We initiate the study of pretty good quantum fractional revival in graphs, a generalization of pretty good quantum state transfer in graphs. We give a complete characterization of pretty good fractional revival in a graph in terms of the eigenvalues and eigenvectors of the adjacency matrix of a graph. This characterization follows from a lemma due to Kronecker on Diophantine approximation, and is similar to the spectral characterization of pretty good state transfer in graphs. Using this, we give complete characterizations of when pretty good fractional revival can occur in paths and in cycles.
Revised:
Accepted:
Published online:
Keywords: Fractional revival, state transfer, quantum, graph
Chan, Ada 1; Drazen, Whitney 2; Eisenberg, Or 3; Kempton, Mark 4; Lippner, Gabor 2
CC-BY 4.0
@article{ALCO_2021__4_6_989_0,
author = {Chan, Ada and Drazen, Whitney and Eisenberg, Or and Kempton, Mark and Lippner, Gabor},
title = {Pretty good quantum fractional revival in paths and cycles},
journal = {Algebraic Combinatorics},
pages = {989--1004},
publisher = {MathOA foundation},
volume = {4},
number = {6},
year = {2021},
doi = {10.5802/alco.189},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.189/}
}
TY - JOUR AU - Chan, Ada AU - Drazen, Whitney AU - Eisenberg, Or AU - Kempton, Mark AU - Lippner, Gabor TI - Pretty good quantum fractional revival in paths and cycles JO - Algebraic Combinatorics PY - 2021 SP - 989 EP - 1004 VL - 4 IS - 6 PB - MathOA foundation UR - https://alco.centre-mersenne.org/articles/10.5802/alco.189/ DO - 10.5802/alco.189 LA - en ID - ALCO_2021__4_6_989_0 ER -
%0 Journal Article %A Chan, Ada %A Drazen, Whitney %A Eisenberg, Or %A Kempton, Mark %A Lippner, Gabor %T Pretty good quantum fractional revival in paths and cycles %J Algebraic Combinatorics %D 2021 %P 989-1004 %V 4 %N 6 %I MathOA foundation %U https://alco.centre-mersenne.org/articles/10.5802/alco.189/ %R 10.5802/alco.189 %G en %F ALCO_2021__4_6_989_0
Chan, Ada; Drazen, Whitney; Eisenberg, Or; Kempton, Mark; Lippner, Gabor. Pretty good quantum fractional revival in paths and cycles. Algebraic Combinatorics, Volume 4 (2021) no. 6, pp. 989-1004. doi : 10.5802/alco.189. https://alco.centre-mersenne.org/articles/10.5802/alco.189/
[1] Pretty good state transfer in qubit chains – the Heisenberg Hamiltonian, J. Math. Phys., Volume 58 (2017) no. 3, Paper no. 0322002, 9 pages | DOI | MR | Zbl
[2] A graph with fractional revival, Phys. Lett. A, Volume 382 (2018) no. 5, pp. 259-264 | DOI | MR | Zbl
[3] Quantum communication through an unmodulated spin chain, Physical Review Letters, Volume 91 (2003) no. 20, Paper no. 207901
[4] Spectra of graphs, Universitext, Springer, New York, 2012, xiv+250 pages | DOI | MR | Zbl
[5] Fundamentals of fractional revival in graphs (2020) (https://arxiv.org/abs/2004.01129)
[6] Quantum fractional revival on graphs, Discrete Appl. Math., Volume 269 (2019), pp. 86-98 | DOI | MR | Zbl
[7] Fractional revivals of the quantum state in a tight-binding chain, Physical Review A, Volume 75 (2007) no. 1, Paper no. 012113, 9 pages | DOI
[8] Perfect transfer of arbitrary states in quantum spin networks, Physical Review A, Volume 71 (2004), Paper no. 032312, 12 pages | DOI
[9] Analytic next-to-nearest-neighbor XX models with perfect state transfer and fractional revival, Physical Review A, Volume 96 (2017) no. 3, Paper no. 032335, 10 pages | DOI
[10] Pretty good state transfer between internal nodes of paths, Quantum Inf. Comput., Volume 17 (2017) no. 9-10, pp. 825-830 | MR
[11] Pretty good quantum state transfer in asymmetric graphs via potential, Discrete Math., Volume 342 (2019) no. 10, pp. 2821-2833 | DOI | MR | Zbl
[12] Pretty good state transfer on double stars, Linear Algebra Appl., Volume 438 (2013) no. 5, pp. 2346-2358 | DOI | MR | Zbl
[13] Exact fractional revival in spin chains, Modern Phys. Lett. B, Volume 30 (2016) no. 26, Paper no. 1650315, 7 pages | DOI | MR
[14] Quantum spin chains with fractional revival, Ann. Physics, Volume 371 (2016), pp. 348-367 | DOI | MR | Zbl
[15] State transfer on graphs, Discrete Math., Volume 312 (2012) no. 1, pp. 129-147 | DOI | MR | Zbl
[16] When can perfect state transfer occur?, Electron. J. Linear Algebra, Volume 23 (2012), pp. 877-890 | DOI | MR | Zbl
[17] Number-theoretic nature of communication in quantum spin systems, Physical Review Letters, Volume 109 (2012) no. 5, Paper no. 050502
[18] Strongly cospectral vertices (2017) (https://arxiv.org/abs/1709.07975)
[19] Perfect, efficient, state transfer and its application as a constructive tool, Int. J. Quantum Inf., Volume 8 (2010) no. 4, pp. 641-676 | DOI | Zbl
[20] Perfect state transfer on graphs with a potential, Quantum Inf. Comput., Volume 17 (2017) no. 3-4, pp. 303-327 | MR
[21] Pretty good quantum state transfer in symmetric spin networks via magnetic field, Quantum Information Processing, Volume 16 (2017) no. 9, p. 16:210 | DOI | MR | Zbl
[22] Vanishing sums of roots of unity, Proceedings, Bicentennial Congress Wiskundig Genootschap (Vrije Univ., Amsterdam, 1978), Part II (Math. Centre Tracts), Volume 101 (1979), pp. 249-268 | MR | Zbl
[23] Pretty good state transfer on circulant graphs, Electron. J. Combin., Volume 24 (2017) no. 2, Paper no. 2.23, 13 pages | DOI | MR | Zbl
[24] Perfect quantum routing in regular spin networks, Physical Review Letters, Volume 106 (2011), Paper no. 020503, 4 pages | DOI
[25] A complete characterization of pretty good state transfer on paths, Quantum Inf. Comput., Volume 19 (2019) no. 7-8, pp. 601-608 | MR
[26] Almost perfect state transfer in quantum spin chains, Physical Review A, Volume 86 (2012), Paper no. 052319, 10 pages | DOI
Cited by Sources: