Quantum Information#

Qubits#

The fundamental unit of information for classical computers is the bit, which can be one of two numbers - zero or one. In contrast, the fundamental unit of information for quantum computers are complex vectors.

Definition: Qubit

A quantum bit (qubit) is a two-dimensional complex vector whose Euclidean norm is \(1\).

Bra-Ket Notation

Qubits are always expressed via bra-ket notation.

Note: Euclidean Norm

The condition that the Euclidean norm must be equal to \(1\) reflects a fundamental truth about the physical nature of quantum systems.

Notation: Common States

Four common qubit states have reserved names and notations:

Name	Notation	State
Zero	\(\left\vert 0 \right\rangle\)	\(\begin{bmatrix}1 \\ 0\end{bmatrix}\)
One	\(\left\vert 1 \right\rangle\)	\(\begin{bmatrix}0 \\ 1\end{bmatrix}\)
Plus	\(\left\vert + \right\rangle\)	\(\begin{bmatrix}\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}\end{bmatrix}\)
Minus	\(\left\vert - \right\rangle\)	\(\begin{bmatrix}\frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}}\end{bmatrix}\)
i	\(\left\vert \mathrm{i} \right\rangle\)	\(\begin{bmatrix}\frac{1}{\sqrt{2}} \\ \mathrm{i}\frac{1}{\sqrt{2}}\end{bmatrix}\)
Minus i	\(\left\vert -\mathrm{i} \right\rangle\)	\(\begin{bmatrix}\frac{1}{\sqrt{2}} \\ -\mathrm{i}\frac{1}{\sqrt{2}}\end{bmatrix}\)

Bloch Sphere#

A single qubit can be visualized intuitively as an arrow pointing from the origin to a point on the so-called Bloch sphere.

Quantum Registers#

Definition: Quantum Register

A quantum register is a sequence of qubits.

Definition: Quantum State

The state of a quantum register with \(n\) qubits \(\left\vert q_{n-1} \right\rangle, \dotsc, \left\vert q_{0} \right\rangle\) or \(\left\langle q_{n-1} \right\vert, \dotsc, \left\langle q_{0} \right\vert\) is their Kronecker product:

\[ \left\vert q_{n-1} \right\rangle \otimes \dotsc \otimes \left\vert q_{0} \right\rangle \qquad \text{or} \qquad \left\langle q_{n-1} \right\vert \otimes \dotsc \otimes \left\langle q_{0} \right\vert \]

Notation

We use some shorthand notations for such products:

\[ \begin{aligned} \left\vert q_{n-1} \right\rangle \left\vert q_{n-2} \right\rangle \cdots \left\vert q_{0} \right\rangle &\qquad \left\langle q_{n-1} \right\vert \left\langle q_{n-2} \right\vert \cdots \left\langle q_{0} \right\vert \\ \left\vert q_{n-1}, q_{n-2}, \dotsc, q_{0} \right\rangle &\qquad \left\langle q_{n-1}, q_{n-2}, \dotsc, q_{0} \right\vert \\ \left\vert q_{n-1} q_{n-2} \cdots q_{0} \right\rangle &\qquad \left\langle q_{n-1} q_{n-2} \cdots q_{0} \right\vert \end{aligned} \]

Example

The state \(\left\vert 10 \right\rangle\) is

\[ \left\vert 10 \right\rangle = \left\vert 1 \right\rangle \left\vert 0 \right\rangle = \left\vert 1 \right\rangle \otimes \left\vert 0 \right\rangle = \begin{bmatrix}0 \cdot \begin{bmatrix}1 \\ 0\end{bmatrix} \\ 1 \cdot \begin{bmatrix}1 \\ 0\end{bmatrix}\end{bmatrix} = \begin{bmatrix}0 \\ 0 \\ 1 \\ 0\end{bmatrix} \]

The state of an \(n\)-qubit quantum register is a complex vector of dimension \(2^n\).

