Partially Coherent Direct Sum Channels

Stefano Chessa and Vittorio Giovannetti

NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy

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We introduce $\textit{Partially Coherent Direct Sum}$ (PCDS) quantum channels, as a generalization of the already known $\textit{Direct Sum}$ quantum channels. We derive necessary and sufficient conditions to identify the subset of those maps which are degradable, and provide a simplified expression for their quantum capacities. Interestingly, the special structure of PCDS allows us to extend the computation of the quantum capacity formula also for quantum channels which are explicitly not degradable (nor antidegradable). We show instances of applications of the results to dephasing channels, amplitude damping channels and combinations of the two.

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► References

[1] C. E. Shannon, Bell System Technical Journal 27, 379 (1948), URL https:/​/​​10.1002/​j.1538-7305.1948.tb01338.x.

[2] A. S. Holevo, Quantum Systems, Channels, Information (De Gruyter, 2019), URL https:/​/​​10.1515/​9783110642490.

[3] J. Watrous, The Theory of Quantum Information (Cambridge University Press, Cambridge, 2018), ISBN 9781107180567, URL https:/​/​​10.1017/​9781316848142.

[4] M. M. Wilde, Quantum Information Theory (Cambridge University Press, Cambridge, 2017), 2nd ed., ISBN 9781107176164, URL https:/​/​​10.1017/​9781316809976.

[5] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, 2010), ISBN 9781107002173, URL https:/​/​​10.1017/​CBO9780511976667.

[6] C. H. Bennett and G. Brassard, Theoretical Computer Science 560, 7 (2014), ISSN 0304-3975, URL https:/​/​​10.1016/​j.tcs.2014.05.025.

[7] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992), URL https:/​/​​10.1103/​PhysRevLett.69.2881.

[8] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993), URL https:/​/​​10.1103/​PhysRevLett.70.1895.

[9] G. Smith and J. Yard, Science 321, 1812 (2008), ISSN 0036-8075, URL https:/​/​​10.1126/​science.1162242.

[10] M. B. Hastings, Nature Physics 5, 255 (2009), ISSN 1745-2481, URL https:/​/​​10.1038/​nphys1224.

[11] A. S. Holevo and V. Giovannetti, Reports on Progress in Physics 75, 046001 (2012), URL https:/​/​​10.1088/​0034-4885/​75/​4/​046001.

[12] L. Gyongyosi, S. Imre, and H. V. Nguyen, IEEE Communications Surveys Tutorials 20, 1149 (2018), URL https:/​/​​10.1109/​COMST.2017.2786748.

[13] A. Muthukrishnan and C. R. Stroud, Phys. Rev. A 62, 052309 (2000), URL https:/​/​​10.1103/​PhysRevA.62.052309.

[14] T. C. Ralph, K. J. Resch, and A. Gilchrist, Phys. Rev. A 75, 022313 (2007), URL https:/​/​​10.1103/​PhysRevA.75.022313.

[15] B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O'Brien, A. Gilchrist, and A. G. White, Nature Physics 5, 134 (2009), ISSN 1745-2481, URL https:/​/​​10.1038/​nphys1150.

[16] Z. Gedik, I. A. Silva, B. Çakmak, G. Karpat, E. L. G. Vidoto, D. O. Soares-Pinto, E. R. deAzevedo, and F. F. Fanchini, Scientific Reports 5, 14671 (2015), ISSN 2045-2322, URL https:/​/​​10.1038/​srep14671.

[17] D. Cozzolino, B. Da Lio, D. Bacco, and L. K. Oxenløwe, Advanced Quantum Technologies 2, 1900038 (2019), URL https:/​/​​10.1002/​qute.201900038.

[18] S. Y. Looi, L. Yu, V. Gheorghiu, and R. B. Griffiths, Phys. Rev. A 78, 042303 (2008), URL https:/​/​​10.1103/​PhysRevA.78.042303.

[19] S. K. Goyal, P. E. Boukama-Dzoussi, S. Ghosh, F. S. Roux, and T. Konrad, Scientific Reports 4, 4543 (2014), ISSN 2045-2322, URL https:/​/​​10.1038/​srep04543.

[20] S. Muralidharan, C.-L. Zou, L. Li, J. Wen, and L. Jiang, New Journal of Physics 19, 013026 (2017), URL https:/​/​​10.1088/​1367-2630/​aa573a.

