Erst-Detektionsprobleme in der Quantenphysik
Theoretische Physik der kondensierten Materie
Zusammenfassung der Projektergebnisse
We investigated quantum systems which are repeatedly projectively measured every τ time units to probe whether they reside in some target state |ψd . The instant of the first successful detection attempt defines the system’s time of first detected arrival. This time can also be interpreted as a quantum search time for the marked state |ψd and generalizes a classical random walker’s first passage time. We computed the distribution of first detection times by means of a renewal equation and related this distribution to the energy spectrum of the system. For systems with continuous energy spectra, the position and strength of their van-Hove singularities determines the asymptotic power-law decay and the superimposed oscillations of the first detection probabilities. For systems with a discrete spectrum the distribution can be formally obtained from an electrostatic analogy. In the Zeno limit τ → 0 of very rapid detection attempts, the process becomes equivalent to a non-Hermitean Schr¨dinger o equation. For discrete energy spectra , the whole distribution can be obtained in this limit by means of an electrostatic theory. We also computed the total detection probability Pdet for systems with a discrete spectrum. This is the fraction of systems in a statistical ensemble which at all arrives at the target. It may be bounded from above in systems with a high degree of symmetry, leading to a very unreliable search process. Furthermore it is bounded from below by the energy fluctuations in the detection state. In some cases, lower and upper bound coincide and thus the exact result is found. Our analysis characterizes initial states for the system which allow for reliable search. Our theory supplies the framework for the analysis of the runtime of future quantum programs.
Projektbezogene Publikationen (Auswahl)
- “First Detected Arrival of a Quantum Walker on an Infinite Line”. Physical Review Letters 120 (2018), 040502
F. Thiel, E. Barkai, and D. A. Kessler
(Siehe online unter https://doi.org/10.1103/PhysRevLett.120.040502) - “Spectral dimension controlling the decay of the quantum first-detection probability”. Physical Review A 97 (2018), 062105
F. Thiel, D. A. Kessler, and E. Barkai
(Siehe online unter https://doi.org/10.1103/PhysRevA.97.062105) - Uncertainty and symmetry bounds for the quantum total detection probability. 2019
F. Thiel et al.
(Siehe online unter https://doi.org/10.1103/PhysRevResearch.2.023392) - “Large fluctuations of the first detected quantum return time”. Physical Review Research 1 (2019), 033086
R. Yin et al.
(Siehe online unter https://doi.org/10.1103/PhysRevResearch.1.033086)
