Extending quantum operations

TL;DR

This paper investigates the conditions under which quantum operations can be extended from given input-output pairs, providing semidefinite programming characterizations and generalizing classical extension theorems in quantum information theory.

Contribution

It introduces a generalized notion of complete positivity on subspaces, formulates it as a semidefinite program, and applies it to quantum channel extensions and the Alberti-Uhlmann theorem.

Findings

Semidefinite programming characterizations of quantum extensions

Generalized extension theorems for operator spaces

Counterexamples to false generalizations of Alberti-Uhlmann

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 in- 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.