Abstract
There exist several graphical languages for quantum information processing, like quantum circuits, ZXCalculus, ZWCalculus, etc. Each of these languages forms a daggersymmetric monoidal category (daggerSMC) and comes with an interpretation functor to the daggerSMC of (finite dimension) Hilbert spaces. In the recent years, one of the main achievements of the categorical approach to quantum mechanics has been to provide several equational theories for most of these graphical languages, making them complete for various fragments of pure quantum mechanics.
We address the question of the extension of these languages beyond pure quantum mechanics, in order to reason on mixed states and general quantum operations, i.e. completely positive maps. Intuitively, such an extension relies on the axiomatisation of a discard map which allows one to get rid of a quantum system, operation which is not allowed in pure quantum mechanics.
We introduce a new construction, the discard construction, which transforms any daggersymmetric monoidal category into a symmetric monoidal category equipped with a discard map. Roughly speaking this construction consists in making any isometry causal.
Using this construction we provide an extension for several graphical languages that we prove to be complete for general quantum operations. However this construction fails for some fringe cases like the Clifford+T quantum mechanics, as the category does not have enough isometries.
BibTeX  Entry
@InProceedings{carette_et_al:LIPIcs:2019:10684,
author = {Titouan Carette and Emmanuel Jeandel and Simon Perdrix and Renaud Vilmart},
title = {{Completeness of Graphical Languages for Mixed States Quantum Mechanics}},
booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
pages = {108:1108:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771092},
ISSN = {18688969},
year = {2019},
volume = {132},
editor = {Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10684},
URN = {urn:nbn:de:0030drops106844},
doi = {10.4230/LIPIcs.ICALP.2019.108},
annote = {Keywords: Quantum Computing, Quantum Categorical Mechanics, Category Theory, Mixed States, Completely Positive Maps}
}
Keywords: 

Quantum Computing, Quantum Categorical Mechanics, Category Theory, Mixed States, Completely Positive Maps 
Collection: 

46th International Colloquium on Automata, Languages, and Programming (ICALP 2019) 
Issue Date: 

2019 
Date of publication: 

04.07.2019 