In quantum mechanics, "bra" and "ket" are two types of mathematical vectors used to represent quantum states in a vector space called a Hilbert space. The terms "bra" and "ket" are derived from the words "bracket" and "ket," respectively, and they represent the two different orientations of vectors.

Ket Vector |ψ⟩: A "ket" vector, denoted by |ψ⟩, is used to represent a quantum state in a Hilbert space. It is a column vector with complex coefficients, and each element of the vector corresponds to a specific state in the quantum system. The elements of the ket vector can be expressed in Dirac notation as |ψ⟩ = [ψ₁, ψ₂, ψ₃, ..., ψ_n], where ψᵢ are the complex coefficients.
For example, if we have a two-level quantum system (qubit), |0⟩ and |1⟩ could be the basis states, and a quantum state |ψ⟩ in this system would be represented as |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex coefficients.
Bra Vector ⟨ψ|: A "bra" vector, denoted by ⟨ψ|, is the complex conjugate transpose of a ket vector. It is a row vector, and it represents the dual vector in the Hilbert space. The elements of the bra vector are the complex conjugates of the corresponding elements of the ket vector. It can be expressed as ⟨ψ| = [ψ₁*, ψ₂*, ψ₃*, ..., ψ_n*], where * represents the complex conjugate.
For the quantum state example |ψ⟩ = α|0⟩ + β|1⟩, the bra vector corresponding to |ψ⟩ would be ⟨ψ| = α*⟨0| + β*⟨1|, where ⟨0| and ⟨1| are the dual basis vectors (row vectors) corresponding to the basis states |0⟩ and |1⟩.

The bra-ket notation, also known as Dirac notation, is a powerful and elegant way to represent quantum states and perform various operations in quantum mechanics, such as computing inner products, calculating probabilities, and describing quantum measurements and transformations. The inner product of a bra and a ket vector, ⟨ψ|ϕ⟩, gives the probability amplitude for transitioning from one state |ϕ⟩ to another state |ψ⟩ in the Hilbert space. The use of bra and ket vectors allows for a clear and concise representation of quantum states and their mathematical manipulations.

The scalar product of two kets |ψ⟩ and |ϕ⟩ is defined as ⟨ψ|ϕ⟩ = ∫ ψ*(x) ϕ(x) dx

The ket and bra correspondence is antilinear

Given two sets |ϕ⟩ and |ψ⟩. Their linear combination is also a new ket.

𝜆1 |ϕ⟩ + 𝜆2 |ψ⟩ is a new set.

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