Mathematical Introduction: Probability – From Classical to Quantum

Required materials

• paper and pencil
• coins, dice, marbles of different colours

Basic definitions of probability theory

A sample space (usually denoted as Ω) is the set of all possible outcomes of a random experiment.
For example, when flipping a coin, the sample space is {Head, Tail}. When rolling a six-sided dice, the sample space is {1, 2, 3, 4, 5, 6}.

Events (E) are certain collections (subsets) of outcomes from the sample space.
An event E is a set of outcomes of a random experiment. For a coin flip, one event could be “getting Head” and another event could be “getting Tail”. There is even an event that includes all possible outcomes: “getting Head or Tail”.

Probability of an event

The probability ${\displaystyle p(E)}$ of an event ${\displaystyle E}$ is a number between 0 and 1 such that:

• ${\displaystyle p(E)=0}$ if and only if ${\displaystyle E}$ is impossible;
• ${\displaystyle p(E)=1}$ if and only if ${\displaystyle E}$ is certain;
• ${\displaystyle 0<p(E)<1}$ in all other cases.

Comparing probabilities

If ${\displaystyle p(A)>p(B)}$, event ${\displaystyle A}$ is more likely to occur than event ${\displaystyle B}$.

Complement of an event

The complement of an event ${\displaystyle E}$, often written as “${\displaystyle \text{not E}}$”, “${\displaystyle \stackrel{¯<!--\; ¯\; -->}{E}}$” or “${\displaystyle -<!--\; -\; -->E}$”, is a set of all possible outcomes of a sample space, except for the event ${\displaystyle E}$. The probabilities of an event and its complement satisfy the equation: ${\displaystyle p(\text{not E})=1-<!--\; -\; -->p(E)}$.

Classical probability theory – examples

Probability theory in quantum physics

In the following, we will use some notations and concepts of quantum physics that are explained in other lessons. You may want to ignore the context, i. e. what the symbols represent, and just look at the mathematical formulas. Instead of talking about a general quantum system, we will immediately take the special example of a qubit.

In quantum physics, a qubit does not simply exist in either the state ${\displaystyle |0⟩}$ or the state ${\displaystyle |1⟩}$– at least not before a measurement is performed (see “Basics of quantum physics”). Instead, it can be in a superposition of the states ${\displaystyle |0⟩}$ and ${\displaystyle |1⟩}$, and one can only make statements about the probabilities that it will be measured in state ${\displaystyle |0⟩}$ or state ${\displaystyle |1⟩}$.

Representation of a qubit

A qubit is usually described by its state ${\displaystyle |\mathrm{\psi <!--\; \psi \; -->}⟩=\mathrm{\alpha <!--\; \alpha \; -->}|0⟩+\mathrm{\beta <!--\; \beta \; -->}|1⟩}$, where ${\displaystyle \mathrm{\alpha <!--\; \alpha \; -->}}$ and ${\displaystyle \mathrm{\beta <!--\; \beta \; -->}}$ are complex numbers with the property ${\displaystyle {\left|\mathrm{\alpha <!--\; \alpha \; -->}\right|}^2+{\left|\mathrm{\beta <!--\; \beta \; -->}\right|}^2=1}$. Refer to the lesson Matrices and how they can be used in quantum computing for the ket notation ${\displaystyle |⟩}$. An introduction to complex numbers can be found in this lesson.

Measuring a qubit

The probability of measuring the qubit to be in the state ${\displaystyle |0⟩}$ is ${\displaystyle {\left|\mathrm{\alpha <!--\; \alpha \; -->}\right|}^2}$.

Similarly, the probability of measuring the qubit to be in the state ${\displaystyle |1⟩}$ is ${\displaystyle {\left|\mathrm{\beta <!--\; \beta \; -->}\right|}^2}$

Like in classical probability theory, these values must sum up to 1, but the presence of complex numbers allows for interference effects, see lesson “Basics of quantum physics”.

Probability in quantum physics – examples

Task for students: Quiz

You can download the quiz for students here:

• Quiz as docx
• Quiz as pdf

Below you find the same questions in an interactive format.
Use this link to send the interactive H5P task to your students.