Finite-Dimensional and Discrete Quantum Systems  

Qubits > s.a. Rebits; decoherence.
* Idea: A qubit is a 2-state system, a quantum system with a 2D Hilbert space; The term was coined by Benjamin Schumacher in 1992, and has become the conceptual tool needed to make progress in quantum computing [@ history, sn(17)jul].
* Properties: Density matrices for 1 qubit are in 1-1 correspondence with points of the 3D solid ball, the Bloch sphere.
* Example: A theoretical example is the two-level atom, and in practice trapped 43Ca+ ions seem to work well (2014).
@ One qubit: Urbantke AJP(91)jun [phases and holonomy]; Slater qp/97 [statistical thermodynamics], qp/00 [and information theory]; Ralph et al FP(98) [solution]; Sassaroli AJP(99)oct [neutrino oscillations]; Bagrov et al JPA(01)qp [V(t)]; Barata & Cortez PLA(02)qp [periodic driving]; An et al JOB(04)qp/05 [coupled to squeezed vacuum field]; Maioli & Sacchetti JSP(05) [+ stochastic perturbation]; Gemmer & Michel PhyE(05)qp [+ environment]; Kato et al qp/06-conf [Holevo capacity from Voronoi diagrams]; Calmet & Calmet PLA(12)-a1201 [in quantum field theory]; Kiktenko & Korotaev PS(13) [coupled to a mixed quantum field state]; Gómez et al a1604 [a qubit is more than a quantum coin]; Liss et al a1812 [topological order]; Amao & Castillo a2001 [geometric algebra approach]; Brandenburger et al a2010 [entropic uncertainty principle]; > s.a. particles; relativistic quantum mechanics.
@ One qubit, in curved spacetime: Palmer et al AP(12); Viennot & Moro a1609 [adiabatic transport].
@ Two qubits: Kummer IJTP(01); Abouraddy et al PRA(01) [decomposition and entanglement]; Avron et al JMP(07); Güngördü et al PRA(12)-a1205 [dynamical invariants of two-level systems]; Santos & Semião PRA(14)-a1311 [+ environment, master equation]; > s.a. composite systems; examples of entanglement.
@ N qubits: Wootters qp/03-conf [generalized Wigner function]; Rigetti et al QIP(04)qp/03 [and information]; Klimov et al JPA(12)-a1007 [phase space]; Coecke & Duncan NJP(11) [graphical calculus]; Giorgi et al PRL(11)-a1108 [three qubits, quantum and classical correlations]; Li et al PRA(14)-a1406 [classifying multi-qubit quantum states]; Rau a2103 [symmetries and geometries].
@ Realizations: Kim Phy(14); > s.a. quantum computing.

Other Types and General Discrete Systems > s.a. Clifford Operators; Discrete Models; Ehrenfest Theorem; wigner functions.
* Qutrits: A qutrit is a three-state quantum system; For example, a single photon passing through a system of three slits, a "Young-type" qutrit.
* Qudits: A qudit is a quantum system with d-dimensional Hilbert space.
* Spin-J system: In the Majorana representation, a state of a spin-J system can be seen as a set of 2J points on the Bloch sphere.
@ General references: Gudder FP(06) [and finite group theory]; Hassan & Joag JPA(07) [combinatorial approach]; Lenz & Veselić MZ(09)-a0709 [discrete Hamiltonians].
@ Relationship with continuum ones: Ruzzi & Galetti JPA(00); Ruzzi JPA(02)qp/01, & Galetti JPA(02); Barker JPA(01), JMP(01) [continuum limits]; Brukner et al PRA(03)qp/02; Kornyak in(09)-a0906 [gauge invariance and continuum limit]; 't Hooft FP(14).
@ Three-level: Sánchez JPA(94); Slater JGP(01)qp/00 [Bures geometry]; Rau & Zhao PRA(05)qp [complete treatment]; Chruściński & Wudarski OSID(11)-a1105, Chruściński & Sarbicki OSID(13)-a1108 [entanglement witnesses]; Jarvis JPA(14)-a1312 [mixed two-qutrit system].
@ One spin: Bruno PRL(12)-a1204 + Niu Phy(12)jun [spin-J system, Majorana's stellar representation]; > s.a. quantum spin models [more spins]; types of quantum states.
@ Other discrete systems: de la Torre & Goyeneche AJP(03)jan-qp/02; Kornyak PPN(13)-a1208; Planat IJGMP(11) [four and eight-level systems]; Domenech et al RPMP(11) [two-valued states over orthomodular lattices]; Vourdas JMP(12) [partial order and T0 topology]; Hanson et al JPA(14) [finite-field frameworks]; van Wonderen & Suttorp a1808 [a thermal bath]; > s.a. cellular automaton; classical systems; graph theory; modified quantum mechanics [discrete underlying space].

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