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Quantum states are usually described using bra–ket notation, also known as Dirac notation.

In Dirac notation, a ket $∣v⟩$ represents a state of a quantum system and is a vector in a complex vector space $C_{n}$. A bra $⟨f∣$ denotes a linear map $f:C_{n}→C$, and the application of $⟨f∣$ to $∣v⟩$ is written as $⟨f∣v⟩$.

The ket $∣v⟩$ can be represented as a column vector:

$∣v⟩=⎣⎢⎢⎢⎢⎡ v_{1}v_{2}⋮v_{n} ⎦⎥⎥⎥⎥⎤ ,$

and the bra $⟨f∣$ as a row vector:

$⟨f∣=[f_{1} f_{2} ⋯ f_{n} ].$

In this representation, a bra next to a ket simply denotes matrix multiplication of a row vector with a column vector:

$⟨f∣v⟩=[f_{1} f_{2} ⋯ f_{n} ]⎣⎢⎢⎢⎢⎡ v_{1}v_{2}⋮v_{n} ⎦⎥⎥⎥⎥⎤ =∑_{i=1}f_{i}v_{i}.$

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