[21] G. Lima, L. Neves, R. Guzmán, E. S. Gómez, W. A. T. Nogueira, A. Delgado, A. Vargas, and C. Saavedra, Opt. Express 19, 3542 (2011), URL https:/​/​​10.1364/​OE.19.003542.

[22] P. B. R. Nisbet-Jones, J. Dilley, A. Holleczek, O. Barter, and A. Kuhn, New Journal of Physics 15, 053007 (2013), URL https:/​/​​10.1088/​1367-2630/​15/​5/​053007.

[23] B. Marques, A. A. Matoso, W. M. Pimenta, A. J. Gutiérrez-Esparza, M. F. Santos, and S. Pádua, Scientific Reports 5, 16049 (2015), ISSN 2045-2322, URL https:/​/​​10.1038/​srep16049.

[24] A. Babazadeh, M. Erhard, F. Wang, M. Malik, R. Nouroozi, M. Krenn, and A. Zeilinger, Phys. Rev. Lett. 119, 180510 (2017), URL https:/​/​​10.1103/​PhysRevLett.119.180510.

[25] M. Kues, C. Reimer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, et al., Nature 546, 622 (2017), ISSN 1476-4687, URL https:/​/​​10.1038/​nature22986.

[26] E. Moreno-Pineda, C. Godfrin, F. Balestro, W. Wernsdorfer, and M. Ruben, Chem. Soc. Rev. 47, 501 (2018), URL http:/​/​​10.1039/​C5CS00933B.

[27] T. Giordani, E. Polino, S. Emiliani, A. Suprano, L. Innocenti, H. Majury, L. Marrucci, M. Paternostro, A. Ferraro, N. Spagnolo, et al., Phys. Rev. Lett. 122, 020503 (2019), URL https:/​/​​10.1103/​PhysRevLett.122.020503.

[28] R. Sawant, J. A. Blackmore, P. D. Gregory, J. Mur-Petit, D. Jaksch, J. Aldegunde, J. M. Hutson, M. R. Tarbutt, and S. L. Cornish, New Journal of Physics 22, 013027 (2020), URL https:/​/​​10.1088/​1367-2630/​ab60f4.

[29] Y.-H. Luo, H.-S. Zhong, M. Erhard, X.-L. Wang, L.-C. Peng, M. Krenn, X. Jiang, L. Li, N.-L. Liu, C.-Y. Lu, et al., Phys. Rev. Lett. 123, 070505 (2019), URL https:/​/​​10.1103/​PhysRevLett.123.070505.

[30] M. Fukuda and M. M. Wolf, Journal of Mathematical Physics 48, 072101 (2007), URL https:/​/​​10.1063/​1.2746128.

[31] L. Gao, M. Junge, and N. LaRacuente, Journal of Mathematical Physics 59, 122202 (2018a), URL https:/​/​​10.1063/​1.5058692.

[32] L. Gao, M. Junge, and N. LaRacuente, Communications in Mathematical Physics 364, 83 (2018b), ISSN 1432-0916, URL https:/​/​​10.1007/​s00220-018-3249-y.

[33] K. Brádler, P. Hayden, and P. Panangaden, Journal of High Energy Physics 2009, 074 (2009), URL https:/​/​​10.1088/​1126-6708/​2009/​08/​074.

[34] K. Brádler, N. Dutil, P. Hayden, and A. Muhammad, Journal of Mathematical Physics 51, 072201 (2010a), URL https:/​/​​10.1063/​1.3449555.

[35] K. Brádler, P. Hayden, D. Touchette, and M. M. Wilde, Phys. Rev. A 81, 062312 (2010b), URL https:/​/​​10.1103/​PhysRevA.81.062312.

[36] K. Brádler, IEEE Transactions on Information Theory 57, 5497 (2011).

[37] K. Brádler, T. Jochym-O'Connor, and R. Jáuregui, Journal of Mathematical Physics 52, 062202 (2011), URL https:/​/​​10.1063/​1.3597233.

[38] V. Siddhu and R. B. Griffiths, arXiv e-prints arXiv:2003.00583 (2020), URL https:/​/​​abs/​2003.00583.

[39] F. Leditzky, D. Leung, and G. Smith, Phys. Rev. Lett. 121, 160501 (2018), URL https:/​/​​10.1103/​PhysRevLett.121.160501.

