## Generalized multiplicative domains and quantum error correction

HTML articles powered by AMS MathViewer

- 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).

## 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