Normal completely positive maps on the space of quantum operations
arXiv:1012.3197 · doi:10.1142/S1230161213500030
Abstract
Quantum supermaps are higher-order maps transforming quantum operations into quantum operations. Here we extend the theory of quantum supermaps, originally formulated in the finite dimensional setting, to the case of higher-order maps transforming quantum operations with input in a separable von Neumann algebra and output in the algebra of the bounded operators on a given separable Hilbert space. In this setting we prove two dilation theorems for quantum supermaps that are the analogues of the Stinespring and Radon-Nikodym theorems for quantum operations. Finally, we consider the case of quantum superinstruments, namely measures with values in the set of quantum supermaps, and derive a dilation theorem for them that is analogue to Ozawa's theorem for quantum instruments. The three dilation theorems presented here show that all the supermaps defined in this paper can be implemented by connecting devices in quantum circuits.
47 pages (in one-column format), including new results about quantum operations on separable von Neumann algebras
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- Symmetries of quantum evolutions
- Semantics for a Quantum Programming Language by Operator Algebras
- Quantum Supermaps are Characterized by Locality
- Quantum and classical dynamical semigroups of superchannels and semicausal channels
- A Profunctorial Semantics for Quantum Supermaps