Evaluation of non-unital qubit channel capacities
S.N. Filippov
Institute of Physics and Technology, Russian Academy of Sciences, Moscow, 117218 Russia
Moscow Institute of Physics and Technology, Dolgoprudny, 141701 Russia
E-mail: sergey.filippov@phystech.edu
Received October 23, 2017
Abstract
We have applied quantum Sinkhorn's theorem to non-unital qubit channels and derived lower and upper bounds on the classical capacity of such channels.
Keywords: qubit channel, non-unital channel, Holevo capacity
Acknowledgements. The study was supported by the Russian Foundation for Basic Research (project no. 16-37-60070 mol-a-dk).
References
1. Holevo A.S. The capacity of quantum channel with general signal states. IEEE Trans. Inf. Theory, 1998, vol. 44, no. 1, pp. 269–273. doi: 10.1109/18.651037.
2. Schumacher B., Westmoreland M.D. Sending classical information via noisy quantum channels. Phys. Rev. A, 1997, vol. 56, pp. 131–138. doi: 10.1103/PhysRevA.56.131.
3. Holevo A.S. Quantum coding theorems. Russ. Math. Surv., 1998, vol. 53, no. 6, pp. 1295–1331. doi: 10.1070/RM1998v053n06ABEH000091.
4. Holevo A.S., Werner R.F. Evaluating capacities of bosonic Gaussian channels. Phys. Rev. A, 2001, vol. 63, no. 3, art. 032312, pp. 1–14. doi: 10.1103/PhysRevA.63.032312.
5. Holevo A.S., Shirokov M.E. Continuous ensembles and the capacity of infinite-dimensional quantum channels. Theory Probab. Appl., 2006, vol. 50, no. 1, pp. 86–98. doi: 10.1137/S0040585X97981470.
6. Holevo A.S., Giovannetti V. Quantum channels and their entropic characteristics. Rep. Prog. Phys., 2012, vol. 75, no. 4, art. 046001, pp. 1–30. doi: 10.1088/0034-4885/75/4/046001.
7. Holevo A.S. Quantum Systems, Channels, Information. Berlin/Boston, Walter de Gruyter GmbH, 2012. xiii, 349 p.
8. Aubrun G., Szarek S.J. Alice and Bob meet Banach: The interface of asymptotic geometry analysis and quantum information theory. In: Mathematical Surveys and Monographs. Book 223. Am. Math. Soc., 2017. 414 p.
9. Gurvits L. Classical complexity and quantum entanglement. J.Comput. Syst. Sci., 2004, vol. 69, no. 3, pp. 448–484. doi: 10.1016/j.jcss.2004.06.003.
10. Filippov S.N., Magadov K.Y. Positive tensor products of maps and n-tensor-stable positive qubit maps. J. Phys. A: Math. Theor., 2017, vol. 50, no. 5, art. 55301, pp. 1–21. doi: 10.1088/1751-8121/aa5301.
11. Filippov S.N., Frizen V.V., Kolobova D.V. Ultimate entanglement robustness of two-qubit states against general local noises. Phys. Rev. A, 2018, vol. 97, no. 1, art. 012322, pp. 1–9. doi: 10.1103/PhysRevA.97.012322.
12. Ruskai M.B., Szarek S., Werner E. An analysis of completely-positive trace-preserving maps on m2. Linear Algebra Its Appl., 2002, vol. 347, nos. 1–3, pp. 159–187. doi: 10.1016/S0024-3795(01)00547-X.
13. King C. Additivity for unital qubit channels. J. Math. Phys., 2002, vol. 43, no. 10, pp. 4641–4653. doi: 10.1063/1.1500791.
14. Wolf M.M., Eisert J., Cubitt T.S., Cirac J.I. Assessing non-Markovian quantum dynamics. Phys. Rev. Lett., 2008, vol. 101, art. 150402. pp. 1–4. doi: 10.1103/PhysRevLett.101.150402.
15. Rivas Á., Huelga S.F., Plenio M.B. Entanglement and non-Markovianity of quantum evolutions. Phys. Rev. Lett., 2010, vol. 105, no. 5, art. 050403, pp. 1–4. doi: 10.1103/PhysRevLett.105.050403.
16. Hall M.J.W., Cresser J.D., Li L., Andersson E. Canonical form of master equations and characterization of non-Markovianity. Phys. Rev. A, 2014, vol. 89, no. 4, art. 042120. doi: 10.1103/PhysRevA.89.042120.
17. Chruściński D., Wudarski F.A. Non-Markovianity degree for random unitary evolution. Phys. Rev. A, 2015, vol. 91, no. 1, art. 012104, pp. 1–5. doi: 10.1103/PhysRevA.91.012104.
18. Filippov S.N., Piilo J., Maniscalco S., Ziman M. Divisibility of quantum dynamical maps and collision models. Phys. Rev. A, 2017, vol. 96, no. 3, art. 032111, pp. 1–13. doi: 10.1103/PhysRevA.96.032111.
19. Benatti F., Chruściński D., Filippov S. Tensor power of dynamical maps and positive versus completely positive divisibility. Phys. Rev. A, 2017, vol. 95, no. 1, art. 012112, pp. 1–5, doi: 10.1103/PhysRevA.95.012112.
20. Moravčíková L., Ziman M. Entanglement-annihilating and entanglement-breaking channels. J. Phys. A: Math. Theor., 2010, vol. 43, no. 27, art. 275306, pp. 1–11. doi: 10.1088/1751-8113/43/27/275306.
21. Filippov S.N., Rybár T., Ziman M. Local two-qubit entanglement-annihilating channels. Phys. Rev. A, 2012, vol. 85, no. 1, art. 012303, pp. 1–9. doi: 10.1103/PhysRevA.85.012303.
22. Filippov S.N., Ziman M. Bipartite entanglement-annihilating maps: Necessary and sufficient conditions. Phys. Rev. A, 2013, vol. 88, no. 3, art. 032316, pp. 1–7. doi: 10.1103/PhysRevA.88.032316.
23. Filippov S.N., Melnikov A.A., Ziman M. Dissociation and annihilation of multipartite entanglement structure in dissipative quantum dynamics. Phys. Rev. A, 2013, vol. 88, no. 6, art. 062328, pp. 1–11. doi: 10.1103/PhysRevA.88.062328.
24. Filippov S.N. PPT-inducing, distillation-prohibiting, and entanglement-binding quantum channels. J. Russ. Laser Res., 2014, vol. 35, no. 5, pp. 484–491. doi: 10.1007/s10946-014-9451-2.
25. Filippov S.N., Magadov K.Y., Jivulescu M.A. Absolutely separating quantum maps and channels. New J. Phys., 2017, vol. 19, art. 083010, pp. 1–19. doi: 10.1088/1367-2630/aa7e06.
26. Nielsen M.A., Chuang I.L. Quantum Computation and Quantum Information. Cambridge, Cambridge Univ. Press, 2000. 700 p.
For citation: Filippov S.N. Evaluation of non-unital qubit channel capacities. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, pp. 258–265.
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