Abstract
We have proven that there exists a quantum state approximating any multi-copy state universally when we measure the error by means of the normalized relative entropy. While the qubit case was proven by Krattenthaler and Slater (IEEE Trans. IT 46, 810–819 (2009)), the general case has been open for more than ten years. For a deeper analysis, we have solved the mini-max problem concerning ‘approximation error’ up to the second order. Furthermore, we have applied this result to quantum lossless data compression, and have constructed a universal quantum lossless data compression.
Similar content being viewed by others
References
Clarke B.S., Barron A.R.: Information-theoretic asymptotics of Bayes methods. IEEE Trans. Inform. Theory 36, 453–471 (1990)
Amari S., Nagaoka H.: Methods of Information Geometry. Providence, RI: Amer. Math. Soc. & Oxford University Press, 2000
Clarke B.S., Barron A.R.: Jeffreys’ prior is asymptotically least favorable under entropy risk. J. Stat. Plan. Inference 41(1), 37–61 (1994)
Krattenthaler C., Slater P.: Asymptotic Redundancies for Universal Quantum Coding. IEEE Trans. Inform. Theory 46, 801–819 (2000)
Hayashi M.: Universal coding for classical-quantum channel. Commun. Math. Phys. 289, 1087–1098 (2009)
Han T.S., Kobayashi K.: Mathematics of Information and Encoding. Providence, RI: Amer Math. Soc. 2002 (Originally written in Japanese in 1999)
Lynch T.J.: Sequence time coding for data compression. Proc. IEEE 54, 1490–1491 (1966)
Davisson L.D.: Comments on ‘Sequence time coding for data compression’. Proc. IEEE 54, 2010 (1966)
Jozsa R., Horodecki M., Horodecki P., Horodecki R.: Universal Quantum Information Compression. Phys. Rev. Lett. 81, 1714 (1998)
Hayashi M.: Exponents of quantum fixed-length pure state source coding. Phys. Rev. A 66, 032321 (2002)
Boström K., Felbinger T.: Lossless quantum data compression and variable-length coding. Phys. Rev. A 65, 032313 (2002)
Hayashi M., Matsumoto K.: Quantum universal variable-length source coding. Phys. Rev. A 66, 022311 (2002)
Koashi M., Imoto N.: Quantum Information is Incompressible Without Errors. Phys. Rev. Lett. 89, 097904 (2002)
Christandl M.: The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography. PhD thesis, February 2006, University of Cambridge. available at http://arXiv.org/abs/quant-ph/0604183v1, 2006
Matsumoto K., Hayashi M.: Universal distortion-free entanglement concentration. Phys. Rev. A 75, 062338 (2007)
Keyl M., Werner R.F.: Estimating the spectrum of a density operator. Phys. Rev. A 64, 052311 (2001)
Schumacher B.: Quantum coding. Phys. Rev. A 51, 2738–2747 (1995)
Hayashi M.: Quantum Information: An Introduction. Berlin: Springer, 2006 (Originally written in Japanese in 2004)
Chou H.-H., Cheng J.: New lower bounds on the average base length of lossless quantum data compression. In: Proceedings of the 8th Quantum Communication, Computing, and Measurement, edited by O. Hirota, J.H. Shapiro, M. Sasaki, Tokyo: NICT Press, 2008, pp. 279–282
Slater P.: Hall normalization constants for the Bures volumes of the n-state quantum systems. J. Phys. A: Math. Gen. 32, 8231–8246 (1999)
Hayashi M.: In preparation
Grosse H., Krattenthaler C., Slater P.B.: Asymptotic Redundancies for Universal Quantum Coding II. In preparation
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. B. Ruskai
Rights and permissions
About this article
Cite this article
Hayashi, M. Universal Approximation of Multi-copy States and Universal Quantum Lossless Data Compression. Commun. Math. Phys. 293, 171–183 (2010). https://doi.org/10.1007/s00220-009-0909-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-009-0909-y