DRAFTS

The above picture of a qubit as two-dimensional vector of length one is almost correct. The only simplification was that the coefficients (the amplitudes) $\alpha$ and $\beta$ were taken to be real numbers $\mathbb{R}$, while in reality they are complex numbers $\mathbb{C}$. This makes the state space of a proper qubit harder (but not impossible) to visualize. None of the examples in this tutorial require this additional structure, so that you can skip this optional part if you like. (The complex number are essential for an efficient formulation of quantum mechanics, though.)

A complex number $\alpha$ can be written as$$
\alpha = \alpha_r+i\cdot \alpha_i
$$with real part $\alpha_r\in\mathbb{R}$ and imaginary part $\alpha_i\in\mathbb{R}$ (yes, the imaginary part is a real number!). The symbol “$i$” denotes the imaginary unit; it is formally defined by the property $i=\sqrt{-1}$ so that $i^2=-1$. The space of complex numbers is labeled by $\mathbb{C}$.

Calculations with complex numbers simply follow from the distributive and associative laws together with $i\cdot i =-1$. For example, adding two complex numbers yields$$
\alpha+\beta=(\alpha_r+i\cdot\alpha_i)+(\beta_r+i\cdot\beta_i)=\underbrace{(\alpha_r+\beta_r)}_{\gamma_r}+i\cdot\underbrace{(\alpha_i+\beta_i)}_{\gamma_i}=\gamma
$$and their product is$$
\alpha\cdot\beta=(\alpha_r+i\cdot\alpha_i)\cdot(\beta_r+i\cdot\beta_i)=\underbrace{(\alpha_r\beta_r-\alpha_i\beta_i)}_{\delta_r}+i\cdot\underbrace{(\alpha_r\beta_i+\alpha_i\beta_r)}_{\delta_i}=\delta\,.
$$Hence one can think of complex numbers as a generalization of real numbers with modified rules for multiplication and addition.

The cool thing about complex numbers is that every polynomial equation has a solution (mathematicians call this algebraically closed)! For example, the equation$$
x^2=-1
$$ has no solution in the real numbers, but it has two complex solutions: $x_1=+i$ and $x_2=-i$. For this and other reasons, both in mathematics and physics, complex numbers $\mathbb{C}$ are the “standard type” of numbers used for the description of many problems, just as the real numbers $\mathbb{R}$ are the “standard numbers” used in school.

• A qubit is a two-dimensional vector of length 1. It is not a classical bit that is in an unkown state 0 or 1.
• The components of the vector are called (probability) amplitudes and can be negative.
• Only the squares of the amplitudes are positive and are interpreted as the probabilities to measure the qubit in a particular state.

TODO

Quantum mechanics is no exception. Therefore the most general quantum state of a qubit can be written as
$$
\ket{\Psi}=\alpha\ket{0}+\beta\ket{1}=\begin{pmatrix}\beta\\\alpha\end{pmatrix}
$$where the two amplitudes $\alpha,\beta\in\mathbb{C}$ are now complex numbers. You cannot visualize this vector as “pointing” in some direction in a two-dimensional space (the deeper reason for this is that complex numbers, unlike real numbers, cannot be ordered from small to large numbers). For example, where would you say the complex vector$$
\begin{pmatrix}i\\0\end{pmatrix}
$$ points to? (I have no clue.) This makes the true quantum states less intuitive than the simplified real ones from above, but not less rigorous – we can still do calculations with them!

The condition that the vector must have length 1 now reads$$
|\alpha|^2+|\beta|^2\stackrel{!}{=}1
$$ where $|\alpha|=\sqrt{\alpha_r^2+\alpha_i^2}$ is called the modulus (or absolute value) of the complex number $\alpha$. Note that $\alpha^2$ does note make sense in the above equation because we want this to be a positive real number (to interpret it as a probability); but for $\alpha=i$ we would get $\alpha^2=-1$, which we cannot read as a probability. By contrast, $|\alpha|^2=\alpha_r^2+\alpha_i^2=0+1^2=1$ is perfectly positive and real.

Do we gain anything by replacing real amplitudes with complex ones? The answer is yes: the space of possible states that a qubit can have becomes in some sense larger when using complex amplitudes. So we should expect that the different states of a qubit can no longer be visualized as a simple unit circle. They can, however, be visualized by a sphere of radius 1, the so called Bloch sphere. To understand how this is possible (it is not obvious!), we first have to state a beautiful equation called Euler’s formula:$$
e^{i\varphi}=\cos(\varphi)+i\cdot \sin(\varphi)
$$Here $\varphi\in\mathbb{R}$ is an arbitrary real number, $i$ is again the imaginary unit, $\sin$ and $\cos$ are the trigonometric functions, and $e=2.718\dots$ is a very special real number called Euler’s number (it shows up all over mathematics).

When following mathematical arguments like the ones that led us here, one can easily loose contact to reality and forget what we actually want to achieve. The use of complex amplitudes that gives rise to the Bloch sphere as the state space of a qubit is so far just a (beautiful) mathematical concept. But in the end we want to use mathematics to describe the actual qubits stored in the orbitals of the electrons in our atoms. Do we really need all this unintuitive complexity (pun intended) to describe their states? The clear answer is: Yes! We can change the states of our qubits in ways that cannot be described by real amplitudes. The states of our qubits really are described by points on the Bloch sphere, and not by points on a unit circle. This tells us a very important lesson: Complex numbers are not just inventions of mathematicans, they are, in some sense, the natural language needed to describe nature.