A1 Refereed original research article in a scientific journal
Extending quantum operations
Authors: Heinosaari T, Jivulescu MA, Reeb D, Wolf MM
Publisher: AMER INST PHYSICS
Publication year: 2012
Journal: Journal of Mathematical Physics
Journal name in source: JOURNAL OF MATHEMATICAL PHYSICS
Journal acronym: J MATH PHYS
Article number: ARTN 102208
Number in series: 10
Volume: 53
Issue: 10
Number of pages: 29
ISSN: 0022-2488
DOI: https://doi.org/10.1063/1.4755845
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/3697752
Abstract
For a given set of input-output pairs of quantum states or observables, we ask the question whether there exists a physically implementable transformation that maps each of the inputs to the corresponding output. The physical maps on quantum states are trace-preserving completely positive maps, but we also consider variants of these requirements. We generalize the definition of complete positivity to linear maps defined on arbitrary subspaces, then formulate this notion as a semidefinite program, and relate it by duality to approximative extensions of this map. This gives a characterization of the maps which can be approximated arbitrarily well as the restriction of a map that is completely positive on the whole algebra, also yielding the familiar extension theorems on operator spaces. For quantum channel extensions and extensions by probabilistic operations we obtain semidefinite characterizations, and we also elucidate the special case of Abelian inputs or outputs. Finally, revisiting a theorem by Alberti and Uhlmann, we provide simpler and more widely applicable conditions for certain extension problems on qubits, and by using a semidefinite programming formulation we exhibit counterexamples to seemingly reasonable but false generalizations of the Alberti-Uhlmann theorem. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4755845]
For a given set of input-output pairs of quantum states or observables, we ask the question whether there exists a physically implementable transformation that maps each of the inputs to the corresponding output. The physical maps on quantum states are trace-preserving completely positive maps, but we also consider variants of these requirements. We generalize the definition of complete positivity to linear maps defined on arbitrary subspaces, then formulate this notion as a semidefinite program, and relate it by duality to approximative extensions of this map. This gives a characterization of the maps which can be approximated arbitrarily well as the restriction of a map that is completely positive on the whole algebra, also yielding the familiar extension theorems on operator spaces. For quantum channel extensions and extensions by probabilistic operations we obtain semidefinite characterizations, and we also elucidate the special case of Abelian inputs or outputs. Finally, revisiting a theorem by Alberti and Uhlmann, we provide simpler and more widely applicable conditions for certain extension problems on qubits, and by using a semidefinite programming formulation we exhibit counterexamples to seemingly reasonable but false generalizations of the Alberti-Uhlmann theorem. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4755845]
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