Dagger Compact Closed Categories and Completely Positive Maps: (Extended Abstract)
By Peter Selinger. QPL, 2007
472 citations recorded by the source corpus.
Dagger compact closed categories were recently introduced by Abramsky and Coecke, under the name ''strongly compact closed categories'', as an axiomatic framework for quantum mechanics. We present a graphical language for dagger compact closed categories, and sketch a proof of its completeness for equational reasoning. We give a general construction, the CPM construction, which associates to each dagger compact closed category its ''category of completely positive maps'', and we show that the resulting category is again dagger compact closed. We apply these ideas to Abramsky and Coecke's interpretation of quantum protocols, and to D'Hondt and Panangaden's predicate transformer semantics.
Subfields: Artificial Intelligence, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/b7db387f8d985640444208f671d0fc63f95c6e1a
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Artificial Intelligence
- Miriam Backens
- Stephen Clark
- Bob Coecke
- Ross Duncan
- Mehrnoosh Sadrzadeh
- Dusko Pavlovic
- Tarmo Uustalu
- Thorsten Altenkirch
In Computational Theory and Mathematics
- Stephen Clark
- Dusko Pavlovic
- Tarmo Uustalu
- Mehrnoosh Sadrzadeh
- Ross Duncan
- Bob Coecke
- Miriam Backens
- Edward Grefenstette
Citations in the corpus
- Mathematical Foundations for a Compositional Distributional Model of Meaning (2010, background): “• Our categorical axiomatics is flexible enough to accommodate mixed states [37], so in principle we are able to study their linguistic significance, and for instance implement the proposals of [3].”
- Quantum picturalism (2009, background): “3: [37, 41] The graphical calculus for dagger”
- The ZX-calculus is complete for stabilizer quantum mechanics (2013, background): “Selinger shows in [17] that this graphical calculus is indeed equivalent to the equational reasoning in dagger compact closed categories, which are the category theoretical framework for quantum mechanics.”
- Monads need not be endofunctors (2010, background): “, in the context of quantum computing we are interested in dagger compact closed categories [9, 10].”
- A Survey of Graphical Languages for Monoidal Categories (2009, unknown): “• A dagger symmetric monoidal category [33] is a dagger braided monoidal category such that the unitary braiding is a symmetry.”
- Interacting quantum observables: categorical algebra and diagrammatics (2009, methodology): “The categorical version of quantum mechanics has become an active area of research, some of the key papers being [1,2,37,15].”
- Interacting Quantum Observables (2008, background): “Within graphical calculus for †-SMCs (see [12]) we depict the morphisms δ and by and , and their adjoints, δ† and † by and , When taking the monoidal structure to be strict—which we will do throughout this paper— classical structures obey the following remarkable theorem [4].”
- Quantum measurements without sums (2006, methodology): “Selinger provides in [31] a multiplicative treatment of density matrices and completely positive maps.”
- Categories in Control (2014, unknown): “nding each linear relation to its adjoint extends to a contravariant functor †: FinRelk →FinRelk, which obeys a list of properties that are summarized by saying that FinRelk is a ‘†-compact’ category [1, 18]. Because two of the operations in the Frobenius monoid (k,+,0,+†,0†) are adjoints of the oth”