Notation

The most common label for quantum states is the Greek letter \(\psi\):

\[ \left\langle \psi \right\vert \qquad \left\vert \psi \right\rangle \]

Computational Bases#

Definition: Computational Basis

An \(n\)-dimensional computational basis is any orthonormal basis for the vector space \(\mathbb{C}^n\) or \(\mathbb{C}^{1 \times n}\).

Example

The states \(\left\vert 0 \right\rangle\) and \(\left\vert 1 \right\rangle\) are a computational basis for \(\mathbb{C}^2\). Similarly, so are \(\left\vert + \right\rangle\) and \(\left\vert - \right\rangle\) as well as \(\left\vert \mathrm{i} \right\rangle\) and \(\left\vert -\mathrm{i} \right\rangle\).

The states \(\left\langle 00 \right\vert\), \(\left\langle 01 \right\vert\), \(\left\langle 10 \right\vert\) and \(\left\langle 11 \right\vert\) are a computational basis for \(\mathbb{C}^{1\times 4}\).

Suppose we have some \(n\)-dimensional quantum state \(\left\vert \psi \right\rangle\) and a computational basis \(\left\vert b_1 \right\rangle, \dotsc, \left\vert b_n \right\rangle\) such that \(\left\vert \psi \right\rangle\) has the representation

\[ \left\vert \psi \right\rangle = \sum_{k = 1}^n c_k \left\vert b_k \right\rangle = c_1 \left\vert b_1 \right\rangle + \cdots c_n \left\vert b_n \right\rangle, \]

where \(c_1, \dotsc, c_k \in \mathbb{C}\).

Definition: Pure State

We say that \(\left\vert \psi \right\rangle\) is a pure state if it is equal to one of \(\left\vert b_1 \right\rangle, \dotsc, \left\vert b_n \right\rangle\).

Definition: Superposition (Mixed State)

We say that \(\left\vert \psi \right\rangle\) is a mixed state or a superposition of \(\left\vert b_1 \right\rangle, \dotsc, \left\vert b_n \right\rangle\) if it is not equal to any of \(\left\vert b_1 \right\rangle, \dotsc, \left\vert b_n \right\rangle\).

Example

The state \(\left\vert + \right\rangle\) is a superposition of \(\left\vert 0 \right\rangle\) and \(\left\vert 1 \right\rangle\). The state \(\left\vert 1 \right\rangle\) is a superposition of \(\left\vert \mathrm{i} \right\rangle\) and \(\left\vert -\mathrm{i} \right\rangle\).

The state \(\left\vert -+ \right\rangle\) is a superposition of \(\left\vert 00 \right\rangle\), \(\left\vert 01 \right\rangle\), \(\left\vert 10 \right\rangle\) and \(\left\vert 11 \right\rangle\).

Notation: Special Bases

A few computational bases have been given names:

\[ \begin{aligned} Z &\overset{\text{def}}{=} \{\left\vert 0\right\rangle, \left\vert 1\right\rangle\} \\ X &\overset{\text{def}}{=} \{\left\vert -\right\rangle, \left\vert +\right\rangle\} \\ Y &\overset{\text{def}}{=} \{\left\vert -\mathrm{i}\right\rangle, \left\vert \mathrm{i} \right\rangle\} \end{aligned} \]

The definitions are analogous for states in \(\mathbb{C}^{1 \times n}\).

Entanglement#

Definition: Product State

An \(n\)-bit quantum state \(\left\vert \Psi \right\rangle \in \mathbb{C}^{2^n}\) is a product state or a separable state if there exist \(n\) qantum states \(\left\vert v_1 \right\rangle, \dotsc, \left\vert v_n \right\rangle \in \mathbb{C}^{2}\) whose Kronecker product is \(\left\vert \Psi \right\rangle\):

\[ \left\vert \Psi \right\rangle = \bigotimes_{k = 1}^n \left\vert v_k \right\rangle = \left\vert v_1 \right\rangle \otimes \cdots \otimes \left\vert v_n \right\rangle \]

Definition: Entangled State

A quantum state is entangled if it is not a product state.

Entangled states are peculiar because measuring some of the qubits of a quantum register in an entangled state directly influences the other.