Bra-Ket Notation

Bra-ket notation is a mathematical notation commonly used in Quantum Computing to represent quantum states and the operations on them. It was introduced by Paul Dirac and is widely used to describe vectors in Hilbert space, which is the mathematical framework for quantum mechanics.

Components of Bra-Ket Notation

• Bra (⟨ϕ|): A bra represents the conjugate transpose (Hermitian conjugate) of a quantum state. It is written as ⟨ϕ|, where ϕ is the name of the state. The bra is the complex conjugate of the ket vector, and it is used to describe the dual vector in the Hilbert space.
• Ket (|ψ⟩): A ket represents a quantum state as a vector in a Hilbert space. The ket is written as |ψ⟩, where ψ is the name of the state. For example, |0⟩ could represent the ground state of a quantum system, and |1⟩ could represent the first excited state.

Inner Product

The inner product (also called the scalar product) between two quantum states |ψ⟩ and |ϕ⟩ is written as:

⟨ϕ|ψ⟩

This represents the amplitude of transitioning from the state |ψ⟩ to the state |ϕ⟩. The result is a complex number.

Outer Product

The outer product between two quantum states |ψ⟩ and |ϕ⟩ is written as:

|ψ⟩⟨ϕ|

This represents an operator, which can act on other quantum states.

Examples

• Inner product: The inner product between |0⟩ and |1⟩ is ⟨0|1⟩ = 0, since the states are orthogonal.
• Outer product: The outer product |0⟩⟨1| represents an operator that maps the state |1⟩ to |0⟩.

Bra-ket notation simplifies many calculations in quantum mechanics and allows for easy representation of quantum states, operators, and their relationships.