bracket with doubled delimiters

Question

I want to typeset Hilbert-Schmidt-Products as the Liouville-Equivalent of the Hilbert Space Braket, with doubled delimiters, like this: \Lbraket{\rho|\rho} results in <<rho|rho>>.

I was trying to adapt the definition of \Braket from the Braket package to do this, ending up with the following code:

{\catcode`\|=\active                                                             
  \xdef\Lbraket{\protect\expandafter\noexpand\csname Lbraket \endcsname}         
  \expandafter\gdef\csname Lbraket \endcsname#1{\begingroup                      
     \ifx\SavedDoubleVert\relax                                                  
       \let\SavedDoubleVert\|\let\|\BraDoubleVert                                
     \fi                                                                         
     \mathcode`\|32768\let|\BraVert                                              
     \left\langle\kern-0.2em\left\langle{#1}\right\rangle\kern-0.2em\right\rangle\endgroup}
}
\def\BraVert{\@ifnextchar|{\|\@gobble}% turn || into \|                          
     {\egroup\,\mid@vertical\,\bgroup}}                                          
\def\BraDoubleVert{\egroup\,\mid@dblvertical\,\bgroup}                           
\let\SavedDoubleVert\relax
\begingroup                                                                      
 \edef\@tempa{\meaning\middle}                                                   
 \edef\@tempb{\string\middle}                                                    
\expandafter \endgroup \ifx\@tempa\@tempb                                        
 \def\mid@vertical{\middle|}                                                     
 \def\mid@dblvertical{\middle\SavedDoubleVert}                                   
\else                                                                            
 \def\mid@vertical{\mskip1mu\vrule\mskip1mu}                                     
 \def\mid@dblvertical{\mskip1mu\vrule\mskip2.5mu\vrule\mskip1mu}                 
\fi

This works, but for very large formulas the double delimiters are not sufficiently tight, and the negative kerning should be larger.

On the other hand, I've found some code that allows to define double delimiters and doesn't seem to suffer from the same problem. The following is from a newsgroup posting, it defines \lleft and \rright so that e.g. \lleft\langle results in <<.

\catcode`@=11                                                                    
\def\lleft#1#2\rright{\begingroup%                                               
   \def\ts@r{\nulldelimiterspace=0pt \mathsurround=0pt}%                         
   \let\@hat=#1%                                                                 
   \def\sht@im{#2}%                                                              
   \def\@t{{\mathchoice{\def\@fen{\displaystyle}\k@fel}%                         
          {\def\@fen{\textstyle}\k@fel}%                                         
          {\def\@fen{\scriptstyle}\k@fel}%                                       
          {\def\@fen{\scriptscriptstyle}\k@fel}}}%                               
   \def\g@rin{\ts@r\left\@hat\vphantom{\sht@im}\right.}%                         
   \def\k@fel{\setbox0=\hbox{$\@fen\g@rin$}\hbox{%                               
      $\@fen \kern.3875\wd0 \copy0 \kern-.3875\wd0%                              
      \llap{\copy0}\kern.3875\wd0$}}%                                            
      \def\pt@h{\mathopen\@t}\pt@h\sht@im%                                       
      \rright}%                                                                  
\def\rright#1{\let\@hat=#1%                                                      
   \def\st@m{\mathclose\@t}%                                                     
   \st@m\endgroup}                                                               
\catcode`@=12

My question is if there's a way to combine to two code snippets to get a correctly resizing Braket with doubled delimiters. Simply doing the following does not work:

{\catcode`\|=\active                                                             
  \xdef\LBraket{\protect\expandafter\noexpand\csname LBraket \endcsname}         
  \expandafter\gdef\csname LBraket \endcsname#1{\begingroup                      
     \ifx\SavedDoubleVert\relax                                                  
       \let\SavedDoubleVert\|\let\|\BraDoubleVert                                
     \fi                                                                         
     \mathcode`\|32768\let|\BraVert                                              
     \lleft\langle{#1}\rright\rangle\endgroup}
}

This results in an error message:

! Extra \middle.
\mid@vertical ->\middle |

My knowledge of lowlevel TeX is not anywhere near sufficient to understand how the implementation of \lleft and \rright works, but maybe someone sees whether the two snippets can be combined.

