Skip to main content
SpringerLink
Log in

The Complexity of Relating Quantum Channels to Master Equations

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Completely positive, trace preserving (CPT) maps and Lindblad master equations are both widely used to describe the dynamics of open quantum systems. The connection between these two descriptions is a classic topic in mathematical physics. One direction was solved by the now famous result due to Lindblad, Kossakowski, Gorini and Sudarshan, who gave a complete characterisation of the master equations that generate completely positive semi-groups. However, the other direction has remained open: given a CPT map, is there a Lindblad master equation that generates it (and if so, can we find its form)? This is sometimes known as the Markovianity problem. Physically, it is asking how one can deduce underlying physical processes from experimental observations.

We give a complexity theoretic answer to this problem: it is NP-hard. We also give an explicit algorithm that reduces the problem to integer semi-definite programming, a well-known NP problem. Together, these results imply that resolving the question of which CPT maps can be generated by master equations is tantamount to solving P = NP: any efficiently computable criterion for Markovianity would imply P = NP; whereas a proof that P = NP would imply that our algorithm already gives an efficiently computable criterion. Thus, unless P does equal NP, there cannot exist any simple criterion for determining when a CPT map has a master equation description.

However, we also show that if the system dimension is fixed (relevant for current quantum process tomography experiments), then our algorithm scales efficiently in the required precision, allowing an underlying Lindblad master equation to be determined efficiently from even a single snapshot in this case.

Our work also leads to similar complexity-theoretic answers to a related long-standing open problem in probability theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Nielsen M.A., Chuang I.L.: Quantum Computation and Quantum Information. Cambridge Univ. Press, Cambridge (2000)

    MATH  Google Scholar 

  2. Carmichael H.J.: Statistical Methods in Quantum Optics, Volume 1. Berlin-Heidelberg-New York: Springer, 2003

  3. Weiss, U.: Quantum dissipative systems. Series in Modern Condensed Matter Physics. River Edge, NJ: World Scientific, 1999

  4. Lindblad G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119 (1976)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Gorini V., Kossakowski A., Sudarshan E.C.G.: Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17, 821 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  6. Holevo A.S.: Statistical structure of quantum theory. Berlin-Heidelberg-New, York: Springer (2001)

    MATH  Google Scholar 

  7. Wolf M.M., Eisert J., Cubitt T.S., Cirac J.I.: Assessing non-Markovian quantum dynamics. Phys. Rev. Lett. 101, 150402 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  8. Boulant, N., Havel, T.F., Pravia, M.A., Cory, D.G.: Robust method for estimating the Lindblad operators of a dissipative quantum process from measurements of the density operator at multiple time points. Phys. Rev. A 67 042322 (2003)

    Google Scholar 

  9. Boulant N., Emerson J., Havel T.F., Cory D.G.: Incoherent noise and quantum information processing. J. Chem. Phys. 121(7), 2955 (2004)

    Article  ADS  Google Scholar 

  10. Howard M. et al.: Quantum process tomography and linblad estimation of a solid-state qubit. New J. Phys 8, 33 (2006)

    Article  ADS  Google Scholar 

  11. Weinstein Y.S. et al.: Quantum process tomography of the quantum fourier transform. J. Chem. Phys 121, 6117 (2004)

    Article  ADS  Google Scholar 

  12. Lidar D.A., Bihary Z., Whaley K.B.: From completely positive maps to the quantum Markovian semigroup master equation. Chem. Phys 268, 35 (2001)

    Article  ADS  Google Scholar 

  13. Papadimitriou, C.H.: Computational Complexity. Reading, MA: Addison Wesley (1993)

  14. Garey, M.R., Johnson D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Sanfrancisco, CA: W.H. Freeman (1979)

  15. Norris J.R.: Markov Chains. Cambridge Univ. Press, Cambridge (1997)

    MATH  Google Scholar 

  16. Elfving, G.: Zur theorie der Markoffschen ketten. Acta Soc. Sei. Fennicae. 2(8) (1937)

  17. Kingman J.F.C.: The imbedding problem for finite Markov chains. Z. Wahr 1, 14 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kingman J.F.C., Williams D.: The combinatorial structure of non-homogeneous Markov chains. Z. Wahr 26, 77 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fuglede B.: On the imbedding problem for stochastic and doubly stochastic matrices. Probab. Th. Rel. Fields, 80, 241 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mukherjea A.: The role of nonnegative idempotent matrices in certain problems in probability. In: Charles R. Johnson, ed., Matrix theory and applications. Providence, RI: Amer. Math. Soc., 1990

  21. Cubitt, T.S.: The embedding problem for stochastic matrices is NP-hard. Manuscript in preparation

  22. Nielsen M.A., Knill E., Laflamme R.: Complete quantum teleportation using nuclear magnetic resonance. Nature 396, 52 (1998)

