This is the appendix and the notation for the Quantum Computing series.
Vector expressed as a ket |v⟩
Vector expressed as a bra〈v|
In component form:
Hilbert space is a vector space with inner product and norm defined and amplitudes belong to complex numbers. (Vector space is just an abstraction of what we learn about vectors in linear algebra.)
The Dirac notation |0⟩ is just a shorthand of:
The result of the inner product is a scalar. The result is independent of which computational bases are used to encode |u⟩ and |v⟩.
The state space of a composite physical system is the tensor product of the state spaces.
The definition of the tensor product is:
Here is another example of a system composed of two states.
Quantum superpositions (Qubits)
The vector form for a 3-qubits:
If the vectors are orthogonal (like the bases),