Skip to main content
SpringerLink
Log in

Quantum Tomography under Prior Information

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

Abstract

We provide a detailed analysis of the question: how many measurement settings or outcomes are needed in order to identify an unknown quantum state which is constrained by prior information? We show that if the prior information restricts the possible states to a set of lower dimensionality, then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the set of all states. Conversely, we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the Minkowski dimension of the set. We apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints. In particular, we show that the minimal number of measurement outcomes (POVM elements) necessary to identify all pure states in a d-dimensional Hilbert space is 4d−3−c(d) α(d) for some \({c(d)\in[1,2]}\) and α(d) being the number of ones appearing in the binary expansion of (d−1).

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. Weigert S.: Pauli problem for a spin of arbitrary length: A simple method to determine its wave function. Phys. Rev. A 45, 7688–7696 (1992)

    Article  ADS  Google Scholar 

  2. Amiet J.-P., Weigert S.: Reconstructing a pure state of a spin s through three Stern-Gerlach measurements. J. Phys. A 32, 2777–2784 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Amiet J.-P., Weigert S.: Reconstructing the density matrix of a spin s through Stern-Gerlach measurements. II. J. Phys. A 32, L269–L274 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Finkelstein J.: Pure-state informationally complete and “really” complete measurements. Phys. Rev. A 70, 052107 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  5. Flammia S., Silberfarb A., Caves C.: Minimal informationally complete measurements for pure states. Found. Phys. 35, 1985–2006 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Gross D., Liu Y.-K., Flammia S.T., Becker S., Eisert J.: Quantum state tomography via compressed sensing. Phys. Rev. Lett. 105, 150401 (2010)

    Article  ADS  Google Scholar 

  7. Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. Amsterdam: North-Holland Publishing Co., 1982

  8. Busch, P., Grabowski, M., Lahti, P.J.: Operational Quantum Physics. Berlin: Springer-Verlag, 1997, second corrected printing

  9. Prugovečki E.: Information-theoretical aspects of quantum measurements. Int. J. Theor. Phys. 16, 321–331 (1977)

    Article  MATH  Google Scholar 

  10. Busch P.: Informationally complete sets of physical quantities. Internat. J. Theoret. Phys. 30(9), 1217–1227 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  11. Caves C.M., Fuchs C.A., Schack R.: Unknown quantum states: the quantum de Finetti representation. J. Math. Phys. 43, 4537–4559 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Cubitt T., Montanaro A., Winter A.: On the dimension of subspaces with bounded Schmidt rank. J. Math. Phys. 49, 022107 (2008)

    Article  MathSciNet  Google Scholar 

  13. Fallat S.M.: Bidiagonal factorizations of totally nonnegative matrices. Amer. Math. Monthly 108, 697–712 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. James I.M.: Some embeddings of projective spaces. Proc. Camb. Phil. Soc. 55, 294–298 (1959)

    Article  ADS  MATH  Google Scholar 

  15. Milgram R.J.: Immersing projective spaces. Ann. of Math. (2) 85, 473–482 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mukherjee A.: Embedding complex projective spaces in Euclidean space. Bull. London Math. Soc. 13, 323–324 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mañé, R.: On the dimension of the compact invariant sets of certain nonlinear maps. In: Dynamical systems and turbulence, Volume 898 of Lecture Notes in Math., Berlin-Heidelberg-Newyork: Springer, 1981, pp. 230–242

  18. Hunt B.R., Kaloshin V.Y.: Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces. Nonlinearity 12, 1263–1275 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Robinson J.C.: Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces. Nonlinearity 22, 711–728 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Ben-Artzi A., Eden A., Foias C., Nicolaenko B.: Hölder continuity for the inverse of Mañé’s projection. J. Math. Anal. Appl. 178, 22–29 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  21. Whitney H.: The self-intersections of a smooth n-manifold in 2n-space. Ann. of Math. 45, 220–246 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  22. James I.M.: Euclidean models of projective spaces. Bull. London Math. Soc. 3, 257–276 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  23. Adachi, M.: Embeddings and immersions, Volume 124 of Translations of Mathematical Monographs. Providence, RI: Amer. Math. Soc., 1993, Translated from the 1984 Japanese original by Kiki Hudson

  24. Chern S.: On the multiplication in the characteristic ring of a sphere bundle. Ann. of Math. 49, 362–372 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  25. Atiyah M.F., Hirzebruch F.: Quelques théorèmes de non-plongement pour les variétés différentiables. Bull. Soc. Math. France 87, 383–396 (1959)

    MathSciNet  MATH  Google Scholar 

  26. Mayer K.H.: Elliptische Differentialoperatoren und Ganzzahligkeitssätze für charakteristische Zahlen. Topology 4, 295–313 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sugawara T.: Nonimmersion and nonembedding theorems for complex Grassmann manifolds. Proc. Japan Acad. Ser. A Math. Sci. 55, 59–64 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  28. Dimitrić I.: A note on equivariant embeddings of Grassmannians. Publ. Inst. Math. (Beograd) 59, 131–137 (1996)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, Turku, Finland

    Teiko Heinosaari

  2. Department of Information Engineering, University of Padova, Via Gradenigo 6/B, 35131, Padova, Italy

    Luca Mazzarella

  3. Department of Mathematics, Technische Universität München, 85748, Garching, Germany

    Michael M. Wolf

Authors
  1. Teiko Heinosaari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luca Mazzarella
    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 Teiko Heinosaari.

Additional information

Communicated by M. B. Ruskai

Rights and permissions

Reprints and permissions

About this article

Cite this article

Heinosaari, T., Mazzarella, L. & Wolf, M.M. Quantum Tomography under Prior Information. Commun. Math. Phys. 318, 355–374 (2013). https://doi.org/10.1007/s00220-013-1671-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-013-1671-8

Keywords

Search

Navigation