Qubit geodesics on the Bloch sphere from optimal-speed Hamiltonian evolutions. (English) Zbl 1522.81054
Summary: In the geometry of quantum evolutions, a geodesic path is viewed as a path of minimal statistical length connecting two pure quantum states along which the maximal number of statistically distinguishable states is minimum. In this paper, we present an explicit geodesic analysis of the dynamical trajectories that emerge from the quantum evolution of a single-qubit quantum state. The evolution is governed by an Hermitian Hamiltonian operator that achieves the fastest possible unitary evolution between given initial and final pure states. Furthermore, in addition to viewing geodesics in ray space as paths of minimal length, we also verify the geodesicity of paths in terms of unit geometric efficiency and vanishing geometric phase. Finally, based on our analysis, we briefly address the main hurdles in moving to the geometry of quantum evolutions for open quantum systems in mixed quantum states.
MSC:
| 81P68 | Quantum computation |
| 53C22 | Geodesics in global differential geometry |
| 81R30 | Coherent states |
| 81P18 | Quantum state tomography, quantum state discrimination |
| 70K42 | Equilibria and periodic trajectories for nonlinear problems in mechanics |
| 81S22 | Open systems, reduced dynamics, master equations, decoherence |
| 81P16 | Quantum state spaces, operational and probabilistic concepts |
References:
| [1] | Beggs, E. J.; Majid, S., Quantum Riemannian Geometry (2020), Cham: Springer, Cham · Zbl 1436.53001 |
| [2] | Majid, S., Quantum gravity on a square graph, Class. Quantum Grav., 36 (2019) · Zbl 1478.83042 · doi:10.1088/1361-6382/ab4975 |
| [3] | Beggs, E. J.; Majid, S., Quantum geodesics in quantum mechanics (2021) |
| [4] | Beggs, E. J.; Majid, S., Quantum geodesic flows and curvature (2022) |
| [5] | Bengtsson, I.; Zyczkowski, K., Geometry of Quantum States (2006), Cambridge: Cambridge University Press, Cambridge · Zbl 1146.81004 |
| [6] | Wootters, W. K., Statistical distance and Hilbert space, Phys. Rev. D, 23, 357 (1981) · doi:10.1103/PhysRevD.23.357 |
| [7] | Braunstein, S. L.; Caves, C. M., Statistical distance and the geometry of quantum states, Phys. Rev. Lett., 72, 3439 (1994) · Zbl 0973.81509 · doi:10.1103/PhysRevLett.72.3439 |
| [8] | Braunstein, S. L.; Caves, C. M.; Belavkin, B. V.; Hirota, O.; Hudson, R. L., Geometry of quantum states, Quantum Communications and Measurement, pp 21-30 (1995), Boston, MA: Springer, Boston, MA · Zbl 0942.81551 |
| [9] | Brody, D. C., Elementary derivation for passage times, J. Phys. A: Math. Gen., 36, 5587 (2003) · Zbl 1042.81011 · doi:10.1088/0305-4470/36/20/314 |
| [10] | Carlini, A.; Hosoya, A.; Koike, T.; Okudaira, Y., Time-optimal quantum evolution, Phys. Rev. Lett., 96 (2006) · doi:10.1103/PhysRevLett.96.060503 |
| [11] | Brody, D. C.; Hook, D. W., On optimum Hamiltonians for state transformations, J. Phys. A: Math. Gen., 39, L167 (2006) · Zbl 1085.81022 · doi:10.1088/0305-4470/39/11/L02 |
| [12] | Brody, D. C.; Hook, D. W., On optimum Hamiltonians for state transformation, J. Phys. A: Math. Theor., 40 (2007) · Zbl 1274.81054 · doi:10.1088/1751-8121/40/35/C01 |
| [13] | Bender, C. M.; Brody, D. C.; Jones, H. F.; Meister, B. K., Faster than Hermitian quantum mechanics, Phys. Rev. Lett., 98 (2007) · Zbl 1228.81027 · doi:10.1103/PhysRevLett.98.040403 |
| [14] | Uhlmann, A., An energy dispersion estimate, Phys. Lett. A, 161, 329 (1992) · doi:10.1016/0375-9601(92)90555-Z |
| [15] | Mostafazadeh, A., Hamiltonians generating optimal-speed evolutions, Phys. Rev. A, 79 (2009) · doi:10.1103/PhysRevA.79.014101 |
| [16] | Diosi, L.; Forgacs, G.; Lukacs, B.; Frisch, H. L., Metricization of thermodynamic-state space and the renormalization group, Phys. Rev. A, 29, 3343 (1984) · doi:10.1103/PhysRevA.29.3343 |
| [17] | Carlini, A.; Hosoya, A.; Koike, T.; Okudaira, Y., Time optimal quantum evolution of mixed states, J. Phys. A: Math. Theor., 41 (2008) · Zbl 1134.81387 · doi:10.1088/1751-8113/41/4/045303 |
| [18] | Campaioli, F.; Sloan, W.; Modi, K.; Pollock, F. A., Algorithm for solving unconstrained unitary quantum brachistochrone problems, Phys. Rev. A, 100 (2019) · doi:10.1103/PhysRevA.100.062328 |
| [19] | O’Connor, E.; Guarnieri, G.; Campbell, S., Action quantum speed limits, Phys. Rev. A, 103 (2021) · doi:10.1103/PhysRevA.103.022210 |
| [20] | Hornedal, N.; Allan, D.; Sonnerborn, O., Extensions of the Mandelstam-Tamm quantum speed limit to systems in mixed states, New J. Phys., 24 (2022) · doi:10.1088/1367-2630/ac688a |
| [21] | Cafaro, C.; Mancini, S., On Grover’s search algorithm from a quantum information geometry viewpoint, Physica A, 391, 1610 (2012) · doi:10.1016/j.physa.2011.09.018 |
| [22] | Cafaro, C., Geometric algebra and information geometry for quantum computational software, Physica A, 470, 154 (2017) · Zbl 1400.81044 · doi:10.1016/j.physa.2016.11.117 |
| [23] | Cafaro, C.; Alsing, P. M., Theoretical analysis of a nearly optimal analog quantum search, Phys. Scr., 94 (2019) · doi:10.1088/1402-4896/ab111f |
| [24] | Cafaro, C.; Alsing, P. M., Continuous-time quantum search and time-dependent two-level quantum systems, Int. J. Quantum Inf., 17 (2019) · Zbl 1426.81024 · doi:10.1142/S0219749919500254 |
| [25] | Gassner, S.; Cafaro, C.; Capozziello, S., Transition probabilities in generalized quantum search Hamiltonian evolutions, Int. J. Geom. Methods Mod. Phys., 17 (2020) · Zbl 07806150 · doi:10.1142/S0219887820500061 |
| [26] | Cafaro, C.; Ray, S.; Alsing, P. M., Geometric aspects of analog quantum search evolutions, Phys. Rev. A, 102 (2020) · doi:10.1103/PhysRevA.102.052607 |
| [27] | Cafaro, C.; Alsing, P. M., Minimum time for the evolution to a nonorthogonal quantum state and upper bound of the geometric efficiency of quantum evolutions, Quantum Rep., 3, 444 (2021) · doi:10.3390/quantum3030029 |
| [28] | Cafaro, C.; Felice, D.; Alsing, P. M., Quantum Groverian geodesic paths with gravitational and thermal analogies, Eur. Phys. J. Plus, 135, 900 (2020) · doi:10.1140/epjp/s13360-020-00914-7 |
| [29] | Cafaro, C.; Ray, S.; Alsing, P. M., Optimal-speed unitary quantum time evolutions and propagation of light with maximal degree of coherence, Phys. Rev. A, 105 (2022) · doi:10.1103/PhysRevA.105.052425 |
| [30] | Anandan, J.; Aharonov, Y., Geometry of quantum evolution, Phys. Rev. Lett., 65, 1697 (1990) · Zbl 0960.81524 · doi:10.1103/PhysRevLett.65.1697 |
| [31] | Chruscinski, D.; Jamiolkowski, A., Geometric Phases in Classical and Quantum Mechanics (2004), Boston, MA: Birkhäuser, Boston, MA · Zbl 1075.81002 |
| [32] | Provost, J. P.; Vallee, G., Riemannian structure on manifolds of quantum states, Commun. Math. Phys., 76, 289 (1980) · Zbl 0451.53053 · doi:10.1007/BF02193559 |
| [33] | Laba, H. P.; Tkachuk, V. M., Geometric characteristics of quantum evolution: curvature and torsion, Condens. Matter Phys., 20 (2017) · doi:10.5488/CMP.20.13003 |
| [34] | Ravicule, M.; Casas, M.; Plastino, A., Information and metrics in Hilbert space, Phys. Rev. A, 55, 1695 (1997) · doi:10.1103/PhysRevA.55.1695 |
| [35] | Dodonov, V. V.; Man’ko, O. V.; Man’ko, V. I.; Wunsche, A., Energy-sensitive and “classical-like” distances between quantum states, Phys. Scr., 59, 81 (1999) · Zbl 1063.81561 · doi:10.1238/Physica.Regular.059a00081 |
| [36] | Birrittella, R. J.; Alsing, P. M.; Gerry, C. C., The parity operator: applications in quantum metrology, AVS Quantum Sci., 3 (2021) · doi:10.1116/5.0026148 |
| [37] | Mukunda, N.; Simon, R., Quantum kinematic approach to the geometric phase. I. General formalism, Ann. Phys., NY, 228, 205 (1993) · Zbl 0791.58121 · doi:10.1006/aphy.1993.1093 |
| [38] | Mittal, V.; Akhilesh, K. S.; Goyal, S. K., Geometric decomposition of geodesics and null-phase curves using Majorana star representation, Phys. Rev. A, 105 (2022) · doi:10.1103/PhysRevA.105.052219 |
| [39] | Nakahara, M., Geometry, Topology and Physics (2003), Bristol: IOP Publishing, Bristol · Zbl 1090.53001 |
| [40] | Eguchi, T.; Gilkey, P. B.; Hanson, A. J., Gravitation, gauge theories and differential geometry, Phys. Rep., 66, 213 (1980) · doi:10.1016/0370-1573(80)90130-1 |
| [41] | Bohm, A.; Boya, L. J.; Kendrick, B., Derivation of the geometric phase, Phys. Rev. A, 43, 1206 (1991) · doi:10.1103/PhysRevA.43.1206 |
| [42] | Samuel, J.; Bhandari, R., General setting for Berry’s phase, Phys. Rev. Lett., 60, 2339 (1988) · doi:10.1103/PhysRevLett.60.2339 |
| [43] | Berry, M. V., Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. A, 392, 45 (1984) · Zbl 1113.81306 · doi:10.1098/rspa.1984.0023 |
| [44] | Simon, B., Holonomy, the quantum adiabatic theorem and Berry’s phase, Phys. Rev. Lett., 51, 2167 (1983) · doi:10.1103/PhysRevLett.51.2167 |
| [45] | Aharonov, Y.; Anandan, J., Phase change during a cyclic quantum evolution, Phys. Rev. Lett., 58, 1593 (1987) · doi:10.1103/PhysRevLett.58.1593 |
| [46] | Pati, A. K., Relation between “phases” and “distance” in quantum evolution, Phys. Lett. A, 159, 105 (1991) · doi:10.1016/0375-9601(91)90255-7 |
| [47] | Pati, A. K., On phases and length of curves in a cyclic quantum evolution, Pramana, 42, 455 (1994) · doi:10.1007/BF02847127 |
| [48] | Pati, A. K., Geometric aspects of noncyclic quantum evolutions, Phys. Rev. A, 52, 2576 (1995) · doi:10.1103/PhysRevA.52.2576 |
| [49] | Rabei, E. M.; Arvind, A.; Mukunda, N.; Simon, R., Bargmann invariants and geometric phases: a generalized connection, Phys. Rev. A, 60, 3397 (1999) · doi:10.1103/PhysRevA.60.3397 |
| [50] | Mukunda, N.; Arvind, A.; Ercolessi, E.; Marmo, G.; Morandi, G.; Simon, R., Bargmann invariants, null phase curves and a theory of the geometric phase, Phys. Rev. A, 67 (2003) · doi:10.1103/PhysRevA.67.042114 |
| [51] | Chaturvedi, S.; Ercolessi, E.; Morandi, G.; Ibort, A.; Marmo, G.; Mukunda, N.; Simon, R., Null phase curves and manifolds in geometric phase theory, J. Math. Phys., 54 (2013) · Zbl 1282.81100 · doi:10.1063/1.4811346 |
| [52] | Petz, D., Monotone metrics on matrix spaces, Linear Algebr. Appl., 244, 81 (1996) · Zbl 0856.15023 · doi:10.1016/0024-3795(94)00211-8 |
| [53] | Petz, D.; Sudar, C.; Barndorff-Nielsen, O. E.; Vendel, E. B., Extending the Fisher metric to density matrices, Geometry in Present Days Science, pp 21-34 (1999), Singapore: World Scientific, Singapore · Zbl 0939.62008 |
| [54] | Taddei, M. M.; Escher, B. M.; Davidovich, L.; de Matos Filho, R. L., Quantum speed limit for physical processes, Phys. Rev. Lett., 110 (2013) · doi:10.1103/PhysRevLett.110.050402 |