Answer 1

A modification of the very clever \lleft macro can do

\documentclass[a4paper]{article}
\makeatletter
\def\LBlleft#1#2|#3\LBrright{\begingroup
   \def\ts@r{\nulldelimiterspace=0pt \mathsurround=0pt}%
   \let\@hat=#1%
   \def\sht@im{\ts@r\left.#2\nonscript\;\middle|\nonscript\;#3\right.}%
   \def\@t{{\mathchoice{\def\@fen{\displaystyle}\k@fel}%
          {\def\@fen{\textstyle}\k@fel}%
          {\def\@fen{\scriptstyle}\k@fel}%
          {\def\@fen{\scriptscriptstyle}\k@fel}}}%
   \def\g@rin{\ts@r\left\@hat\vphantom{\sht@im}\right.}%
   \def\k@fel{\setbox0=\hbox{$\@fen\g@rin$}\hbox{%
      $\@fen \kern.3875\wd0 \copy0 \kern-.3875\wd0
      \llap{\copy0}\kern.3875\wd0$}}%
      \def\pt@h{\mathopen\@t}\pt@h\sht@im
      \LBrright}%
\def\LBrright#1{\let\@hat=#1%
   \def\st@m{\mathclose\@t}%
   \st@m\endgroup}
\DeclareRobustCommand\LBraket[1]{\LBlleft<#1\LBrright>}
\makeatother
\begin{document}
$\displaystyle\LBraket{\frac{a}{b}|b}$
$\textstyle\LBraket{\frac{a}{b}|b}$
$\scriptstyle\LBraket{\frac{a}{b}|b}$
$\scriptscriptstyle\LBraket{\frac{a}{b}|b}$
\end{document}

ADDED (Support for multiple bars)

\makeatletter

\def\LB@add@@{\aftergroup\LB@add@}
\def\LB@add@{\aftergroup\LB@add}
\def\LB@add{\advance\@tempcnta\@ne}
\def\LB@barsl{\ifcase\@tempcnta\or
  \kern-\nulldelimiterspace\left.\or
  \kern-2\nulldelimiterspace\left.\left.\fi}
\def\LB@barsr{\ifcase\@tempcnta\or
  \right.\kern-\nulldelimiterspace\or
  \right.\right.\kern-2\nulldelimiterspace\fi}

\def\LBlleft#1#2\LBrright{\begingroup
  % count the number of bars
  \mathcode`|="8000
  \begingroup\lccode`~=`| \lowercase{\endgroup\def~}{\aftergroup\LB@add@@}
  \@tempcnta=\z@\setbox\z@=\hbox{$\left<#2\right>$}
  % now \@tempcnta has the number of bars and we redefine |
  \begingroup\lccode`~=`| \lowercase{\endgroup\def~}{\nonscript\;\middle|\nonscript\;}
  \def\ts@r{\nulldelimiterspace=0pt \mathsurround=0pt}%
  \let\@hat=#1%
  \def\sht@im{\LB@barsl#2\LB@barsr}%
  \def\@t{{\mathchoice{\def\@fen{\displaystyle}\k@fel}%
          {\def\@fen{\textstyle}\k@fel}%
          {\def\@fen{\scriptstyle}\k@fel}%
          {\def\@fen{\scriptscriptstyle}\k@fel}}}%
   \def\g@rin{\ts@r\left\@hat\vphantom{\sht@im}\right.}%
   \def\k@fel{\setbox0=\hbox{$\@fen\g@rin$}\hbox{%
      $\@fen \kern.3875\wd0 \copy0 \kern-.3875\wd0
      \llap{\copy0}\kern.3875\wd0$}}%
      \def\pt@h{\mathopen\@t}\pt@h\sht@im
      \LBrright}%
\def\LBrright#1{\let\@hat=#1%
   \def\st@m{\mathclose\@t}%
   \st@m\endgroup}
\DeclareRobustCommand\LBraket[1]{\LBlleft<#1\LBrright>}
\makeatother

It supports at most two bars, but if needed one can extend the macros \LB@barsl and \LB@barsr.