[40] S. Chessa and V. Giovannetti, Communications Physics 4, 22 (2021), ISSN 2399-3650, URL https:/​/​​10.1038/​s42005-021-00524-4.

[41] I. Devetak and P. W. Shor, Communications in Mathematical Physics 256, 287 (2005), ISSN 1432-0916, URL https:/​/​​10.1007/​s00220-005-1317-6.

[42] M.-D. Choi, Linear Algebra and its Applications 10, 285 (1975), ISSN 0024-3795, URL https:/​/​​10.1016/​0024-3795(75)90075-0.

[43] K. Kraus, Annals of Physics 64, 311 (1971), ISSN 0003-4916, URL https:/​/​​10.1016/​0003-4916(71)90108-4.

[44] W. F. Stinespring, Proceedings of the American Mathematical Society 6, 211 (1955), full publication date: Apr., 1955, URL https:/​/​​10.2307/​2032342.

[45] F. Caruso and V. Giovannetti, Phys. Rev. A 74, 062307 (2006), URL https:/​/​​10.1103/​PhysRevA.74.062307.

[46] S. Lloyd, Phys. Rev. A 55, 1613 (1997), URL https:/​/​​10.1103/​PhysRevA.55.1613.

[47] P. W. Shor, in lecture notes, MSRI Workshop on Quantum Computation (Quantum Information and Cryptography) (2002), URL https:/​/​​workshops/​203/​schedules/​1181.

[48] I. Devetak, IEEE Transactions on Information Theory 51, 44 (2005), URL https:/​/​​10.1109/​TIT.2004.839515.

[49] M. Keyl, Physics Reports 369, 431 (2002), ISSN 0370-1573, URL https:/​/​​10.1016/​S0370-1573(02)00266-1.

[50] S. Khatri, K. Sharma, and M. M. Wilde, Phys. Rev. A 102, 012401 (2020), URL https:/​/​​10.1103/​PhysRevA.102.012401.

[51] C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal, Phys. Rev. Lett. 83, 3081 (1999), URL https:/​/​​10.1103/​PhysRevLett.83.3081.

[52] C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal, IEEE Transactions on Information Theory 48, 2637 (2002), URL https:/​/​​10.1109/​TIT.2002.802612.

[53] C. H. Bennett, I. Devetak, P. W. Shor, and J. A. Smolin, Phys. Rev. Lett. 96, 150502 (2006), URL https:/​/​​10.1103/​PhysRevLett.96.150502.

[54] A. D'Arrigo, G. Benenti, and G. Falci, New Journal of Physics 9, 310 (2007), URL https:/​/​​10.1088/​1367-2630/​9/​9/​310.

[55] M. Tomamichel, M. M. Wilde, and A. J. Winter, IEEE International Symposium on Information Theory, ISIT 2015, Hong Kong, China, June 14-19, 2015 pp. 2386–2390 (2015), URL https:/​/​​10.1109/​ISIT.2015.7282883.

[56] V. Giovannetti and R. Fazio, Phys. Rev. A 71, 032314 (2005), URL https:/​/​​10.1103/​PhysRevA.71.032314.

[57] A. D'Arrigo, G. Benenti, G. Falci, and C. Macchiavello, Phys. Rev. A 88, 042337 (2013), URL https:/​/​​10.1103/​PhysRevA.88.042337.

[58] G. Smith, Phys. Rev. A 78, 022306 (2008), URL https:/​/​​10.1103/​PhysRevA.78.022306.

Cited by

[1] Stefano Chessa and Vittorio Giovannetti, "Quantum capacity analysis of multi-level amplitude damping channels", Communications Physics 4 1, 22 (2021).

[2] Laszlo Gyongyosi, Sandor Imre, Philip R. Hemmer, and Alan L. Migdall, Quantum Computing, Communication, and Simulation II 71 (2022) ISBN:9781510649019.

[3] Rajiuddin Sk and Prasanta K. Panigrahi, "Information capacity analysis of fully correlated multilevel amplitude damping channels", Physical Review A 109 3, 032425 (2024).

[4] Stefano Chessa and Vittorio Giovannetti, "Resonant Multilevel Amplitude Damping Channels", Quantum 7, 902 (2023).

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