Introduction
\(\newcommand{\myket}[1]{{\ \color{blue} \ket{#1}}}\) In the existing literature, not all authors use the same notation and conventions.
In addition, to make things as simple and understandable as possible, I sometimes deviate from the standard notation.
This page summarizes the most important mathematical notations and conventions used throughout this website.
Complex Numbers
In my videos, complex numbers are introduced as tuples \((r, \varphi)\) where \(r\) is the absolute value (or weight factor) and \(\varphi\) is the phase angle of the complex number.
To highlight the probability interpretation of complex numbers in quantum mechanics, the weight factor is often written as a square root of a percent value. In addition, to make phase angles easily understandable, they are often written in degrees. So an example for a complex number written that way would be \((\sqrt{50\%}, 180°)\).
However, to be closer to the standard notation, I decided to use the following notation on this website:
\[ (r, \varphi) = r \cdot e^{i \cdot \varphi} \]
In addition, there are the following shortcuts for certain phase factors:
\[e^{i \cdot 0°} = 1 \]
\[e^{i \cdot 90°} = i \]
\[e^{i \cdot 180°} = -1 \]
\[e^{i \cdot 270°} = -i \]
Vectors
The term vector is used with many different meanings. See my article What is a Vector? for details.
But on this website, the general meaning of the term vector is the following:
A vector is an element from a vector space. A vector space is a set, whose elements can be added and scaled by numbers. Depending on the concrete vector space, these numbers could be real numbers or complex numbers. The operations of vector addition and scalar multiplication must satisfy the vector space axioms.
The concrete vector spaces, which are relevant for this website, are:
- Quantum states
- Displacements in 2D or 3D physical space
- Component vectors, which are used to encode the above-mentioned things
Quantum States
Quantum states are written in Dirac ket notation. A general quantum state is usually denoted \(\myket{\psi}\).
If we want to denote a specific state, we write a ket with a specific label inside. For example, for a single photon system, we write \(\myket{H}\) for a horizontally polarized photon and \(\myket{V}\) for a vertically polarized photon. For a two-photon-system, we write for example \(\myket{VH}\) for a state, where photon 1 is polarized vertically and photon 2 is polarized horizontally.
In the context of quantum computation, one often uses certain physical states to encode abstract logical values (0 and 1). One could, for example, define to use \(\myket{H}\) to represent a logical 0, and \(\myket{V}\) to represent a logical 1. After this decision was made, the states are usually written as \(\myket{0}\) and \(\myket{1}\). Of course, this can also be applied to multi-particle states. For example, the above-mentioned two-photon state would then be written as \(\myket{10}\).
Linear Combinations of Basis States
Quantum states can be treated as vectors. So one can create new quantum states by computing linear combinations of given quantum states. For example, if \(\myket{H}\) and \(\myket{V}\) are quantum states, then \(\sqrt{50\%} \myket{H} + \sqrt{50\%} \myket{V}\) is also a quantum state.
One can define a set of so-called basis states, such that every possible quantum state can be expressed as a linear combination of these basis states. For example, for a one-photon system, every state can be expressed in the following form, where \(a_0\) and \(a_1\) are complex numbers.
\[ \myket{\psi} = a_0 \myket{H} + a_1 \myket{V} \]
In this context, the factors \(a_0\) and \(a_1\) are called amplitudes.
Standard Bases and Component Vectors
Once we have defined a certain ordered set of basis vectors, we only need the corresponding amplitudes to represent the state. These amplitudes are often written as a so-called component vector. For example, if we choose the ordered set \(\{\{ \myket{H}, \myket{V} \}\}\) as our standard basis, the above-mentioned state \(\myket{\psi}\) can be represented by the following component vector \(\psi\):
\[ \psi = \begin{pmatrix} a_0 \\ a_1 \end{pmatrix} \]
The standard bases used on this website are the following:
- For one-photon-systems: \(\{\{ \myket{H}, \myket{V} \}\}\)
- For two-photon-systems: \(\{\{ \myket{HH}, \myket{HV}, \myket{VH}, \myket{VV} \}\}\)
- For one-qbit-systems: \(\{\{ \myket{0}, \myket{1} \}\}\)
- For two-qbit-systems: \(\{\{ \myket{00}, \myket{01}, \myket{10}, \myket{11} \}\}\)
Transformation Matrices
Once we have defined an ordered set of basis vectors, we can also introduce matrices to define transformations of states. For example, the following matrix transforms a one-photon state by introducing a 180° phase shift of the \(a_1\) component.
\[ Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \]
Such a matrix is applied to a component vector by multiplying it from the left. The resulting component vector represents the new state.
The Conjugate Transpose
There is one important operation to calculate with component vectors and transformation matrices: the conjugate transpose.
If \(X\) is a complex number, vector, or matrix, the conjugate transpose \(X^*\) is obtained by transposing the item (in case it is a vector or matrix), and then applying complex conjugation to each element.
This operation may also be called the star operation.
Here are some examples:
\[ ( r e^{i \varphi} ) ^* = r e^{-i \varphi} \]
\[ \begin{pmatrix} a_0 \\ a_1 \end{pmatrix} ^* = \begin{pmatrix} a_0^* & a_1^* \end{pmatrix} \]
\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} ^* = \begin{pmatrix} a^* & c^* \\ b^* & d^* \end{pmatrix} \]
Please note that there are many different names and notations for the star operation. It is also called “hermitian conjugate”, “hermitian transpose”, “adjoint”, “transjugate”, or “dagger”. And it is also written \(X^H\) or \(X^\dagger\).
Data Types of Variables
To understand a mathematical formula, it is essential to know the “data types” of the contained variables. To make this easier, there is usually a convention which letter or symbol is used for which kind of data. Here are the conventions used on this website.
| Symbol | Meaning / data type |
|---|---|
| \(r, p\) | A real number larger or equal to zero |
| \(\alpha, \beta, \varphi, \theta, \tau\) | An angle, usually between 0° and 360° |
| \(a, b, c, d, z\) | A complex number |
| \(\myket{\psi}, \myket{H}, \myket{V}\) … | A quantum-mechanical state vector |
| \(\psi\) | A complex component vector representing a quantum state |
| \(P, R, X, Y, Z, H\) | A transformation matrix consisting of complex numbers |
| \(\vec{x}, \vec{y}, \vec{z}\) | A unit vector in 2D or 3D space |
Not all letters are used as variables. Here are the symbols which denote specific mathematical constants:
| Symbol | Meaning |
|---|---|
| \(e\) | Euler’s number, a real number with value 2.71828… |
| \(i\) | A shortcut for \(e^{i \cdot 90°}\) called imaginary unit |