Bra-Ket Notation

The standard mathematical notation for quantum states, invented by Dirac, using angle brackets and vertical bars.

Bra-ket notation (also called Dirac notation) is the standard language for writing quantum states and operations. Once you learn it, quantum mechanics becomes much easier to read and write.

The Basics

Kets |⟩

A ket $|⟩$ represents a quantum state (a column vector):

$$|0⟩=\left(\begin{array}{c}1\\ 0\end{array}\right),|1⟩=\left(\begin{array}{c}0\\ 1\end{array}\right)$$

$$|\psi ⟩=\alpha |0⟩+\beta |1⟩=\left(\begin{array}{c}\alpha \\ \beta \end{array}\right)$$

You can put any label inside: $|0⟩$, $|1⟩$, $|\psi ⟩$, $|+⟩$, $|\text{cat}⟩$, etc.

Bras ⟨|

A bra $⟨|$ is the conjugate transpose of a ket (a row vector):

$$⟨\psi |=|\psi {⟩}^{†}=({\alpha }^{\ast }{\beta }^{\ast })$$

where ${\alpha }^{\ast }$ is the complex conjugate of $\alpha$.

Operations

Inner Product (Bra-Ket)

Combining a bra and a ket gives a number:

$$⟨\varphi |\psi ⟩=\text{inner product (a complex number)}$$

This is also called a bracket (bra + ket = bracket).

For computational basis states: $$⟨0|0⟩=1,⟨1|1⟩=1,⟨0|1⟩=0,⟨1|0⟩=0$$

Outer Product (Ket-Bra)

Combining a ket and a bra gives an operator (matrix):

$$|0⟩⟨1|=\left(\begin{array}{c}1\\ 0\end{array}\right)(01)=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$$

Common Notation

Multi-Qubit States

For multiple qubits, we use tensor products:

$$|0⟩\otimes |1⟩=|01⟩=|0,1⟩$$

All three notations mean the same thing. The tensor product symbol $\otimes$ is often omitted.

Why This Notation?

1. Compact: $|\psi ⟩$ is shorter than writing column vectors
2. Flexible: Works in any dimension, any basis
3. Intuitive: Inner products $⟨\varphi |\psi ⟩$ look like what they compute
4. Standard: Universal in quantum physics and computing

Pronunciation

• $|0⟩$: “ket zero”
• $⟨\psi |$: “bra psi”
• $⟨\varphi |\psi ⟩$: “bra phi ket psi” or “phi inner psi” or “bracket phi psi”

See also: Quantum State, Hilbert Space, Density Matrix