\[
\newcommand{\BB}{\mathbb{B}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\HH}{\mathbb{H}}
\newcommand{\KK}{\mathbb{K}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\ii}{\mathrm{i}}
\newcommand{\jj}{\mathrm{j}}
\newcommand{\kk}{\mathrm{k}}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\FT}{\mathcal{F}} % Fourier Transform
\newcommand{\tto}{\twoheadrightarrow}
\newcommand{\inv}{^{-1}}
\newcommand{\RF}{\mathrm{RF}}
\newcommand{\sys}{\mathrm{sys}}
\newcommand{\diag}{\mathrm{diag}}
\newcommand{\cx}{\mathrm{CX}}
\newcommand{\cy}{\mathrm{CY}}
\newcommand{\cz}{\mathrm{CZ}}
\newcommand{\cat}{\ket{\mathrm{cat}}}
\newcommand{\catp}[1]{\ket{\mathrm{cat}_{#1}}}
\newcommand{\calE}{\mathcal{E}}
\newcommand{\calF}{\mathcal{F}}
\newcommand{\calH}{\mathcal{H}}
\newcommand{\calR}{\mathcal{R}}
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\norm}[1]{\|#1\|}
\newcommand{\sprod}[2]{\langle#1|#2\rangle} % deprecated, use braket instead.
\newcommand{\braket}[2]{\langle#1|#2\rangle} % scalar product
\newcommand{\ptrace}[2]{\mathrm{tr}_{#1}\left(#2\right)}
\newcommand{\trace}[1]{\mathrm{tr}\left(#1\right)}
\newcommand{\rank}[1]{\mathrm{rank}\left(#1\right)}
\newcommand{\floor}[1]{\lfloor#1\rfloor}
\newcommand{\ceil}[1]{\lceil#1\rceil}
\newcommand{\bra}[1]{\langle#1|}
\newcommand{\ket}[1]{|#1\rangle}
\newcommand{\proj}[1]{\ket{#1}\bra{#1}}
\newcommand{\mean}[1]{\langle#1\rangle}
\newcommand{\wt}[1]{\mathrm{wt}\left(#1\right)}
\newcommand{\prob}[1]{\mathrm{Prob}\left[#1\right]}
\newcommand{\orac}{\mathrm{Orac}}
\newcommand{\?}{} % sometimes I need just a separator other than whitespace
\]
For two hermitian operators we write:
\[ A \leq B \]
if \(\langle\psi|B - A|\psi\rangle \geq 0\) for all \(|\psi\rangle\). In the special case that one
operator is just a multiple of the Identity, e.g. \(A=aI\), we write: