ALGEBRAIC COMBINATORICS

no. 6
Pretty good quantum fractional revival in paths and cycles
Chan, Ada1; Drazen, Whitney2; Eisenberg, Or3; Kempton, Mark4; Lippner, Gabor2
1 Department of Mathematics and Statistics York University Toronto ON Canada
2 Department of Mathematics Northeastern University Boston MA USA
3 Department of Mathematics Harvard University Cambridge MA USA
4 Department of Mathematics Brigham Young University Provo UT USA
Algebraic Combinatorics, Volume 4 (2021) no. 6, pp. 989-1004.

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.

Received:
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Accepted:
Published online:
DOI: 10.5802/alco.189
Classification: 05E30, 05C50, 33C05, 15A16, 81P40
Keywords: Fractional revival, state transfer, quantum, graph
Chan, Ada 1; Drazen, Whitney 2; Eisenberg, Or 3; Kempton, Mark 4; Lippner, Gabor 2

1 Department of Mathematics and Statistics York University Toronto ON Canada
2 Department of Mathematics Northeastern University Boston MA USA
3 Department of Mathematics Harvard University Cambridge MA USA
4 Department of Mathematics Brigham Young University Provo UT USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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] Banchi, Leonardo; Coutinho, Gabriel; Godsil, Chris; Severini, Simone 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] Bernard, Pierre-Antoine; Chan, Ada; Loranger, Érika; Tamon, Christino; Vinet, Luc A graph with fractional revival, Phys. Lett. A, Volume 382 (2018) no. 5, pp. 259-264 | DOI | MR | Zbl

[3] Bose, Sougato Quantum communication through an unmodulated spin chain, Physical Review Letters, Volume 91 (2003) no. 20, Paper no. 207901

[4] Brouwer, Andries E.; Haemers, Willem H. Spectra of graphs, Universitext, Springer, New York, 2012, xiv+250 pages | DOI | MR | Zbl

[5] Chan, Ada; Coutinho, Gabriel; Drazen, Whitney; Eisenberg, Or; Godsil, Chris; Lippner, Gabor; Kempton, Mark; Tamon, Christino; Zhan, Hanmeng Fundamentals of fractional revival in graphs (2020) (https://arxiv.org/abs/2004.01129)

[6] Chan, Ada; Coutinho, Gabriel; Tamon, Christino; Vinet, Luc; Zhan, Hanmeng Quantum fractional revival on graphs, Discrete Appl. Math., Volume 269 (2019), pp. 86-98 | DOI | MR | Zbl

[7] Chen, Bing; Song, Zexi; Sun, Chang-Pu 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] Christandl, Matthias; Datta, Nilanjana; Dorlas, Tony; Ekert, Artur; Kay, Alastair; Landahl, Andrew Perfect transfer of arbitrary states in quantum spin networks, Physical Review A, Volume 71 (2004), Paper no. 032312, 12 pages | DOI

[9] Christandl, Matthias; Vinet, Luc; Zhedanov, Alexei 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] Coutinho, Gabriel; Guo, Krystal; van Bommel, Christopher M. Pretty good state transfer between internal nodes of paths, Quantum Inf. Comput., Volume 17 (2017) no. 9-10, pp. 825-830 | MR

[11] Eisenberg, Or; Kempton, Mark; Lippner, Gabor Pretty good quantum state transfer in asymmetric graphs via potential, Discrete Math., Volume 342 (2019) no. 10, pp. 2821-2833 | DOI | MR | Zbl

[12] Fan, Xiaoxia; Godsil, Chris Pretty good state transfer on double stars, Linear Algebra Appl., Volume 438 (2013) no. 5, pp. 2346-2358 | DOI | MR | Zbl

[13] Genest, Vincent X.; Vinet, Luc; Zhedanov, Alexei Exact fractional revival in spin chains, Modern Phys. Lett. B, Volume 30 (2016) no. 26, Paper no. 1650315, 7 pages | DOI | MR

[14] Genest, Vincent X.; Vinet, Luc; Zhedanov, Alexei Quantum spin chains with fractional revival, Ann. Physics, Volume 371 (2016), pp. 348-367 | DOI | MR | Zbl

[15] Godsil, Chris State transfer on graphs, Discrete Math., Volume 312 (2012) no. 1, pp. 129-147 | DOI | MR | Zbl

[16] Godsil, Chris When can perfect state transfer occur?, Electron. J. Linear Algebra, Volume 23 (2012), pp. 877-890 | DOI | MR | Zbl

[17] Godsil, Chris; Kirkland, Stephen; Severini, Simone; Smith, Jamie Number-theoretic nature of communication in quantum spin systems, Physical Review Letters, Volume 109 (2012) no. 5, Paper no. 050502

[18] Godsil, Chris; Smith, Jamie Strongly cospectral vertices (2017) (https://arxiv.org/abs/1709.07975)

[19] Kay, Alastair 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] Kempton, Mark; Lippner, Gabor; Yau, Shing-Tung Perfect state transfer on graphs with a potential, Quantum Inf. Comput., Volume 17 (2017) no. 3-4, pp. 303-327 | MR

[21] Kempton, Mark; Lippner, Gabor; Yau, Shing-Tung 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] Lenstra, Hendrik W. 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] Pal, Hiranmoy; Bhattacharjya, Bikash 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] Pemberton-Ross, Peter J.; Kay, Alastair Perfect quantum routing in regular spin networks, Physical Review Letters, Volume 106 (2011), Paper no. 020503, 4 pages | DOI

[25] van Bommel, Christopher M. A complete characterization of pretty good state transfer on paths, Quantum Inf. Comput., Volume 19 (2019) no. 7-8, pp. 601-608 | MR

[26] Vinet, Luc; Zhedanov, Alexei Almost perfect state transfer in quantum spin chains, Physical Review A, Volume 86 (2012), Paper no. 052319, 10 pages | DOI

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