle$ $=$ $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$

We call these ket vectors, enclosed within a $|$ and $\rangle$. Ket vectors are column vectors, meaning they have a single column with multiple rows.

There are also bra vectors. Bra vectors are represented by $\langle$ and $|$. For example: $\langle0|$ and $\langle1|$. As you would guess, bra vectors are row vectors, meaning they have one row but multiple columns.

Computational basis

Now that we have seen that the $|0\rangle$ and $|1\rangle$ states are essentially vectors, we will discuss another unique property of these two states.

They are called the computational basis states. We will talk about the computational part of this term in a future lesson but let’s see why they are called a basis.

These two vectors are the basis of the vector space that represents single-qubit quantum states.

Now, what is a basis? A basis is a set of vectors that can span the entire vector space. This essentially means that every vector in the state space can be written as a linear combination of the $|0\rangle$ and $|1\rangle$ vectors.

An example of this would be an arbitrary state $|\phi\rangle$ written as $0.7 *|0\rangle+0.9*|1\rangle$ in bra-ket notation.

In vector notation, it would look like this:

$$|\phi\rangle = 0.7*\begin{bmatrix} 1 \\ 0 \end{bmatrix}+ 0.9 * \begin{bmatrix} 0 \\ 1 \end{bmatrix}$$ ...