Extending quantum operations

Teiko  Heinosaari; Maria A.  Jivulescu; David  Reeb; Michael M.  Wolf

doi:10.1063/1.4755845

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.

ACKNOWLEDGMENTS

The authors thank Francesco Buscemi, Keiji Matsumoto, and David Pérez-García for very valuable discussions. T.H. acknowledges support from the Academy of Finland (Grant No. 138135). M.A.J. would like to thank for financial support the European project QUEVADIS, the Rectorate of University Politehnica Timişoara, and the strategic grant POSDRU/21/1.5/G/13798 co-financed by the European Social Fund - Investing in People, within the Sectorial Operational Programme Human Resources Development 2007–2013. D.R. and M.M.W. acknowledge support from the European projects COQUIT and QUEVADIS, the CHIST-ERA/BMBF project CQC, and from the Alfried Krupp von Bohlen und Halbach-Stiftung.

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