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Imagine doing mechanics without a precise notion of time, or thermodynamics without a definition of temperature. There is a huge recent upspring of "complex systems" research, with research institutes, journals and conferences devoted to it. Yet, there is no commonly agreed notion of what actually is "complexity". Can one give an operational definition of what is complexity, so that one can at least decide objectively and unambiguously whether a human is more complex than a bacterium? Or at least more complex than a stone? In my talk I want to give a review of attempts made during the last 30 years to define "complexity" in such a way that it agrees with the intuitive notion shared by most natural scientists. It will turn out that there are close connections to similar notions in computer science ("complexity of an algorithm") and in information theory ("algorithmic complexity"). There are subtleties which make it very unlikely that the above questions can ever be answered in the affirmative, but existing definitions of complexity can be useful when restricted to more narrowly limited problems.

In this talk I'll discuss recent joint work with Raymond Laflamme, David Poulin and Maia Lesosky in which a unified approach to quantum error correction is presented, called "operator quantum error correction". This scheme relies on a generalized notion of noiseless subsystems and includes the known techniques for the error correction of quantum operations --i.e., the standard model, the method of decoherence-free subspaces, and the noiseless subsystem method--as special cases. Correctable codes in this approach take the form of operator algebras and operator semigroups. The condition from the standard model is shown to be necessary for all of the known methods of error correction, and we'll see this as part of a discussion on conditions that characterize correctability for the general case.