Dynamical Decomposition of Bilinear Control Systems Subject to Symmetries
Tóm tắt
We describe a method to analyze and decompose the dynamics of a bilinear control system subject to symmetries. The method is based on the concept of generalized Young symmetrizers of representation theory. It naturally applies to the situation where the system evolves on a tensor product space and there exists a finite group of symmetries for the dynamics which interchanges the various factors. This is the case for quantum mechanical multipartite systems, such as spin networks, where each factor of the tensor product represents the state of one of the component systems. We present several examples of application.
Tài liệu tham khảo
Abramowitz M, Stegun IA. Handbook of Mathematical Functions. New York: Dover Publications; 1965.
Alcock-Zeilinger J, Weigert H. Compact Hermitian projection operators. J. Math. Phys. 2017;58(5):051702.
Albertini F, D’Alessandro D. Controllability of symmetric spin networks. J Math Phys 2018;59:052102.
Altafini C. Controllability of quantum mechanical systems by root space decomposition of su(n). J Math Phys 2002;43(5):2051–2062.
Arenz C, Gualdi G, Burgarth D. Control of open quantum systems: case study of the central spin model. New J Phys 2014;16:065023.
Butkovskiy AG, Samoilenko YI. 1990. Control of Quantum-Mechanical Processes and Systems, Mathematics and its Applications 56, Kluwer Academic Publisher.
Chen J, Zhou H, Duan C, Peng X. 2017. Preparing GHZ and W states on a long-range Ising spin model by global control. Physical Review A.
D’Alessandro D. Introduction to Quantum Control and Dynamics. Boca Raton: CRC Press; 2007.
D’Alessandro D. 2010. Constructive decomposition of the controllability Lie algebra for Quantum systems. IEEE Trans Autom Control, pp 1416–1421.
de Graaf WA, Algebras Lie. 2000. Theory and Algorithms. Amsterdam the netherlands: North holland.
Dixon JD. 1967. Problems in Group Theory, Dover Publications 2007, Mineola N. Y., reprinted from Blaidshell Publishing Company, Waltham, MA.
Dür W, Vidal G, Cirac JI. Three qubits can be entangled in two inequivalent ways. Phys Rev. A 2000;62:062314.
Greenberger DM, Horne MA, Zeilinger A. Going beyond Bell’s theorem. Bell’s Theorem, Quantum Theory and the Conceptions of the Universe. Dordrecht: Kluwer Academics; 1989. p. 73–76.
Fulton W, Harris J, Vol. 129. Representation Theory; a First Course Graduate Texts in Mathematics. New York: Springer; 2004.
Gauthier JP, Kupka I, Sallet G. Controllability of right invariant systems on real simple Lie groups. Syst Control Lett 1984;5(3):187–190.
Goodman R, Wallach NR. 2009. Symmetry, Representations and Invariants, Springer Graduate Texts in Mathematics.
Isaacs IM. Character Theory of Finite Groups. New York: Dover; 1976.
Keppeler S, Sjödal M. Hermitian Young Operators. J Math Phys 2014;55: 021702.
Jurdjevic V. 1996. Geometric control theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press.
Jurdjević V, Sussmann H. Control systems on Lie groups. J Differ Equations 1972;12:313–329.
Lloyd S. Almost any quantum logic gate is universal. Phys Rev Lett 1995;75:346.
Long CT. Elementary Introduction to Number Theory, 2nd ed. Lexington: D. C. Heath and Company; 1972.
Polack T, Suchowski H, Tannor D. Uncontrollable quantum systems: A classification scheme based on Lie subalgebras. Phys Rev A 2009;79:053403.
Rotman J. 1995. An introduction to the theory of groups. Springer-Verlag.
Sengupta AN. 2012. Representing finite groups; A semisimple introduction. Springer.
Serre J-P. 1977. Linear Representation of Finite Groups, Graduate texts in Mathematics, No. 42.
Tung WK. Group Theory in Physics. Singapore: World Scientific; 1985.
Wang X, Burgarth D, Schirmer SG. Subspace controllability of spin \(\frac {1}{2}\) chains with symmetries. Phys Rev A 2016;94:052319.
Wang X, Pemberton-Ross P, Schirmer SG. Symmetry and controllability for spin networks with a single node control. IEEE Trans Autom Control 2012;57(8): 1945–1956.
Woit P. 2017. Quantum theory, Groups and Representations. Springer.
Zeier R, Schulte-Herbruggen T. Symmetry principles in quantum systems theory. J Math Phys 2011;52:113510.
Zimborás Z, Zeier R, Schulte-Herbrüggen T, Burgarth D. Symmetry criteria for quantum simulability of effective interactions. Phys Rev A 2015;92:042309.