Quantum complementarity and operator structures (pp0067-0083)
          David W. Kribs, Jeremy Livick, Mike I. Nelson, Rajesh Perira, and Mizanur Rahaman
          doi: https://doi.org/10.26421/QIC19.1-2-5
Abstracts: We establish operator structure identities for quantum channels and their error-correcting and private codes, emphasizing the complementarity relationship between the two perspectives. Relevant structures include correctable and private operator algebras, and operator spaces such as multiplicative domains and nullspaces of quantum channels and their complementary maps. For the case of privatizing to quantum states, we also derive dimension inequalities on the associated operator algebras that further quantify the trade-off between correction and privacy.
Key words: completely positive map, quantum channel, complementary maps, operator algebra, operator system, private quantum code, quantum error correcting code, private algebra, correctable algebra, multiplicative domain