  We present a multiresolution approach to the theory of quantum information.
It arose from an effort to develop a systematic mathematical approach to the
analysis of an infinite array of qubits, i.e., a structure that may be
interpreted as a quantum metamaterial. Foundational to our approach are two
mathematical constructions with classical roots: the Borel isomorphism and the
Haar basis. Here, these constructions are intertwined to establish an
identification between $L_2(0,1]$ and the Hilbert space of an infinite array of
qubits and to enable analysis of operators that act on arrays of qubits (either
finite or infinite). The fusion of these two concepts empowers us to represent
quantum operations and observables through geometric operators. As an
unexpected upshot, we observe that the fundamental concept of calculus is
inherent in an infinite array of qubits; indeed, the antiderivative arises as a
natural and indispensable operator in this context.
