Finite and Profinite Quantum Systems, Softcover reprint of the original 1st ed. 2017 Quantum Science and Technology Series

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.

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(p^e).

9 p-adic numbers and profinite groups.

11 A quantum system with positions in the profinite group ℤ.

Index.

Provides a unique overview of finite quantum systems in a pedagogical form suitable for advanced students and researchers alike, presenting theory complemented with examples

Offers an introduction to the field at a time of current interest and emerging applications in quantum information and related areas

Takes an interdisciplinary approach combining advanced topics in mathematics and condensed matter and quantum physics

Includes supplementary material: sn.pub/extras