Coherent information of a quantum channel or its complement is generically positive

Satvik Singh and Nilanjana Datta

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom

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The task of determining whether a given quantum channel has positive capacity to transmit quantum information is a fundamental open problem in quantum information theory. In general, the coherent information needs to be computed for an unbounded number of copies of a channel in order to detect a positive value of its quantum capacity. However, in this paper, we show that the coherent information of a $\textit{single copy}$ of a $\textit{randomly selected channel}$ is positive almost surely if the channel's output space is larger than its environment. Hence, in this case, a single copy of the channel typically suffices to determine positivity of its quantum capacity. Put differently, channels with zero coherent information have measure zero in the subset of channels for which the output space is larger than the environment. On the other hand, if the environment is larger than the channel's output space, identical results hold for the channel's complement.

If a quantum channel's output space is larger than its environment, then the information leakage by the channel to its environment is expected to be smaller in comparison to the amount of information that is sent to the output. Hence, such a channel should be able to transmit quantum information at a net positive rate. Surprisingly, this intuition fails to hold in general, and examples of quantum channels with large output spaces are known to exist that nevertheless have no capacity to transmit quantum information. However, we show that even though this intuition is not always correct, it is 'almost always' correct. In other words, whenever the output space of a channel is larger than its environment, one can be 'almost sure' that the channel has the ability to transmit quantum information at a strictly positive rate.

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[1] Howard Barnum, M. A. Nielsen, and Benjamin Schumacher. Information transmission through a noisy quantum channel. Phys. Rev. A, 57:4153–4175, Jun 1998. doi:10.1103/​PhysRevA.57.4153.

[2] Hellmut Baumgärtel. Analytic perturbation theory for matrices and operators. Birkhäuser Verlag, 1985.

[3] Charles H. Bennett, Gilles Brassard, Sandu Popescu, Benjamin Schumacher, John A. Smolin, and William K. Wootters. Purification of noisy entanglement and faithful teleportation via noisy channels. Phys. Rev. Lett., 76:722–725, Jan 1996. doi:10.1103/​PhysRevLett.76.722.

[4] Charles H. Bennett, David P. DiVincenzo, and John A. Smolin. Capacities of quantum erasure channels. Phys. Rev. Lett., 78:3217–3220, Apr 1997. doi:10.1103/​PhysRevLett.78.3217.

[5] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. Mixed-state entanglement and quantum error correction. Phys. Rev. A, 54:3824–3851, Nov 1996. doi:10.1103/​PhysRevA.54.3824.

[6] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. Mixed-state entanglement and quantum error correction. Phys. Rev. A, 54:3824–3851, Nov 1996. doi:10.1103/​PhysRevA.54.3824.

[7] Charles H. Bennett, Peter W. Shor, John A. Smolin, and Ashish V. Thapliyal. Entanglement-assisted classical capacity of noisy quantum channels. Phys. Rev. Lett., 83:3081–3084, Oct 1999. doi:10.1103/​PhysRevLett.83.3081.

[8] Samuel L. Braunstein and Peter van Loock. Quantum information with continuous variables. Rev. Mod. Phys., 77:513–577, Jun 2005. doi:10.1103/​RevModPhys.77.513.

[9] N. Cai, A. Winter, and R. W. Yeung. Quantum privacy and quantum wiretap channels. Problems of Information Transmission, 40(4):318–336, October 2004. doi:10.1007/​s11122-005-0002-x.

[10] Man-Duen Choi. Completely positive linear maps on complex matrices. Linear Algebra and its Applications, 10(3):285–290, June 1975. doi:10.1016/​0024-3795(75)90075-0.

[11] John B Conway. A Course in Functional Analysis. Graduate Texts in Mathematics. Springer, New York, NY, 2 edition, January 1994.

[12] Toby Cubitt, David Elkouss, William Matthews, Maris Ozols, David Pérez-García, and Sergii Strelchuk. Unbounded number of channel uses may be required to detect quantum capacity. Nature Communications, 6(1), March 2015. doi:10.1038/​ncomms7739.

[13] Toby S. Cubitt, Mary Beth Ruskai, and Graeme Smith. The structure of degradable quantum channels. Journal of Mathematical Physics, 49(10):102104, 2008. arXiv:https:/​/​​10.1063/​1.2953685, doi:10.1063/​1.2953685.

[14] I. Devetak. The private classical capacity and quantum capacity of a quantum channel. IEEE Transactions on Information Theory, 51(1):44–55, 2005. doi:10.1109/​TIT.2004.839515.

