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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Generalized multiplicative domains and quantum error correction
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by Nathaniel Johnston and David W. Kribs PDF
Proc. Amer. Math. Soc. 139 (2011), 627-639 Request permission

Abstract:

Given a completely positive map, we introduce a set of algebras that we refer to as its generalized multiplicative domains. These algebras are generalizations of the traditional multiplicative domain of a completely positive map, and we derive a characterization of them in the unital, trace-preserving case, in other words the case of unital quantum channels, that extends Choi’s characterization of the multiplicative domains of unital maps. We also derive a characterization that is in the same flavour as a well-known characterization of bimodules, and we use these algebras to provide a new representation-theoretic description of quantum error-correcting codes that extends previous results for unitarily-correctable codes, noiseless subsystems and decoherence-free subspaces.
References
  • S. A. Aly, A. Klappenecker, Subsystem code constructions. IEEE International Symposium on Information Theory, ISIT (2008), 369-373.
  • D. Bacon, Operator quantum error-correcting subsystems for self-correcting quantum memories. Phys. Rev. A, 73 012340 (2006).
  • Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters, Mixed-state entanglement and quantum error correction, Phys. Rev. A (3) 54 (1996), no. 5, 3824–3851. MR 1418618, DOI 10.1103/PhysRevA.54.3824
  • C. Beny, A. Kempf, D. W. Kribs, Generalization of Quantum Error Correction via the Heisenberg Picture. Phys. Rev. Lett., 98 100502 (2007).
  • C. Beny, D. W. Kribs, A. Pasieka, Algebraic formulation of quantum error correction. Int. J. Quantum Inf., 6 (2008), 597-603.
  • R. Blume-Kohout, H.K. Ng, D. Poulin, L. Viola, Characterizing the structure of preserved information in quantum processes. Phys. Rev. Lett. 100, 030501 (2008).
  • Man Duen Choi, A Schwarz inequality for positive linear maps on $C^{\ast } \$-algebras, Illinois J. Math. 18 (1974), 565–574. MR 355615
  • Man Duen Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10 (1975), 285–290. MR 376726, DOI 10.1016/0024-3795(75)90075-0
  • Man-Duen Choi, Nathaniel Johnston, and David W. Kribs, The multiplicative domain in quantum error correction, J. Phys. A 42 (2009), no. 24, 245303, 15. MR 2515540, DOI 10.1088/1751-8113/42/24/245303
  • M. D. Choi, D. W. Kribs, Method to Find Quantum Noiseless Subsystems. Phys. Rev. Lett. 96, 050501 (2006).
  • Kenneth R. Davidson, $C^*$-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1402012, DOI 10.1090/fim/006
  • L.-M. Duan G.-C. Guo, Preserving Coherence in Quantum Computation by Pairing Quantum Bits. Phys. Rev. Lett. 79, 1953 (1997).
  • Daniel Gottesman, An introduction to quantum error correction, Quantum computation: a grand mathematical challenge for the twenty-first century and the millennium (Washington, DC, 2000) Proc. Sympos. Appl. Math., vol. 58, Amer. Math. Soc., Providence, RI, 2002, pp. 221–235. MR 1922900, DOI 10.1090/psapm/058/1922900
  • Daniel Gottesman, Class of quantum error-correcting codes saturating the quantum Hamming bound, Phys. Rev. A (3) 54 (1996), no. 3, 1862–1868. MR 1450567, DOI 10.1103/PhysRevA.54.1862
  • John A. Holbrook, David W. Kribs, and Raymond Laflamme, Noiseless subsystems and the structure of the commutant in quantum error correction, Quantum Inf. Process. 2 (2003), no. 5, 381–419 (2004). MR 2065970, DOI 10.1023/B:QINP.0000022737.53723.b4
  • J. Kempe, D. Bacon, D. A. Lidar, K. B. Whaley, Theory of decoherence-free fault-tolerant universal quantum compuation. Phys. Rev. A 63, 42307 (2001).
  • Andreas Klappenecker and Pradeep Kiran Sarvepalli, On subsystem codes beating the quantum Hamming or Singleton bound, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007), no. 2087, 2887–2905. MR 2360182, DOI 10.1098/rspa.2007.0028
  • E. Knill, Protected realizations of quantum information, Phys. Rev. A (3) 74 (2006), no. 4, 042301, 11. MR 2287923, DOI 10.1103/PhysRevA.74.042301
  • E. Knill, R. Laflamme, A. Ashikhmin, H.N. Barnum, L. Viola, W.H. Zurek, Introduction to quantum error correction, Los Alamos Science, November 27, 2002.
  • Emanuel Knill, Raymond Laflamme, and Lorenza Viola, Theory of quantum error correction for general noise, Phys. Rev. Lett. 84 (2000), no. 11, 2525–2528. MR 1745959, DOI 10.1103/PhysRevLett.84.2525
  • D. Kribs, R. Laflamme, D. Poulin, Unified and Generalized Approach to Quantum Error Correction. Phys. Rev. Lett. 94, 180501 (2005).
  • David W. Kribs, Raymond Laflamme, David Poulin, and Maia Lesosky, Operator quantum error correction, Quantum Inf. Comput. 6 (2006), no. 4-5, 382–398. MR 2255992
  • D. W. Kribs, R. W. Spekkens, Quantum Error Correcting Subsystems as Unitarily Recoverable Subsystems. Phys. Rev. A 74, 042329 (2006).
  • D. A. Lidar, I. L. Chuang, K. B. Whaley, Decoherence-free subspaces for quantum computation. Phys. Rev. Lett. 81, 2594 (1998).
  • Michael A. Nielsen and Isaac L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge, 2000. MR 1796805
  • G. Massimo Palma, Kalle-Antti Suominen, and Artur K. Ekert, Quantum computers and dissipation, Proc. Roy. Soc. London Ser. A 452 (1996), no. 1946, 567–584. MR 1378845, DOI 10.1098/rspa.1996.0029
  • Vern Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR 1976867
  • D. Poulin, Stabilizer Formalism for Operator Quantum Error Correction. Phys. Rev. Lett. 95, 230504 (2005).
  • A. Shabani, D.A. Lidar, Theory of Initialization-Free Decoherence-Free Subspaces and Subsystems. Phys. Rev. A 72, 042303 (2005).
  • P. W. Shor, Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, R2493 (1995).
  • M. Silva, E. Magesan, D. W. Kribs, J. Emerson, Scalable protocol for identification of correctable codes. Phys. Rev. A 78, 012347 (2008).
  • A. M. Steane, Error correcting codes in quantum theory, Phys. Rev. Lett. 77 (1996), no. 5, 793–797. MR 1398854, DOI 10.1103/PhysRevLett.77.793
  • Paolo Zanardi, Stabilizing quantum information, Phys. Rev. A (3) 63 (2001), no. 1, 012301, 4. MR 1816609, DOI 10.1103/PhysRevA.63.012301
  • P. Zanardi, M. Rasetti, Noiseless quantum codes. Phys. Rev. Lett. 79, 3306 (1997).
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Additional Information
  • Nathaniel Johnston
  • Affiliation: Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada N1G 2W1
  • David W. Kribs
  • Affiliation: Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
  • Received by editor(s): July 20, 2009
  • Received by editor(s) in revised form: March 17, 2010
  • Published electronically: July 26, 2010
  • Communicated by: Marius Junge
  • © Copyright 2010 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 627-639
  • MSC (2010): Primary 46L05, 47L05, 46N50
  • DOI: https://doi.org/10.1090/S0002-9939-2010-10556-7
  • MathSciNet review: 2736344