Uncertainty and symmetry bounds for the quantum total detection probability
We investigate a generic discrete quantum system prepared in state |ψ_{in}〉 under repeated detection attempts, aimed to find the particle in state |d〉, for example, a quantum walker on a finite graph searching for a node. For the corresponding classical random walk, the total detection probability P...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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American Physical Society
2020-06-01
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| Series: | Physical Review Research |
| Online Access: | http://doi.org/10.1103/PhysRevResearch.2.023392 |
| _version_ | 1797211422144331776 |
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| author | Felix Thiel Itay Mualem David A. Kessler Eli Barkai |
| author_facet | Felix Thiel Itay Mualem David A. Kessler Eli Barkai |
| author_sort | Felix Thiel |
| collection | DOAJ |
| description | We investigate a generic discrete quantum system prepared in state |ψ_{in}〉 under repeated detection attempts, aimed to find the particle in state |d〉, for example, a quantum walker on a finite graph searching for a node. For the corresponding classical random walk, the total detection probability P_{det} is unity. Due to destructive interference, one may find initial states |ψ_{in}〉 with P_{det}<1. We first obtain an uncertainty relation which yields insight on this deviation from classical behavior, showing the relation between P_{det} and energy fluctuations: ΔPVar[H[over ̂]]_{d}≥|〈d|[H[over ̂],D[over ̂]]|ψ_{in}〉|^{2}, where ΔP=P_{det}−|〈ψ_{in}|d〉|^{2} and D[over ̂]=|d〉〈d| is the measurement projector. Secondly, exploiting symmetry we show that P_{det}≤1/ν, where the integer ν is the number of states equivalent to the initial state. These bounds are compared with the exact solution for small systems, obtained from an analysis of the dark and bright subspaces, showing the usefulness of the approach. The upper bound works well even in large systems, and we show how to tighten the lower bound in this case. |
| first_indexed | 2024-04-24T10:26:14Z |
| format | Article |
| id | doaj.art-b0c733a9113e400d85eb9655762f34be |
| institution | Directory Open Access Journal |
| issn | 2643-1564 |
| language | English |
| last_indexed | 2024-04-24T10:26:14Z |
| publishDate | 2020-06-01 |
| publisher | American Physical Society |
| record_format | Article |
| series | Physical Review Research |
| spelling | doaj.art-b0c733a9113e400d85eb9655762f34be2024-04-12T16:56:14ZengAmerican Physical SocietyPhysical Review Research2643-15642020-06-012202339210.1103/PhysRevResearch.2.023392Uncertainty and symmetry bounds for the quantum total detection probabilityFelix ThielItay MualemDavid A. KesslerEli BarkaiWe investigate a generic discrete quantum system prepared in state |ψ_{in}〉 under repeated detection attempts, aimed to find the particle in state |d〉, for example, a quantum walker on a finite graph searching for a node. For the corresponding classical random walk, the total detection probability P_{det} is unity. Due to destructive interference, one may find initial states |ψ_{in}〉 with P_{det}<1. We first obtain an uncertainty relation which yields insight on this deviation from classical behavior, showing the relation between P_{det} and energy fluctuations: ΔPVar[H[over ̂]]_{d}≥|〈d|[H[over ̂],D[over ̂]]|ψ_{in}〉|^{2}, where ΔP=P_{det}−|〈ψ_{in}|d〉|^{2} and D[over ̂]=|d〉〈d| is the measurement projector. Secondly, exploiting symmetry we show that P_{det}≤1/ν, where the integer ν is the number of states equivalent to the initial state. These bounds are compared with the exact solution for small systems, obtained from an analysis of the dark and bright subspaces, showing the usefulness of the approach. The upper bound works well even in large systems, and we show how to tighten the lower bound in this case.http://doi.org/10.1103/PhysRevResearch.2.023392 |
| spellingShingle | Felix Thiel Itay Mualem David A. Kessler Eli Barkai Uncertainty and symmetry bounds for the quantum total detection probability Physical Review Research |
| title | Uncertainty and symmetry bounds for the quantum total detection probability |
| title_full | Uncertainty and symmetry bounds for the quantum total detection probability |
| title_fullStr | Uncertainty and symmetry bounds for the quantum total detection probability |
| title_full_unstemmed | Uncertainty and symmetry bounds for the quantum total detection probability |
| title_short | Uncertainty and symmetry bounds for the quantum total detection probability |
| title_sort | uncertainty and symmetry bounds for the quantum total detection probability |
| url | http://doi.org/10.1103/PhysRevResearch.2.023392 |
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