3 Quantum systems

With the basics for a probabilistic system defined, we now look into describing a quantum computer mathematically. In the following table you can see the analogy from the quantum world to the probabilistic world.

Probabilistic world	Quantum world
Probability distributions	Quantum states
Probabilities	Amplitudes
Deterministic possibilities	Classical possibilities
Stochastic matrix as process	Unitary matrix as process

3.1 Classical possibilities

Like in the probabilistic systems we need to define all outcomes for a quantum system. For example, a photon can be in the state up or down. We call these possibilities classical possibilities. Note that we will only be using a finite number of possibilities.

Example: Random bit

Imagine you have a random bit generator, which outputs one bit. The classical possibilities of this generator are \(0\) and \(1\).

We will always assume the classical possibilities to be ordered in some way (even if it is an arbitrary one), like the deterministic possibilities. In the example above, the classical possibilities are \(0\), \(1\) not \(1\), \(0\). We will need this to know the order of entries in vectors and matrices later.

3.2 Quantum states

One of the most important element of the quantum world is a quantum state. A quantum state describes the state of a quantum system as a vector. Each entry of the vector represents a classical possibility (similar to the deterministic possibilities in a probability distribution). The entries of a quantum state are called amplitude. In contrast to a probabilistic system, these entries can be negative and are also complex numbers.

These amplitudes tell us the probability of the quantum state being in the corresponding classical possibility. To calculate the probabilities from the amplitude, we can take the square of the absolute value of the amplitude.

This means that for the classical possibility \(x\) and a quantum state \(\psi\) the probability for \(x\) is \(\Pr[x] = |\psi|^2\). To have valid probabilities, the sum of all probabilities need to sum up to \(1\). From this we get the formal definition of a quantum state:

Definition 3.1 (Quantum State) A quantum state is a vector \(\psi \in \mathbb{C}^n\) with \(\sqrt{\sum_{i=1}^n |\psi_i|^2} = 1\).

Example: Some Quantum states

The following vectors are valid quantum states with the classical possibilities \(0\) and \(1\): \[ \ket{0} \coloneqq \begin{pmatrix} 1 \\ 0 \end{pmatrix},\quad \ket{1} \coloneqq \begin{pmatrix} 0 \\ 1 \end{pmatrix},\quad \ket{+} \coloneqq \begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{pmatrix},\quad \ket{-} \coloneqq \begin{pmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{pmatrix}. \] Note that the symbol \(\ket{}\) is not yet introduced, so just understand it as some label at this point. The probabilities for each state can be calculated as follows: \[ \begin{aligned} \ket{0}:&& \Pr[0] &= |1|^2 = 1 \quad &&\Pr[1] = |0|^2 = 0,\\ \ket{1}:&& \Pr[0] &= |0|^2 = 0 \quad &&\Pr[1] = |1|^2 = 1,\\ \ket{+}:&& \Pr[0] &= \lvert\tfrac{1}{\sqrt{2}}\rvert^2 = \tfrac{1}{2} &&\Pr[1] = \lvert\tfrac{1}{\sqrt{2}}\rvert^2 = \tfrac{1}{2},\\ \ket{-}:&& \Pr[0] &= \lvert\tfrac{1}{\sqrt{2}}\rvert^2 = \tfrac{1}{2} &&\Pr[1] = \lvert\tfrac{-1}{\sqrt{2}}\rvert^2 = \tfrac{1}{2}. \end{aligned} \] We can see here that two different quantum states (\(\ket{+}\) and \(\ket{-}\)) can have the same probabilities for all classical possibilities.

3.3 Unitary transformation

We now have defined quantum states and need a way to describe some processes, which we want to apply on the quantum states. In the probabilistic world, we have stochastic matrices for this, but unfortunately we can not use these matrices on quantum states, since the output of applying these on a quantum state is not guaranteed to be a quantum state again. We therefore look for a different property of a matrix for which the outcome of applying that matrix is guaranteed to be a quantum state. The following Lemma is therefore useful.

Lemma 3.1 (Unitary matrix) For a square matrix \(U\), the following are equivalent:

\(U\) maps every quantum state to a quantum state,
\(U^\dagger U = I\) and \(UU^\dagger = I\),
\(U^\dagger U = I\),
all columns are quantum states and mutually orthogonal.

Definition 3.2 (Unitary transformation) A matrix \(U\) is called unitary iff \(U^\dagger U = I\) and \(UU^\dagger = I\).

Then the evolution of a quantum state is always described by a unitary matrix. So if the current state is \(\psi\) and we apply the transformation matrix \(U\), the state is \(U\psi\) afterwards.

A unitary matrix is by definition invertible, therefore we can undo all unitary transformations by applying \(U^\dagger\).

Example: Some Unitary transformations

The following matrices are examples for unitary transformations: \[ X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},\quad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix},\quad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. \] These matrices are called Pauli-matrices, we will get to know them later on.

As an example for applying a unitary on a quantum state, we apply the Pauli \(X\) matrix on the quantum state \(\ket{0}\):

\[ X\ket{0} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \ket{1}. \]