Dagger Compact Closed Categories and Completely Positive Maps
Dagger Compact Closed Categories and Completely Positive Maps
Selinger uses the CPM construction to extend dagger compact semantics from pure states to completely positive maps and mixed-state quantum computation.
Paper information (2007, ENTCS; early version at QPL 2005) — Peter Selinger. DOI; author page
Work
The paper studies how to turn a dagger compact category for pure quantum processes into a model for mixed states and noisy processes. Its central CPM construction maps a dagger compact category to a category of completely positive maps, bringing density operators, mixed states, and open-system processes into one compositional and graphical semantics.
Contributions
- A CPM construction from pure-state categories to categories of completely positive maps.
- An extension of dagger compact protocol semantics to mixed-state quantum computation.
- A compositional and graphical account of completely positive processes.
- An abstract basis for noise, discarding, and open-system semantics in quantum programming.
Limits
The CPM construction is a semantic construction, not noise calibration, a hardware compiler, or a performance model. A concrete category, scalar interpretation, and classical structure are still needed for a particular implementation; expressive categorical structure does not by itself provide an efficient algorithm or a machine-checked proof.
Connections
The paper extends the pure-protocol perspective of A Categorical Semantics of Quantum Protocols and provides important background for the completely positive maps used in Denotational Semantics. It can also be compared with QML: QML focuses on language constructs, while CPM focuses on extending pure-state semantics to mixed-state processes.
Original paper
Peter Selinger, Dagger Compact Closed Categories and Completely Positive Maps, Electronic Notes in Theoretical Computer Science 170, 139–163 (2007), DOI: 10.1016/j.entcs.2006.12.018. Author page