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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 21st 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexbrief entry _[[superoperator]]_, just for completeness

   brief entry superoperator, just for completeness

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 12th 2023
   • (edited Sep 12th 2023)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added pointer to: * J. K. Vizzotto, [[Thorsten Altenkirch]], A. Sabry, *Structuring quantum effects: superoperators as arrows*, Mathematical Structures in Computer Science **16** 3 (2006) 453-468 &lbrack;[arXiv:quant-ph/0501151](https://arxiv.org/abs/quant-ph/0501151), [doi:10.1017/S0960129506005287](https://doi.org/10.1017/S0960129506005287)&rbrack; What I was looking for is the answer to the following question: In a compact closed category, "superoperators" $$ \big(\mathscr{H} \multimap \mathscr{H}\big) \longrightarrow \big(\mathscr{K} \multimap \mathscr{K}\big) $$ hence $$ \mathscr{H} \otimes \mathscr{H}^\ast \longrightarrow \mathscr{K} \otimes \mathscr{K}^\ast $$ may equivalently be understood as $$ \mathscr{H} \otimes \mathscr{K} \longrightarrow \mathscr{H} \otimes \mathscr{K} $$ and then be composed as such. Is there a standard name for this re-identification and/or the resulting notion of composition of superoperators? <a href="https://ncatlab.org/nlab/revision/diff/superoperator/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/superoperator/3">v3</a>, <a href="https://ncatlab.org/nlab/show/superoperator">current</a>

   I have added pointer to:

   • J. K. Vizzotto, Thorsten Altenkirch, A. Sabry, Structuring quantum effects: superoperators as arrows, Mathematical Structures in Computer Science 16 3 (2006) 453-468 [arXiv:quant-ph/0501151, doi:10.1017/S0960129506005287]

   What I was looking for is the answer to the following question:

   In a compact closed category, “superoperators”

   $$\big(\mathscr{H} \multimap \mathscr{H}\big) \longrightarrow \big(\mathscr{K} \multimap \mathscr{K}\big)$$

   hence

   $$\mathscr{H} \otimes \mathscr{H}^\ast \longrightarrow \mathscr{K} \otimes \mathscr{K}^\ast$$

   may equivalently be understood as

   $$\mathscr{H} \otimes \mathscr{K} \longrightarrow \mathscr{H} \otimes \mathscr{K}$$

   and then be composed as such. Is there a standard name for this re-identification and/or the resulting notion of composition of superoperators?

   diff, v3, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeSep 12th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded to the entry ([here](https://ncatlab.org/nlab/show/superoperator#Details)) a few lines with a tad more details <a href="https://ncatlab.org/nlab/revision/diff/superoperator/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/superoperator/4">v4</a>, <a href="https://ncatlab.org/nlab/show/superoperator">current</a>

   added to the entry (here) a few lines with a tad more details

   diff, v4, current