Finite and Profinite Quantum Systems This monograph provides an introduction to finite quantum systems a field at the interface between quantum information and number theory with applications in quantum computation and condensed matter physics. The first major part of this monograph studies the so called `qubits and `qudits systems with periodic finite lattice as position space. It also discusses the so called mutually unbiased bases which have applications in quantum information and quantum cryptography. Quantum logic and its applications to quantum gates is also studied. The second part studies finite quantum systems where the position takes values in a Galois field. This combines quantum mechanics with Galois theory. The third part extends the discussion to quantum systems with variables in profinite groups considering the limit where the dimension of the system becomes very large. It uses the concepts of inverse and direct limit and studies quantum mechanics on p adic numbers. Applications of the formalism include quantum optics and quantum computing two dimensional electron systems in magnetic fields and the magnetic translation group the quantum Hall effect other areas in condensed matter physics and Fast Fourier Transforms. The monograph combines ideas from quantum mechanics with discrete mathematics algebra and number theory. It is suitable for graduate students and researchers in quantum physics mathematics and computer science. Mathematical symbols 1 Introduction 2 Partial orders and Pontryagin duality 3 The ring ℤ (d) 4 Quantum systems with variables in ℤ (d) 5 Finite Geometries and Mutually Unbiased Bases 6 Quantum logic of finite quantum systems 7 Galois fields 8 Quantum systems with variables in GF(pe) 9 p adic numbers and profinite groups 10 A quantum system with positions in the profinite group ℤp 11 A quantum system with positions in the profinite group ℤ Index.