| [55] | del Campo, A.; Egusquiza, I. L.; Plenio, M. B.; Huelga, S. F., Quantum speed limits in open system dynamics, Phys. Rev. Lett., 110 (2013) · doi:10.1103/PhysRevLett.110.050403 |
| [56] | Deffner, S.; Lutz, E., Quantum speed limit for non-Markovian dynamics, Phys. Rev. Lett., 111 (2013) · doi:10.1103/PhysRevLett.111.010402 |
| [57] | Brody, D. C.; Longstaff, B., Evolution speed of open quantum dynamics, Phys. Rev. Res., 1 (2019) · doi:10.1103/PhysRevResearch.1.033127 |
| [58] | Bengtsson, I.; Weis, S.; Zyczkowski, K.; Kielanowski, P.; Ali, S.; Odzijewicz, A.; Schlichenmaier, M.; Voronov, T., Geometry of the set of mixed quantum states: an apophatic approach, Geometric Methods in Physics (Trends in Mathematics) (2013), Basel: Birkhäuser, Basel |
| [59] | Uhlmann, A., Geometric phases and related structures, Rep. Math. Phys., 36, 461 (1995) · Zbl 0891.46051 · doi:10.1016/0034-4877(96)83640-8 |
| [60] | Gibilisco, P.; Isola, T., Wigner-Yanase information on quantum state space: the geometric approach, J. Math. Phys., 44, 3752 (2003) · Zbl 1062.81019 · doi:10.1063/1.1598279 |
| [61] | Cai, X.; Zheng, Y., Quantum dynamical speedup in a nonequilibrium environment, Phys. Rev. A, 95 (2017) · doi:10.1103/PhysRevA.95.052104 |
| [62] | Bures, D., An extension of Kakutani’s theorem on infinite product measures to the tensor product of semifinite \(####\)-algebras, Trans. Am. Math. Soc., 135, 199 (1969) · Zbl 0176.11402 · doi:10.1090/S0002-9947-1969-0236719-2 |
| [63] | Uhlmann, A., The “transition probability” in the state space of a *-algebra, Rep. Math. Phys., 9, 273 (1976) · Zbl 0355.46040 · doi:10.1016/0034-4877(76)90060-4 |
| [64] | Hübner, M., Explicit computation of the Bures distance for density matrices, Phys. Lett. A, 163, 239 (1992) · doi:10.1016/0375-9601(92)91004-B |
| [65] | Dittmann, J., On the Riemannian metric on the space of density matrices, Rep. Math. Phys., 36, 309 (1995) · Zbl 0886.53033 · doi:10.1016/0034-4877(96)83627-5 |
| [66] | Wigner, E. P.; Yanase, M. M., Information content of distributions, Proc. Natl Acad. Sci. USA, 49, 910 (1963) · Zbl 0128.14104 · doi:10.1073/pnas.49.6.910 |
| [67] | Luo, S., Wigner-Yanase skew information and uncertainty relations, Phys. Rev. Lett., 91 (2003) · doi:10.1103/PhysRevLett.91.180403 |
| [68] | Sjöqvist, E., Geometry along evolution of mixed quantum states, Phys. Rev. Res., 2 (2020) · doi:10.1103/PhysRevResearch.2.013344 |
| [69] | Cafaro, C.; Alsing, P. M., Complexity of pure and mixed qubit geodesic paths on curved manifolds, Phys. Rev. D, 106 (2022) · doi:10.1103/PhysRevD.106.096004 |
| [70] | Cafaro, C.; Ali, S. A., Jacobi fields on statistical manifolds of negative curvature, Physica D, 234, 70 (2007) · Zbl 1129.53019 · doi:10.1016/j.physd.2007.07.001 |
| [71] | Cafaro, C.; Mancini, S., Quantifying the complexity of geodesic paths on curved statistical manifolds through information geometric entropies and Jacobi fields, Physica D, 240, 607 (2011) · Zbl 1217.37079 · doi:10.1016/j.physd.2010.11.013 |
| [72] | Bertlmann, R. A.; Krammer, P., Bloch vectors for qudits, J. Phys. A: Math. Theor., 41 (2008) · Zbl 1140.81321 · doi:10.1088/1751-8113/41/23/235303 |
| [73] | Kurzynski, P.; Kolodziejski, A.; Laskowski, W.; Markiewicz, M., Three-dimensional visualization of a qutrit, Phys. Rev. A, 93 (2016) · doi:10.1103/PhysRevA.93.062126 |
| [74] | Goyal, S. K.; Neethi Simon, B.; Singh, R.; Simon, S., Geometry of the generalized Bloch ball for qutrits, J. Phys. A: Math. Theor., 49 (2016) · Zbl 1342.81068 · doi:10.1088/1751-8113/49/16/165203 |
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