[15] I. Devetak and P. W. Shor. The capacity of a quantum channel for simultaneous transmission of classical and quantum information. Communications in Mathematical Physics, 256(2):287–303, March 2005. doi:10.1007/​s00220-005-1317-6.

[16] David P. DiVincenzo, Peter W. Shor, and John A. Smolin. Quantum-channel capacity of very noisy channels. Phys. Rev. A, 57:830–839, Feb 1998. doi:10.1103/​PhysRevA.57.830.

[17] G. Edgar. Measure, Topology, and Fractal Geometry. Undergraduate Texts in Mathematics. Springer New York, 2008. URL: https:/​/​​books?id=6DpyQgAACAAJ.

[18] Jean Ginibre. Statistical ensembles of complex, quaternion, and real matrices. Journal of Mathematical Physics, 6(3):440–449, March 1965. doi:10.1063/​1.1704292.

[19] Vittorio Giovannetti and Rosario Fazio. Information-capacity description of spin-chain correlations. Phys. Rev. A, 71:032314, Mar 2005. doi:10.1103/​PhysRevA.71.032314.

[20] M. Grassl, Th. Beth, and T. Pellizzari. Codes for the quantum erasure channel. Phys. Rev. A, 56:33–38, Jul 1997. doi:10.1103/​PhysRevA.56.33.

[21] Leonid Gurvits. Classical deterministic complexity of Edmonds' problem and quantum entanglement. In Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, STOC '03, page 10–19, New York, NY, USA, 2003. Association for Computing Machinery. doi:10.1145/​780542.780545.

[22] Erkka Haapasalo, Michal Sedlák, and Mário Ziman. Distance to boundary and minimum-error discrimination. Phys. Rev. A, 89:062303, Jun 2014. doi:10.1103/​PhysRevA.89.062303.

[23] P.R. Halmos. Measure Theory. Graduate Texts in Mathematics. Springer New York, 1976. URL: https:/​/​​books?id=-Rz7q4jikxUC.

[24] Klemens Hammerer, Anders S. Sørensen, and Eugene S. Polzik. Quantum interface between light and atomic ensembles. Rev. Mod. Phys., 82:1041–1093, Apr 2010. doi:10.1103/​RevModPhys.82.1041.

[25] M. B. Hastings. Superadditivity of communication capacity using entangled inputs. Nature Physics, 5(4):255–257, March 2009. doi:10.1038/​nphys1224.

[26] Patrick Hayden, Sepehr Nezami, Xiao-Liang Qi, Nathaniel Thomas, Michael Walter, and Zhao Yang. Holographic duality from random tensor networks. Journal of High Energy Physics, 2016(11), November 2016. doi:10.1007/​jhep11(2016)009.

[27] Patrick Hayden and Andreas Winter. Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1. Communications in Mathematical Physics, 284(1):263–280, September 2008. doi:10.1007/​s00220-008-0624-0.

[28] Alexander S. Holevo. Quantum Systems, Channels, Information. De Gruyter, November 2012. doi:10.1515/​9783110273403.

[29] A.S. Holevo. The capacity of the quantum channel with general signal states. IEEE Transactions on Information Theory, 44(1):269–273, 1998. doi:10.1109/​18.651037.

[30] Paweł Horodecki, Michał Horodecki, and Ryszard Horodecki. Binding entanglement channels. Journal of Modern Optics, 47(2-3):347–354, February 2000. doi:10.1080/​09500340008244047.

[31] Pavan Hosur, Xiao-Liang Qi, Daniel A. Roberts, and Beni Yoshida. Chaos in quantum channels. Journal of High Energy Physics, 2016(2), February 2016. doi:10.1007/​jhep02(2016)004.

[32] A. Jamiołkowski. Linear transformations which preserve trace and positive semidefiniteness of operators. Reports on Mathematical Physics, 3(4):275–278, December 1972. doi:10.1016/​0034-4877(72)90011-0.

[33] Youn-Chang Jeong, Jong-Chan Lee, and Yoon-Ho Kim. Experimental implementation of a fully controllable depolarizing quantum operation. Phys. Rev. A, 87:014301, Jan 2013. doi:10.1103/​PhysRevA.87.014301.

[34] C. King. The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory, 49(1):221–229, 2003. doi:10.1109/​TIT.2002.806153.

[35] C. King, K. Matsumoto, M. Nathanson, and M.B. Ruskai. Properties of conjugate channels with applications to additivity and multiplicativity. Markov Processes And Related Fields, 13(2):391–423, 2007.

[36] Dennis Kretschmann, Dirk Schlingemann, and Reinhard F. Werner. The information-disturbance tradeoff and the continuity of Stinespring's representation. IEEE Transactions on Information Theory, 54(4):1708–1717, 2008. doi:10.1109/​TIT.2008.917696.

