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Completeness of Graphical Languages for Mixed States Quantum Mechanics (Track B: Automata, Logic, Semantics, and Theory of Programming)

Authors Titouan Carette, Emmanuel Jeandel , Simon Perdrix , Renaud Vilmart

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Author Details

Titouan Carette
  • Université de Lorraine, CNRS, Inria, LORIA, F 54000 Nancy, France
Emmanuel Jeandel
  • Université de Lorraine, CNRS, Inria, LORIA, F 54000 Nancy, France
Simon Perdrix
  • Université de Lorraine, CNRS, Inria, LORIA, F 54000 Nancy, France
Renaud Vilmart
  • Université de Lorraine, CNRS, Inria, LORIA, F 54000 Nancy, France

Funding

This work is funded by ANR-17-CE25-0009 SoftQPro and PIA-GDN/Quantex.

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Titouan Carette, Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. Completeness of Graphical Languages for Mixed States Quantum Mechanics (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 108:1-108:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.108

Abstract

There exist several graphical languages for quantum information processing, like quantum circuits, ZX-Calculus, ZW-Calculus, etc. Each of these languages forms a dagger-symmetric monoidal category (dagger-SMC) and comes with an interpretation functor to the dagger-SMC 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 dagger-symmetric 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing
  • Theory of computation → Quantum computation theory
  • Theory of computation → Logic
Keywords
  • Quantum Computing
  • Quantum Categorical Mechanics
  • Category Theory
  • Mixed States
  • Completely Positive Maps

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