Entanglement in Bipartite Quantum Systems with Fast Local Unitary Control
Abstract: The well-known Schmidt decomposition, or equivalently, the complex singular value decomposition, states that a pure quantum state of a bipartite system can always be brought into a "diagonal" form using local unitary transformations. In this work we consider a finite-dimensional closed bipartite system with fast local unitary control. In this setting one can define a reduced control system on the singular values of the state which is equivalent to the original control system. We explicitly describe this reduced control system and prove equivalence to the original system. Moreover, using the reduced control system, we prove that the original system is controllable and stabilizable and we deduce quantum speed limits. We also treat the fermionic and bosonic cases in parallel, which are related to the Autonne-Takagi and Hua factorization respectively.
- Control Theory from the Geometric Viewpoint. Encyclopaedia of Mathematical Sciences. Springer, Heidelberg, 2004.
- Differential Inclusions: Set-Valued Maps and Viability Theory. Springer, Berlin Heidelberg, 1984.
- Autonne, L. Sur les Matrices Hypohermitiennes et sur les Matrices Unitaires. Ann. Univ. Lyon Nouvelle Ser. 38, 1 (1915).
- Geometry of Quantum States: An Introduction to Quantum Entanglement, 2nd ed. Cambridge University Press, Cambridge, 2017.
- Quantum Cryptography: Public Key Distribution and Coin Tossing. Theoretical Computer Science 560 (2014), 7–11. Theoretical Aspects of Quantum Cryptography – celebrating 30 years of BB84.
- D’Alessandro, D. Introduction to Quantum Control and Dynamics, 2nd ed. Chapman and Hall/CRC, New York, 2021.
- Quantum Sensing. Rev. Mod. Phys. 89 (2017), 035002.
- Lie Theory for Quantum Control. GAMM-Mitteilungen 31 (2008), 59–93.
- Reachable Sets from Toy Models to Controlled Markovian Quantum Systems. Proc. IEEE Conf. Decision Control (IEEE-CDC) 58 (2019), 2322.
- Elliott, D. Bilinear Control Systems: Matrices in Action. Springer, London, 2009.
- Quantum Cryptography. Rev. Mod. Phys. 74 (2002), 145–195.
- Helgason, S. Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York, 1978.
- Matrix Analysis, 2nd ed. Cambridge University Press, Cambridge, 2012.
- Hua, L. K. On the Theory of Automorphic Functions of a Matrix Variable. I: Geometrical Basis. Amer. J. Math. 66 (1944), 470–488.
- Bipartite Entanglement, Spherical Actions, and Geometry of Local Unitary Orbits. J. Math. Phys. 54, 2 (2013), 022202.
- Jurdjevic, V. Geometric Control Theory. Cambridge University Press, Cambridge, 1997.
- Kleinsteuber, M. Jacobi-Type Methods on Semisimple Lie Algebras: A Lie Algebraic Approach to the Symmetric Eigenvalue Problem. PhD thesis, TU Munich, 2006.
- Liu, W. An Approximation Algorithm for Nonholonomic Systems. SIAM J. Control Optim. 35 (1997), 1328–1365.
- Analytic, Differentiable and Measurable Diagonalizations in Symmetric Lie Algebras, 2022. arXiv:2212.00713.
- Reduced Control Systems on Symmetric Lie Algebras, 2023. arXiv:2307.13664.
- Optimal Control of Bipartite Quantum Systems. In preparation.
- Reachability, Coolability, and Stabilizability of Open Markovian Quantum Systems with Fast Unitary Control, 2023. arXiv:2308.00561, submitted to SIAM J. Control Optim.
- Steering the Eigenvalues of the Density Operator in Hamiltonian-Controlled Quantum Lindblad Systems. IEEE Trans. Automat. Contr. 63 (2018), 672–681.
- Exploring the Limits of Open Quantum Dynamics I: Motivation, First Results from Toy Models to Applications. In Proc. MTNS (2022), p. 1069. arXiv:2003.06018.
- Optimal Control of Quantum Dissipative Dynamics: Analytic Solution for Cooling the Three-Level ΛΛ\Lambdaroman_Λ System. Phys. Rev. A 69 (2004), 053408.
- Smirnov, G. Introduction to the Theory of Differential Inclusions. Amer. Math. Soc., Providence, Rhode Island, 2002.
- Takagi, T. On an Algebraic Problem Related to an Analytic Theorem of Carathéodory and Fejér and on an Allied Theorem of Landau. Jap. J. Math. 83, 1 (1925).
- Vidal, G. Efficient Classical Simulation of Slightly Entangled Quantum Computations. Phys. Rev. Lett. 91, 14 (2003), 147902.
- vom Ende, F. Reachability in Controlled Markovian Quantum Systems: An Operator-Theoretic Approach. PhD thesis, TU Munich, 2020.
- Exploring the Limits of Controlled Markovian Quantum Dynamics with Thermal Resources. Open Syst. Inf. Dyn. 30, 1 (2023), 2350005.
- Yuan, H. Characterization of Majorization Monotone Quantum Dynamics. IEEE Trans. Automat. Contr. 55 (2010), 955–959.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.