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A single classical system is characterized by its manifold of states; and to combine several systems, we take the product of manifolds. A single quantum system is characterized by its Hilbert space of states; and to combine several systems, we take the tensor product of Hilbert spaces. But what if we choose to combine an infinite number of systems? A naive attempt to describe such combinations fails, for there is apparently no natural notion of an infinite product of manifolds; nor of an infinite tensor product of Hilbert spaces.

The Problem of Time in Quantum Gravity and Cosmology

There are two notions that play a central role in the mathematical theory of computation. One is that of a computable problem, i.e., of a problem that can, in principle, be solved by an (idealized) computer. It is known that there exist problems that \'have answers\', but for which those answers are not computable. The other is that of the difficulty of a computation, i.e. of the number of (idealized) steps required actually to carry out that computation.

Teleportation, quantum key distribution, and quantum algorithms.