Abstract
This paper presents a series of results on the interplay between quantum estimation, cloning and finite de Finetti theorems. First, we consider the measure-and-prepare channel that uses optimal estimation to convert M copies into k approximate copies of an unknown pure state and we show that this channel is equal to a random loss of all but s particles followed by cloning from s to k copies. When the number k of output copies is large with respect to the number M of input copies the measure-and-prepare channel converges in diamond norm to the optimal universal cloning. In the opposite case, when M is large compared to k, the estimation becomes almost perfect and the measure-and-prepare channel converges in diamond norm to the partial trace over all but k systems. This result is then used to derive de Finetti-type results for quantum states and for symmetric broadcast channels, that is, channels that distribute quantum information to many receivers in a permutationally invariant fashion. Applications of the finite de Finetti theorem for symmetric broadcast channels include the derivation of diamond-norm bounds on the asymptotic convergence of quantum cloning to state estimation and the derivation of bounds on the amount of quantum information that can be jointly decoded by a group of k receivers at the output of a symmetric broadcast channel.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Gisin, N., Massar, S.: Phys. Rev. Lett. 79, 2153 (1997)
Werner, R.F.: Phys. Rev. AÂ 58, 1827 (1998)
Bruß, D., Ekert, A., Macchiavello, C.: Phys. Rev. Lett. 81, 2598 (1998)
Keyl, M., Werner, R.F.: J. Math. Phys. 40, 3283 (1999)
Bruß, D., Cinchetti, M., DAriano, G.M., Macchiavello, C.: Phys. Rev. A 62, 12302 (2000)
D’Ariano, G.M., Macchiavello, C.: Phys. Rev. A 67, 042306 (2003)
Bae, J., AcÃn, A.: Phys. Rev. Lett. 97, 30402 (2006)
Chiribella, G., D’Ariano, G.M.: Phys. Rev. Lett. 97, 250503 (2006)
Keyl, M.: Problem 22 of the list, http://www.imaph.tu-bs.de/qi/problems/
Christandl, M., Koenig, R., Mitchison, G., Renner, R.: Comm. Math. Phys. 273, 473 (2007)
Renner, R.: Nature Physics 3, 645 (2007)
Koenig, R., Mitchison, G.: J. Math. Phys. 50, 12105 (2009)
Caves, C.M., Fuchs, C.A., Schack, R.: J. Math. Phys. 43, 4537 (2002)
de Finetti, B.: Theory of Probability. Wiley, New York (1990)
Massar, S., Popescu, S.: Phys. Rev. Lett. 74, 1259 (1995)
Scarani, V., Iblisdir, S., Gisin, N., AcÃn, A.: Rev. Mod. Phys. 77, 1225 (2005)
Aharonov, D., Kitaev, A., Nisan, N.: Quantum Circuits with Mixed States. In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC). ACM, New York (1998)
Paulsen, V.I.: Completely bounded maps and dilations. Longman Scientific and Technical (1986)
Askey, R.: Orthogonal polynomials and special functions, Philadelphia, PA. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 21 (1975)
Yard, J., Hayden, P., Devetak, I.: arXiv:quant-ph/0603098v1
Chiribella, G., D’Ariano, G.M., Perinotti, P.: J. Math. Phys. 50, 42101 (2009)
Leung, D., Smith, G.: Comm. Math. Phys. 292, 201 (2009)
Holevo, A.S., Werner, R.F.: Phys. Rev. AÂ 3, 32312 (2001)
Klee, V.: Canad. J. Math. 16, 517 (1963)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chiribella, G. (2011). On Quantum Estimation, Quantum Cloning and Finite Quantum de Finetti Theorems. 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_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-18073-6_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-18072-9
Online ISBN: 978-3-642-18073-6
eBook Packages: Computer ScienceComputer Science (R0)