0 Nomenclature and notation¶

0.0 Linear algebra and quantum mechanics¶

A positive operator \(A\) is one for which \(\expval{A}{\psi} \geq 0\) for all \(\ket\psi\).
A positive definite operator \(A\) is one for which \(\expval{A}{\psi} > 0\) for all \(\ket\psi \neq 0\).
The support of an operator is defined to be the vector space orthogonal to its kernel.
Hermitian operator: the vector space spanned by its eigenvectors has non-zero eigenvalues.
\(U \rightarrow\) unitary operator or matrix.
For two-level quantum systems used as qubits, we shall usually identify the state \(\ket0\rightarrow(1,0)\) and \(\ket1\rightarrow(0,1)\).
We use the notations \(I, X, Y, Z\) to denote \(\sigma_0, \sigma_1, \sigma_2, \sigma_3\) respectively.

Probability distribution refers to a finite set of real numbers, \(p_x\), such that \(p_x \geq 0\) and \(\Sigma_xp_x = 1\).
The relative entropy of a positive operator \(A\) with respect to another positive operator \(B\) is defined by:

\[ S(A\|B) \equiv \tr(A\log A) - \tr(A\log B) \]