Abstract
The Quantum Reverse Shannon Theorem states that any quantum channel can be simulated by an unlimited amount of shared entanglement and an amount of classical communication equal to the channel’s entanglement assisted classical capacity. In this extended abstract, we summarize a new and conceptually simple proof of this theorem [journal reference: arXiv.org:quant-ph/0912.3805], which has previously been proved in [Bennett et al., arXiv.org:quant-ph/0912.5537]. Our proof is based on optimal one-shot Quantum State Merging and the Post-Selection Technique for quantum channels.
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References
Abeyesinghe, A., Devetak, I., Hayden, P., Winter, A.: The mother of all protocols: Restructuring quantum information’s family tree. Proc. R. Soc. A 465(2108), 2537 (2009), arXiv.org:quant-ph/0606225
Bennett, C.H., Devetak, I., Harrow, A.W., Shor, P.W., Winter, A.: The quantum reverse Shannon theorem (2006), arXiv.org:quant-ph/0912.5537
Bennett, C.H., Shor, P.W., Smolin, J.A., Thapliyal, A.V.: Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem. IEEE Trans. Inf. Theory 48(10), 2637 (2002), arXiv.org:quant-ph/0106052
Berta, M.: Single-shot quantum state merging, Diploma thesis ETH Zurich (2008), arXiv.org:quant-ph/0912.4495
Berta, M., Christandl, M., Renner, R.: A Conceptually Simple Proof of the Quantum Reverse Shannon Theorem (2009), arXiv.org:quant-ph/0912.3805, submitted to Comm. Math. Phys. (2009)
Berta, M., Dupuis, F., Renner, R., Wullschleger, J.: Optimal decoupling (2010) (in preparation)
Christandl, M., König, R., Renner, R.: Post-selection technique for quantum channels with applications to quantum cryptography. Phys. Rev. Lett 102, 20504 (2009), arXiv.org:quant-ph/0809.3019
Datta, N.: Min- and max- relative entropies and a new entanglement monotone. IEEE Trans. Inf. Theory 55(6), 2816 (2009), arXiv.org:quant-ph/0803.2770
Devetak, I.: The private classical capacity and quantum capacity of a quantum channel. IEEE Trans. Inf. Theory 51, 44 (2005), arXiv:quant-ph/0304127
Dupuis, F.: The Decoupling Approach to Quantum Information Theory. PhD thesis, Université de Montréal (2009), arXiv.org:quant-ph/1004.1641
Harrow, A.W.: Entanglement spread and clean resource inequalities. Proc. XVI Int. Cong. Math. Phys. 536 (2009), arXiv.org:quant-ph/0909.1557
Holevo, A.S.: The capacity of the quantum communication channel with general signal states. IEEE Trans. Inf. Theory 44, 269 (1998), arXiv.org:quant-ph/9611023
Horodecki, M., Oppenheim, J., Winter, A.: Partial quantum information. Nature 436, 673–676 (2005), arXiv.org:quant-ph/0505062
Horodecki, M., Oppenheim, J., Winter, A.: Quantum state merging and negative information. Comm. Math. Phys 269, 107 (2006), arXiv.org:quant-ph/0512247
Kitaev, A.: Quantum computations: algorithms and error correction. Russian Math. Surveys 52, 1191 (1997)
König, R., Renner, R., Schaffner, C.: The operational meaning of min- and max-entropy. IEEE Trans. Inf. Theory 55(9), 4337 (2009), arXiv.org:quant-ph/0807.1338
Lloyd, S.: Capacity of the noisy quantum channel. Phys. Rev. A 55, 1613 (1997), arXiv.org:quant-ph/9604015
Oppenheim, J.: State redistribution as merging: introducing the coherent relay (2008), arXiv.org:quant-ph/0805.1065
Renner, R.: Security of Quantum Key Distribution. PhD thesis, ETH Zurich (2005), arXiv.org:quant-ph/0512258
Renner, R.S., König, R.: Universally Composable Privacy Amplification Against Quantum Adversaries. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 407–425. Springer, Heidelberg (2005)
Renner, R., Wolf, S.: Smooth Rényi entropy and applications. In: Proc. IEEE Int. Symp. Inf. Theory, vol. 233 (2004)
Schumacher, B., Westmoreland, M.D.: Sending classical information via noisy quantum channels. Phys. Rev. A 56, 131 (1997)
Shannon, C.E.: A mathematical theory of communication. Bell. Syst. Tech. J. 423, 379–423, 623–656 (1948)
Shor, P.W.: The quantum channel capacity and coherent information. In: Lecture notes, MSRI Workshop on Quantum Computation (2002)
Slepian, D., Wolf, J.: Noiseless coding of correlated information sources. IEEE Trans. Inf. Theory 19, 461 (1971)
Stinespring, W.: Positive function on C*-algebras. Proc. Amer. Math. Soc. 6, 211 (1955)
Tomamichel, M., Colbeck, R., Renner, R.: A fully quantum asymptotic equipartition property. IEEE Trans. Inf. Theory 55(12), 5840 (2009), arXiv.org:quant-ph/0811.1221
Tomamichel, M., Colbeck, R., Renner, R.: Duality between smooth min- and max-entropies. IEEE Trans. Inf. Theory 56(9), 4674 (2010), arXiv.org:quant-ph/0907.5238
van Dam, W., Hayden, P.: Universal entanglement transformations without communication. Phys. Rev. A, Rapid Comm. 67, 060302(R) (2003), arXiv.org:quant-ph/0201041
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Berta, M., Christandl, M., Renner, R. (2011). A Conceptually Simple Proof of the Quantum Reverse Shannon Theorem. In: van Dam, W., Kendon, V.M., Severini, S. (eds) Theory of Quantum Computation, Communication, and Cryptography. TQC 2010. Lecture Notes in Computer Science, vol 6519. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18073-6_11
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DOI: https://doi.org/10.1007/978-3-642-18073-6_11
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