Nov 15, 2020

How to read the bra–ket notation?

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:

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

See also

On the nature of the wave function

The wave function is an abstract mathematical concept and cannot be "measured" directly. So what is it then?

The Hadamard gate

The definition of the Hadamard gate and some of its properties.

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