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Quantum error correction on infinite-dimensional Hilbert spaces. (English) Zbl 1216.81055
Summary: We present a generalization of quantum error correction to infinite-dimensional Hilbert spaces. We find that, under relatively mild conditions, much of the structure known from systems in finite-dimensional Hilbert spaces carries straightforwardly over to infinite dimensions. We also find that, at least in principle, there exist qualitatively new classes of quantum error correcting codes that have no finite-dimensional counterparts. We begin with a shift of focus from states to algebras of observables. Standard subspace codes and subsystem codes are seen as the special case of algebras of observables given by finite-dimensional von Neumann factors of type I. The new classes of codes that arise in infinite dimensions are shown to be characterized by von Neumann algebras of types II and III, for which we give in-principle physical examples.
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81P70 Quantum coding (general)
46L60 Applications of selfadjoint operator algebras to physics
81R15 Operator algebra methods applied to problems in quantum theory
94B60 Other types of codes
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