%
\newcommand\spinvec{{\bf\sigma}}
\newcommand\spina{\sigma_a}
\newcommand\spinb{\sigma_b}
\newcommand\spinc{\sigma_c}
\newcommand\spinx{\sigma_x}
\newcommand\spiny{\sigma_y}
\newcommand\spinz{\sigma_z}
%
\newcommand\unita{{\bf\hat a}}
\newcommand\unitb{{\bf\hat b}}
\newcommand\unitc{{\bf\hat c}}
\newcommand\unitz{{\bf\hat z}}
%
\newcommand{\Ham}{\mathcal{H}}
%
\newcommand\identity{\mathbb{I}}
%
\newcommand\lorentz{
  \sqrt{1 - \beta^2} }
%
%
\newcommand{\G}{G{\"o}del}
\newcommand{\na}{na{\"i}ve}
\newcommand{\Sch}{Schr{\"o}dinger}
%
%
\newcommand{\ket}[1]{|\,#1\,\rangle}
\newcommand{\bra}[1]{\langle\, #1\,|}
\newcommand{\braket}[2]{\langle #1|#2\rangle}
\newcommand{\ketbra}[2]{\ket{#1}\bra{#2}}
\newcommand{\amplitude}[3]{\langle\,#1\, |\, #2\, |\,#3\,\rangle}
\newcommand{\proj}[2]{\ket{#1}\bra{#2}}
\newcommand{\expect}[1]{\langle\, #1\, \rangle}
\newcommand{\xvec}{\mathbf{x}}
%
%
%\newcommand{\curl}[1]{\text{curl \textbf{#1}}}
%\newcommand{\div}[1]{\text{div \textbf{#1}}}
%
%
\newcommand{\twobyone}[2]{
  \begin{pmatrix}
    #1 \\
    #2
  \end{pmatrix}
   }
\newcommand{\onebytwo}[2]{
  \begin{pmatrix}
    #1 & #2
  \end{pmatrix}
  }
\newcommand{\threebyone}[3]{
  \begin{pmatrix}
   #1 & #2 & #3
  \end{\pmatrix}
  }
\newcommand{\onebythree}[3]{
  \begin{pmatrix}
    #1 \\
    #2 \\
    #3
  \end{pmatrix}
  }
\newcommand{\twobytwo}[4]{
  \begin{pmatrix}
    #1 & #2 \\
    #3 & #4
  \end{pmatrix}
  }
\newcommand{\threebythree}[9]{
  \begin{pmatrix}
    #1 & #2 & #3 \\
    #4 & #5 & #6 \\
    #7 & #8 & #9
  \end{pmatrix}
  }
\newcommand{\CNOT}{
  \begin{pmatrix}
   1  &  0  &  0  &  0  \\
   0  &  1  &  0  &  0  \\
   0  &  0  &  0  &  1  \\
   0  &  0  &  1  &  0
  \end{pmatrix}
  }
\newcommand{\Bphi}{
  \begin{pmatrix}
    1  &  0  &  0  &  0 \\
    0  &  1  &  0  &  0 \\
    0  &  0  &  1  &  0 \\
    0  &  0  &  0  &  e^{i\phi}
  \end{pmatrix}
  }
\newcommand{\X}{\twobytwo{0}{1}{1}{0}  }
\newcommand{\Y}{\twobytwo{0}{-i}{i}{0}}
\newcommand{\Z}{\twobytwo{1}{0}{0}{-1}}
\newcommand{\Id}{\twobytwo{1}{0}{0}{1}}
\newcommand{\Had}{\frac{1}{\sqrt{2}}\twobytwo{1}{1}{1}{-1}}
\newcommand{\shift}{\twobytwo{1}{0}{0}{e^{i\phi}}}
\newcommand{\norm}{\frac{1}{\sqrt{2}}}
%
%
\newcommand{\Bell}[2]{\ket{\beta_{#1 #2}}}
\newcommand{\Bellbra}[2]{\bra{\beta_{#1 #2}}}
\newcommand{\Bellamp}{\braket{\beta_{ab}}{\beta_{cd}}}
%