Dirac Notation: The Language of Quantum States

Dirac notation is compact and expressive for quantum mechanics and quantum computing.

Kets and Bras

• Ket: $|\psi\rangle$, a column vector.
• Bra: $\langle\psi|$, the conjugate transpose row vector.

Inner product:

$$\langle\phi|\psi\rangle$$

Outer product:

$$|\phi\rangle\langle\psi|$$

Basis States

For one qubit:

$$|0\rangle = \begin{bmatrix}1 \\ 0\end{bmatrix}, \quad |1\rangle = \begin{bmatrix}0 \\ 1\end{bmatrix}$$

Any state is a linear combination of basis kets.

Operators and Expectation

An observable or gate is an operator $A$ acting on kets:

$$A|\psi\rangle$$

Expectation value in state $|\psi\rangle$:

$$\langle A \rangle = \langle\psi|A|\psi\rangle$$

Multi Qubit States

Tensor products combine systems:

$$|ab\rangle = |a\rangle \otimes |b\rangle$$

Examples:

• $|00\rangle$
• $|01\rangle$
• $|10\rangle$
• $|11\rangle$

Dirac notation keeps these expressions readable as systems scale.

Projectors and Completeness

Projectors extract components along basis directions. For computational basis:

$$P_0 = |0\rangle\langle0|, \quad P_1 = |1\rangle\langle1|$$

They satisfy:

$$P_0 + P_1 = I$$

This identity (completeness) is why probabilities sum to 1 in projective measurement.

Change of Basis in Bra-Ket Form

Suppose $\{|u_k\rangle\}$ is another orthonormal basis. Any state can be expanded as:

$$|\psi\rangle = \sum_k |u_k\rangle\langle u_k|\psi\rangle$$

The coefficient $\langle u_k|\psi\rangle$ is exactly the amplitude in that basis. This is the compact Dirac way to express basis transforms used in many algorithms.

Tensor Product Ordering Matters

For multi qubit systems, qubit ordering is part of the definition. For example, $|01\rangle$ can mean qubit 0 is $|0\rangle$ and qubit 1 is $|1\rangle$ (or the opposite), depending on convention.

When reading circuit outputs, always verify endianness used by the framework or paper.