2010/12/31 by Robert Zeier, Thomas Schulte‐Herbrüggen, T. Schulte-Herbrueggen
Chemistry · Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Applied mathematics #Combinatorics #Controllability #Homogeneous space #Irreducibility #Irreducible representation #Lattice (music) #Lie algebra #Mathematics #Molecular spectroscopy and chirality #Physics #Pure mathematics #Quantum #Quantum mechanics #Rank (graph theory) #Simple (philosophy) #Symmetry (geometry) #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1063/1.3657939
published as J. Math. Phys. 52, 113510 (2011) · For complete table with all irreducible simple subalgebras of su(N) [N up to 2^15] download ancillary file; v2: further results on spin chains with limited local control, minor corrections, and updated references; v3: minor amendments in Sec. 10.6, Appendix C more reader-friendly (see C.3, C.4), electronic supplement list corrected in 2 cases (dim=560, 4928) out of 32768 instances
arxiv created 2011/07/21 · openalex publication_date 2011/11/01 · arxiv updated 2011/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
General dynamic properties such as controllability and simulability of spin systems, fermionic and bosonic systems are investigated in terms of symmetry. Symmetries may be due to the interaction topology or due to the structure and representation of the system and control Hamiltonians. In either case, they obviously entail constants of motion. Conversely, the absence of symmetry implies irreducibility and provides a convenient necessary condition for full controllability much easier to assess than the well-established Lie-algebra rank condition. We give a complete lattice of irreducible simple subalgebras of \documentclass[12pt]minimal\begindocument\mathfrak su(2n)\enddocumentsu(2n) for up to n = 15 qubits. It complements the symmetry condition by allowing for easy tests solving homogeneous linear equations to filter irreducible representations of other candidate algebras of classical type as well as of exceptional types. Moreover, here we give the first single necessary and sufficient symmetry condition for full controllability. The lattice of irreducible simple subalgebras given also determines mutual simulability of dynamic systems of spin or fermionic or bosonic nature. We illustrate how controlled quadratic fermionic (and bosonic) systems can be simulated by spin systems and in certain cases also vice versa.