HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information
Skip to Main content Skip to Navigation
Conference papers

Completeness of Graphical Languages for Mixed States Quantum Mechanics

Titouan Carette 1 Simon Perdrix 1 Renaud Vilmart 1 Emmanuel Jeandel 1
1 MOCQUA - Designing the Future of Computational Models
Inria Nancy - Grand Est, LORIA - FM - Department of Formal Methods
Abstract : There exist several graphical languages for quantum information processing, like quantum circuits, ZX-Calculus, ZW-Calculus, etc. Each of these languages forms a †-symmetric monoidal category (†-SMC) and comes with an interpretation functor to the †-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 †-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.
Complete list of metadata

Cited literature [44 references]  Display  Hide  Download

Contributor : Renaud Vilmart Connect in order to contact the contributor
Submitted on : Tuesday, February 19, 2019 - 6:07:15 PM
Last modification on : Thursday, April 28, 2022 - 4:12:03 PM
Long-term archiving on: : Monday, May 20, 2019 - 5:18:51 PM


Files produced by the author(s)




Titouan Carette, Simon Perdrix, Renaud Vilmart, Emmanuel Jeandel. Completeness of Graphical Languages for Mixed States Quantum Mechanics. ICALP, 2019, Patras, Greece. ⟨10.4230/LIPIcs.ICALP.2019.108⟩. ⟨hal-02025720⟩



Record views


Files downloads