Notation

Below is a brief list of the notation used in this book. In general, the Dirac notation is used for abstract vectors and operators that are expressed in a coordinate free manner, and traditional matrix algebra notation is used when a vector or an operator is expressed in terms of coordinates of a specific basis.

N is the dimension of a Hilbert space

a, b, c, x, y, z are scalars which can be complex numbers

X, Y, P, Q are matrices

diag[X] is a diagonal matrix formed from the N × 1 column matrix X

α, β, γ are often used to represent N × 1 column matrices of amplitudes

αi is one coordinate value of α; that is, a single amplitude

X† is the Hermitian transpose of X

X−1 is the inverse of the full rank matrix X

If α is an N × 1 column matrix, then α† is an 1 × N row matrix of conjugate values

(α† · β) is the inner product of two N × 1 column matrices (α, β)

ψ · φ† is the outer product matrix of two N × 1 column matrices (ψ, φ)

Tr[X] is the trace of the square matrix X

X ⊗ Y is the Kronecker product of two matrices

V = {|Vi⟩, i = 1, N} orthonormal basis, or W = {|Wi⟩, i = 1, N} for another one

|X⟩ is an abstract vector; it can be represented by a N × 1 matrix α once you choose a basis