Why do we divide by $\sqrt2$ in the qubit states $\lvert\pm\rangle=\frac{1}{\sqrt2}(\lvert0\rangle\pm\lvert1\rangle)$?

Question

I have a very basic question. I have found qubits are represented as complex vectors. I get it totally. I understand bracket notation and vector\matrix algebra. However, I cannot move further from here. It is often referred like the equation below.

$$ |+\rangle=\frac{1}{\sqrt{2}} ( |0\rangle+ |1\rangle) $$

$$ |-\rangle=\frac{1}{\sqrt{2}} ( |0\rangle- |1\rangle) $$

My questions are:

1. Why is divided by $\frac{1}{\sqrt{2}} $ ?
2. What does this symbol mean $|+\rangle$, $|-\rangle$? I understand what $|0\rangle$ and $|1\rangle$ means.

Answer 1

#1: the $1/\sqrt{2}$ is a normalization which ensures that the ``length'' of the vector is one.

#2: The notation $|\pm\rangle$ is just a label for the two states defined above. Since the states $|0\rangle, |1\rangle$ are elements of a vector space, you can take linear combinations and therefore construct the states $|\pm\rangle$

Answer 2

The $\frac{1}{\sqrt{2}}$ is due to the normalization condition which says that sum of the squares of the amplitudes of the must be equal to one while the square of the amplitude refers to the probability of getting that particular state when the qubits are measured

The vectors $|+⟩$ and $|-⟩$ are known as the eigenvectors for the Hadamard gate. When we apply the $H$ gate on the $|0⟩$ and $|1⟩$, we get $|+⟩$ and $|-⟩$ respectively.

That is, $$H|0⟩=|+⟩$$ and $$H|1⟩=|-⟩$$