[37] Ryszard Kukulski, Ion Nechita, Łukasz Pawela, Zbigniew Puchała, and Karol Życzkowski. Generating random quantum channels. Journal of Mathematical Physics, 62(6):062201, Jun 2021. doi:10.1063/​5.0038838.

[38] Felix Leditzky, Debbie Leung, and Graeme Smith. Dephrasure channel and superadditivity of coherent information. Phys. Rev. Lett., 121:160501, Oct 2018. doi:10.1103/​PhysRevLett.121.160501.

[39] Debbie Leung and Graeme Smith. Continuity of quantum channel capacities. Communications in Mathematical Physics, 292(1):201–215, May 2009. doi:10.1007/​s00220-009-0833-1.

[40] Sheng-Kai Liao, Hai-Lin Yong, Chang Liu, Guo-Liang Shentu, Dong-Dong Li, Jin Lin, Hui Dai, Shuang-Qiang Zhao, Bo Li, Jian-Yu Guan, Wei Chen, Yun-Hong Gong, Yang Li, Ze-Hong Lin, Ge-Sheng Pan, Jason S. Pelc, M. M. Fejer, Wen-Zhuo Zhang, Wei-Yue Liu, Juan Yin, Ji-Gang Ren, Xiang-Bin Wang, Qiang Zhang, Cheng-Zhi Peng, and Jian-Wei Pan. Long-distance free-space quantum key distribution in daylight towards inter-satellite communication. Nature Photonics, 11(8):509–513, July 2017. doi:10.1038/​nphoton.2017.116.

[41] Seth Lloyd. Capacity of the noisy quantum channel. Phys. Rev. A, 55:1613–1622, Mar 1997. doi:10.1103/​PhysRevA.55.1613.

[42] László Lovász. Singular spaces of matrices and their application in combinatorics. Boletim da Sociedade Brasileira de Matemática, 20(1):87–99, October 1989. doi:10.1007/​bf02585470.

[43] I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, and N. Gisin. Long-distance teleportation of qubits at telecommunication wavelengths. Nature, 421(6922):509–513, January 2003. doi:10.1038/​nature01376.

[44] B. Marques, A. A. Matoso, W. M. Pimenta, A. J. Gutiérrez-Esparza, M. F. Santos, and S. Pádua. Experimental simulation of decoherence in photonics qudits. Scientific Reports, 5(1), November 2015. doi:10.1038/​srep16049.

[45] Francesco Mezzadri. How to generate random matrices from the classical compact groups. Notices of the American Mathematical Society, 54(5):592 – 604, May 2007.

[46] Ashley Montanaro. Weak multiplicativity for random quantum channels. Communications in Mathematical Physics, 319(2):535–555, January 2013. doi:10.1007/​s00220-013-1680-7.

[47] Ramis Movassagh and Jeffrey Schenker. Theory of ergodic quantum processes, 2020. arXiv:2004.14397.

[48] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, USA, 10th edition, 2011.

[49] Cheng-Zhi Peng, Tao Yang, Xiao-Hui Bao, Jun Zhang, Xian-Min Jin, Fa-Yong Feng, Bin Yang, Jian Yang, Juan Yin, Qiang Zhang, Nan Li, Bao-Li Tian, and Jian-Wei Pan. Experimental free-space distribution of entangled photon pairs over 13 km: Towards satellite-based global quantum communication. Phys. Rev. Lett., 94:150501, Apr 2005. doi:10.1103/​PhysRevLett.94.150501.

[50] F. Rellich and J. Berkowitz. Perturbation Theory of Eigenvalue Problems. New York University. Institute of Mathematical Sciences. Gordon and Breach, 1969.

[51] M. Ricci, F. De Martini, N. J. Cerf, R. Filip, J. Fiurášek, and C. Macchiavello. Experimental purification of single qubits. Phys. Rev. Lett., 93:170501, Oct 2004. doi:10.1103/​PhysRevLett.93.170501.

[52] Tobias Schmitt-Manderbach, Henning Weier, Martin Fürst, Rupert Ursin, Felix Tiefenbacher, Thomas Scheidl, Josep Perdigues, Zoran Sodnik, Christian Kurtsiefer, John G. Rarity, Anton Zeilinger, and Harald Weinfurter. Experimental demonstration of free-space decoy-state quantum key distribution over 144 km. Phys. Rev. Lett., 98:010504, Jan 2007. doi:10.1103/​PhysRevLett.98.010504.

