Conditions for the absence of local extrema in problems of quantum coherent control

N. B. Il'in ; A. N. Pechen

Informatics and Automation, Complex analysis, mathematical physics, and applications, Tome 301 (2018), pp. 119-123.

Résumé

We consider a terminal control problem for quantum systems which is formulated as the problem of maximizing the objective functional at some fixed finite time. Within the framework of this problem, we discuss known results on the local maxima of the objective functional that are not global. This question is important for quantum control, since such local maxima could make it difficult to find the global maximum by local search in numerical optimization or under laboratory conditions.

Bibliographie

[1] Accardi L., Lu Y. G., Volovich I., Quantum theory and its stochastic limit, Springer, Berlin, 2002 | MR | Zbl

[2] Breuer H.-P., Petruccione F., The theory of open quantum systems, Oxford Univ. Press, Oxford, 2002 | MR | Zbl

[3] Brif C., Chakrabarti R., Rabitz H., “Control of quantum phenomena”, Advances in chemical physics, 148, eds. S. A. Rice, A. R. Dinner, J. Wiley Sons, New York, 2012, 1–76

[4] Brockett R. W., “Least squares matching problems”, Linear Algebra Appl., 122–124 (1989), 761–777 | DOI | MR | Zbl

[5] Brumer P., Shapiro M., Principles of the quantum control of molecular processes, J. Wiley Sons, Hoboken, NJ, 2003

[6] D'Alessandro D., Introduction to quantum control and dynamics, Chapman Hall, Boca Raton, FL, 2008 | MR | Zbl

[7] De Fouquieres P., Schirmer S. G., “A closer look at quantum control landscapes and their implication for control optimization”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 16:3 (2013), 1350021 | DOI | MR | Zbl

[8] Glaser S. J., Boscain U., Calarco T., Koch C. P., Köckenberger W., Kosloff R., Kuprov I., Luy B., Schirmer S., Schulte-Herbrüggen T., Sugny D., Wilhelm F. K., “Training Schrödinger's cat: quantum optimal control – Strategic report on current status, visions and goals for research in Europe”, Eur. Phys. J. D, 69:12 (2015), 279 | DOI

[9] Glaser S. J., Schulte-Herbrüggen T., Sieveking M., Schedletzky O., Nielsen N. C., Sørensen O. W., Griesinger C., “Unitary control in quantum ensembles: Maximizing signal intensity in coherent spectroscopy”, Science, 280:5362 (1998), 421–424 | DOI

[10] Ho T.-S., Rabitz H., “Why do effective quantum controls appear easy to find?”, J. Photochem. Photobiol. A, 180:3 (2006), 226–240 | DOI

[11] Holevo A. S., Statistical structure of quantum theory, Lect. Notes Phys. Monogr., 67, Springer, Berlin, 2011 | MR

[12] Letokhov V. S., Laser control of atoms and molecules, Oxford Univ. Press, Oxford, 2007

[13] Lyakhov K., Lee H.-J., Pechen A., “Some features of Boron isotopes separation by the laser-assisted retardation of condensation method in multipass irradiation cell implemented as a resonator”, IEEE J. Quantum Electron., 52:12 (2016), 1400208 | DOI

[14] J. Neumann, “Some matrix-inequalities and metrization of matric-space”, Izv. Nauchno-Issled. Inst. Mat. Mekh. Tomsk. Gos. Univ. im. V. V. Kuibysheva, 1:3 (1937), 286–300 | Zbl

[15] S. Attal, A. Joye, C.-A. Pillet (eds.), Open quantum systems, v. II, Lect. Notes Math., 1881, The Markovian approach, Springer, Berlin, 2006 | MR | Zbl

[16] Pechen A., Il'in N., “Trap-free manipulation in the Landau–Zener system”, Phys. Rev. A, 86:5 (2012), 052117 | DOI

[17] A. N. Pechen, N. B. Il'in, “Coherent control of a qubit is trap-free”, Proc. Steklov Inst. Math., 285 (2014), 233–240 | DOI | DOI | MR | Zbl

[18] A. N. Pechen, N. B. Il'in, “On critical points of the objective functional for maximization of qubit observables”, Russ. Math. Surv., 70:4 (2015), 782–784 | DOI | DOI | MR | Zbl

[19] A. N. Pechen, N. B. Il'in, “On the problem of maximizing the transition probability in an $n$-level quantum system using nonselective measurements”, Proc. Steklov Inst. Math., 294 (2016), 233–240 | DOI | DOI | MR | Zbl

[20] A. N. Pechen, N. B. Il'in, “On extrema of the objective functional for short-time generation of single-qubit quantum gates”, Izv. Math., 80:6 (2016), 1200–1212 | DOI | DOI | MR | Zbl

[21] Pechen A., Il'in N., “Control landscape for ultrafast manipulation by a qubit”, J. Phys. A: Math. Theor., 50:7 (2017), 075301 | DOI | MR | Zbl

[22] Pechen A. N., Tannor D. J., “Control of quantum transmission is trap free”, Can. J. Chem., 92:2 (2014), 157–159 | DOI

[23] Rabitz H., Hsieh M., Rosenthal C., “Landscape for optimal control of quantum-mechanical unitary transformations”, Phys. Rev. A, 72:5 (2005), 052337 | DOI

[24] Rabitz H. A., Hsieh M. M., Rosenthal C. M., “Quantum optimally controlled transition landscapes”, Science, 303:5666 (2004), 1998–2001 | DOI

[25] Rach N., Müller M. M., Calarco T., Montangero S., “Dressing the chopped-random-basis optimization: A bandwidth-limited access to the trap-free landscape”, Phys. Rev. A, 92:6 (2015), 062343 | DOI

[26] Rice S. A., Zhao M., Optical control of molecular dynamics, J. Wiley Sons, New York, 2000

[27] Tannor D. J., Introduction to quantum mechanics: A time-dependent perspective, Univ. Sci. Books, Sausalito, CA, 2007

[28] Trushechkin A., “Semiclassical evolution of quantum wave packets on the torus beyond the Ehrenfest time in terms of Husimi distributions”, J. Math. Phys., 58:6 (2017), 062102 | DOI | MR | Zbl

[29] Trushechkin A. S., Volovich I. V., “Perturbative treatment of inter-site couplings in the local description of open quantum networks”, Europhys. Lett., 113:3 (2016), 30005 | DOI

[30] Volovich I. V., “Cauchy–Schwarz inequality-based criteria for the non-classicality of sub-Poisson and antibunched light”, Phys. Lett. A, 380:1–2 (2016), 56–58 | DOI | MR | Zbl

[31] I. V. Volovich, S. V. Kozyrev, “Manipulation of states of a degenerate quantum system”, Proc. Steklov Inst. Math., 294 (2016), 241–251 | DOI | DOI | MR | Zbl

[32] Wu R., Pechen A., Rabitz H., Hsieh M., Tsou B., “Control landscapes for observable preparation with open quantum systems”, J. Math. Phys., 49:2 (2008), 022108 | DOI | MR | Zbl