    Article  ADS  Google Scholar 

  23. Vandersypen L.M.K., Chuang I.L.: NMR techniques for quantum control and computation. Rev. Mod. Phys 76, 1037 (2004)

    Article  ADS  Google Scholar 

  24. Emerson J. et al.: Symmetrized characterization of noisy quantum processes. Science, 317, 1893 (2007)

    Article  ADS  Google Scholar 

  25. Riebe M. et al.: Process tomography of ion trap quantum gates. Phys. Rev. Lett 97, 220407 (2006)

    Article  ADS  Google Scholar 

  26. Monz T. et al.: Realization of the quantum Toffoli gate with trapped ions. Phys. Rev. Lett. 102, 040501 (2009)

    Article  ADS  Google Scholar 

  27. O’Brien J.L. et al.: Quantum process tomography of a controlled-not gate. Phys. Rev. Lett. 93, 080502 (2004)

    Article  Google Scholar 

  28. Lundeen J.S. et al.: Tomography of quantum detectors. Nature Phys 5, 27 (2009)

    Article  ADS  Google Scholar 

  29. Howard M. et al.: Quantum process tomography and Linblad estimation of a solid-state qubit. New J. Phys 8, 33 (2006)

    Article  ADS  Google Scholar 

  30. Choi M.D.: Completely positive linear maps on complex matrices. Lin. Alg. Appl 10, 285 (1975)

    Article  MATH  Google Scholar 

  31. Jamiolkowski A.: Linear transformations which preserve trace and positive semidefiniteness of operators. Rep. Math. Phys. 3, 275 (1972)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. Wolf M.M., Cirac J.I.: Dividing quantum channels. Commun. Math. Phys. 279, 147 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. Denisov L.V.: Infinitely divisible markov mappings in quantum probability theory. Th. Prob. Appl 33, 392 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  34. Gurvits, L.: Classical deterministic complexity of Edmonds problem and quantum entanglement. In: Proceedings of the thirty-fifth ACM symposium on Theory of computing. New York: ACM Press, 2003, pp. 10–19

  35. Ioannou L.M.: Computational complexity of the quantum separability problem. Quant. Inf. Comp. 7(4), 335–370 (2007)

    MathSciNet  MATH  Google Scholar 

  36. Vandenberghe L., Boyd S.: Semidefinite programming. SIAM Rev. 38(1), 49–95 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  37. Horn R.A., Johnson C.R.: Topics in Matrix Analysis. Cambridge Univ. Press, Cambridge (1994)

    MATH  Google Scholar 

  38. Kato, T.: Perturbation Theory for Linear Operators. Berlin-Heidelberg-New York: Springer, Second edition, 1976

  39. Weilenmann J.: Continuity properties of fractional powers, of the logarithm, and of holomorphic semigroups. J. Func. Anal. 27, 1–20 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  40. Moler C., Van Loan C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45, 3–49 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the 10th Annual ACM Symposium on Theory of Computing (STOC’78), New York:ACM Press, 1978, p. 216

  42. Stewart G.W.: Error and perturbation bounds for subspaces associated with certain eigenvalue problems. SIAM Rev. 15(4), 727–764 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  43. Golub, G.H., van Loan, C.F.: Matrix Computations. Baltimore, MD: Johns Hopkins University Press, Third edition, 1996

  44. Horn R.A., Johnson C.R.: Matrix Analysis. Cambridge Univ. Press, Cambridge (1990)

    MATH  Google Scholar 

  45. Schrijver A.: Theory of Linear and Integer Programming. Wiley, New York (1986)

    MATH  Google Scholar 

  46. Porkolab, L., Khachiyan, L.: Computing integral points in convex semi-algebraic sets. In: Proceedings of the 38th Annual Symposium on Foundations of Computer Science (FOCS ’97), Discataway, NJ: IEEE 1997, p. 162

  47. Porkolab, L.: On the Complexity of Real and Integer Semidefinite Programming. PhD thesis, Rutgers, (1996)

  48. Cuthbert J.R.: The logarithm function for finite-state Markov semi-groups. J. London Math. Soc. 6(2), 524 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  49. Johansen S.: Some results on the imbedding problem for finite markov chains. J. London Math. Soc. 8(2), 345 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  50. Carette, P.: Characterizations of embeddable 3x3 stochastic matrices with a negative eigenvalue. New York J. Math. 1, 120 (1995)

    Google Scholar 

  51. Cuthbert J.R.: On uniqueness of the logarithm for Markov semi-groups. J. London Math. Soc. 4(2), 623 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  52. Davies E.B.: Embeddable Markov matrices. Electronic J. Prob 15, 1474 (2010)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK

    Toby S. Cubitt

  2. Departamento de Análisis Matemático, Universidad Complutense de Madrid, Plaza de Ciencias 3, Ciudad Universitaria, 28040, Madrid, Spain

    Toby S. Cubitt

  3. Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195, Berlin, Germany

    Jens Eisert

  4. Institute for Physics and Astronomy, Potsdam University, 14476, Potsdam, Germany

    Jens Eisert

  5. Niels Bohr Institute, Blegdamsvej 17, 2100, Copenhagen, Denmark

    Michael M. Wolf

  6. Zentrum Mathematik, Technische Universität München, 85748, Garching, Germany

    Michael M. Wolf

Authors
  1. Toby S. Cubitt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jens Eisert
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael M. Wolf
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Toby S. Cubitt.

Additional information

Communicated by M. B. Ruskai

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cubitt, T.S., Eisert, J. & Wolf, M.M. The Complexity of Relating Quantum Channels to Master Equations. Commun. Math. Phys. 310, 383–418 (2012). https://doi.org/10.1007/s00220-011-1402-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-011-1402-y

Keywords

Search

Navigation