[53] Benjamin Schumacher and Michael D. Westmoreland. Sending classical information via noisy quantum channels. Phys. Rev. A, 56:131–138, Jul 1997. doi:10.1103/​PhysRevA.56.131.

[54] A. Shaham and H. S. Eisenberg. Realizing controllable depolarization in photonic quantum-information channels. Phys. Rev. A, 83:022303, Feb 2011. doi:10.1103/​PhysRevA.83.022303.

[55] Peter Shor. The quantum channel capacity and coherent information. MSRI Workshop on Quantum Computation, 2002.

[56] Peter W. Shor. Equivalence of additivity questions in quantum information theory. Communications in Mathematical Physics, 246(3):453–472, April 2004. doi:10.1007/​s00220-003-0981-7.

[57] Vikesh Siddhu. Entropic singularities give rise to quantum transmission. Nat. Commun., 12(1), October 2021. URL: https:/​/​​10.1038/​s41467-021-25954-0.

[58] Satvik Singh and Nilanjana Datta. Detecting positive quantum capacities of quantum channels. npj Quantum Information, 8(1), May 2022. doi:10.1038/​s41534-022-00550-2.

[59] Satvik Singh and Nilanjana Datta. Fully undistillable quantum states are separable. preprint arXiv:2207.05193, 2022.

[60] Sergei Slussarenko and Geoff J. Pryde. Photonic quantum information processing: A concise review. Applied Physics Reviews, 6(4):041303, December 2019. doi:10.1063/​1.5115814.

[61] G. Smith and J. Yard. Quantum communication with zero-capacity channels. Science, 321(5897):1812–1815, September 2008. doi:10.1126/​science.1162242.

[62] Graeme Smith and John A. Smolin. Detecting incapacity of a quantum channel. Phys. Rev. Lett., 108:230507, Jun 2012. doi:10.1103/​PhysRevLett.108.230507.

[63] W. Forrest Stinespring. Positive functions on C$^*$-algebras. Proceedings of the American Mathematical Society, 6(2):211–216, 1955. doi:10.1090/​s0002-9939-1955-0069403-4.

[64] David Sutter, Volkher B. Scholz, Andreas Winter, and Renato Renner. Approximate degradable quantum channels. IEEE Transactions on Information Theory, 63(12):7832–7844, 2017. doi:10.1109/​TIT.2017.2754268.

[65] Hiroki Takesue, Sae Woo Nam, Qiang Zhang, Robert H. Hadfield, Toshimori Honjo, Kiyoshi Tamaki, and Yoshihisa Yamamoto. Quantum key distribution over a 40-dB channel loss using superconducting single-photon detectors. Nature Photonics, 1(6):343–348, June 2007. doi:10.1038/​nphoton.2007.75.

[66] Rupert Ursin, Thomas Jennewein, Markus Aspelmeyer, Rainer Kaltenbaek, Michael Lindenthal, Philip Walther, and Anton Zeilinger. Quantum teleportation across the Danube. Nature, 430(7002):849–849, August 2004. doi:10.1038/​430849a.

[67] Shun Watanabe. Private and quantum capacities of more capable and less noisy quantum channels. Phys. Rev. A, 85:012326, Jan 2012. doi:10.1103/​PhysRevA.85.012326.

[68] Christian Weedbrook, Stefano Pirandola, Raúl García-Patrón, Nicolas J. Cerf, Timothy C. Ralph, Jeffrey H. Shapiro, and Seth Lloyd. Gaussian quantum information. Rev. Mod. Phys., 84:621–669, May 2012. doi:10.1103/​RevModPhys.84.621.

[69] R. F. Werner and A. S. Holevo. Counterexample to an additivity conjecture for output purity of quantum channels. Journal of Mathematical Physics, 43(9):4353–4357, September 2002. doi:10.1063/​1.1498491.

[70] Mark M. Wilde. Quantum Information Theory. Cambridge University Press, 2013. doi:10.1017/​cbo9781139525343.

[71] Paolo Zanardi and Namit Anand. Information scrambling and chaos in open quantum systems. Phys. Rev. A, 103:062214, Jun 2021. doi:10.1103/​PhysRevA.103.062214.

Cited by

[1] Satvik Singh and Nilanjana Datta, "Fully Undistillable Quantum States Are Separable", IEEE Transactions on Information Theory 69 10, 6564 (2023).

[2] Satvik Singh and Nilanjana Datta, "Detecting positive quantum capacities of quantum channels", npj Quantum Information 8 1, 50 (2022).

[3] D. -S. Wang, "On quantum channel capacities: an additive refinement", arXiv:2205.07205, (2022).

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