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Quantum systems carry information. Quantum theory supports at least twodistinct kinds of information (classical and quantum), and a variety ofdifferent ways to encode and preserve information in physical systems. Asystem's ability to carry information is constrained and defined by the noisein its dynamics. This paper introduces an operational framework, usinginformation-preserving structures to classify all the kinds of information thatcan be perfectly (i.e., with zero error) preserved by quantum dynamics. Weprove that every perfectly preserved code has the same structure as a matrixalgebra, and that preserved information can always be corrected. We alsoclassify distinct operational criteria for preservation (e.g., "noiseless","unitarily correctible", etc.) and introduce two new and natural criteria formeasurement-stabilized and unconditionally preserved codes. Finally, forseveral of these operational critera, we present efficient (polynomial in thestate-space dimension) algorithms to find all of a channel'sinformation-preserving structures.

Information preserving structures: A general framework for quantum zero-error information