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Pseudo-unitary freedom in the operator-sum representation, positive maps, and quantum error correction

Mark Byrd, Southern Illinois University

(Session 10b : Saturday from 4:00pm-4:30pm)

Abstract. A dynamical map is a map which takes one density operator to another. Such a map can be written in an operator-sum representation (OSR) using a spectral decomposition. The method of the construction applies to more general maps which need not be completely positive. The OSR is not unique; there is a freedom to choose a different set of operators in the OSR, yet still obtain the same map. Here we show that, whereas the freedom for completely positive maps is unitary, the freedom for maps which are not necessarily completely positive is pseudo-unitary. Those that are genuinely different must therefore differ by the number and type of spectral decomposition. Moreover, our theorem enables us to prove a necessary and sufficient condition for error correcting codes which can correct errors due to maps which are not completely positive.