## Abstract

This paper attempts to develop a theory of sufficiency in the setting of non-commutative algebras parallel to the ideas in classical mathematical statistics. Sufficiency of a coarse-graining means that all information is extracted about the mutual relation of a given family of states. In the paper sufficient coarse-grainings are characterized in several equivalent ways and the non-commutative analogue of the factorization theorem is obtained. As an application we discuss exponential families. Our factorization theorem also implies two further important results, previously known only in finite Hilbert space dimension, but proved here in generality: the Koashi-Imoto theorem on maps leaving a family of states invariant, and the characterization of the general form of states in the equality case of strong subadditivity.

### Similar content being viewed by others

## References

Accardi, L., Cecchini, C.: Conditional expectations in von Neumann algebras and a theorem of Takesaki. J. Funct. Anal.

**45**, 245–273 (1982)Barndorff-Nielsen, O.:

*Information and exponential families in statistical theory*. Wiley Series in Probability and Mathematical Statistics. Chichester: John Wiley & Sons, Ltd., 1978Barndorff-Nielsen, O.E., Gill, R., Jupp, P.E.: On quantum statistical inference. J. R. Stat. Soc. Ser. B Stat. Methodol.

**65**, 775–816 (2003)Blank, J., Exner, P., Havliček, M.:

*Hilbert space operators in quantum physics*. New York: American Institute of Physics, 1994Bratteli, O., Robinson, D.W.:

*Operator algebras and quantum statistical mechanics. 1. C*- and W*–algebras, symmetry groups, decomposition of states*. 2nd ed., Texts and Monographs in Physics, New York: Springer Verlag, 1987Hayden, P., Jozsa, R., Petz, D., Winter, A.: Structure of states which satisfy strong subadditivity of quantum entropy with equality. Commun. Math. Phys.

**246**, 359–374 (2004)Koashi, M., Imoto, N.: Operations that do not disturb partially known quantum states. Phys. Rev. A,

**66**, 022318 (2002)Lieb, E.H., Ruskai, M.B.: Proof of the strong subadditivity of quantum mechanical entropy. J. Math. Phys.

**14**, 1938–1941 (1973)Mosonyi, M., Petz, D.: Structure of sufficient quantum coarse-grainings. Lett. Math. Phys.

**68**, 19–30 (2004)Nielsen, M.A., Chuang, I.L.:

*Quantum Computation and Quantum Information*. Cambridge: Cambridge University Press, 2000Nielsen, M.A., Petz, D.: A simple proof of the strong subadditivity inequality. Quantum Information & Computation

**5**, 507–513 (2005)Ohya, M., Petz, D.:

*Quantum Entropy and Its Use*. Heidelberg:Springer-Verlag, 1993, 2nd edition, 2004Petz, D.: Direct integral of multifunctions into von Neumann algebras. Studia Sci. Math. Hungar.

**18**, 239–245 (1978)Petz, D.: Sufficient subalgebras and the relative entropy of states of a von Neumann algebra. Commun. Math. Phys.

**105**, 123–131 (1986)Petz, D.: Quasi-entropies for finite quantum systems. Rep. Math. Phys.

**21**, 57–65 (1986)Petz, D.: Sufficiency of channels over von Neumann algebras. Quart. J. Math. Oxford,

**39**, 907–1008 (1988)Petz, D.: Geometry of canonical correlation on the state space of a quantum system. J. Math. Phys.

**35**, 780–795 (1994)Petz, D.: Discrimination between states of a quantum system by observations. J Funct. Anal.

**120**, 82–97 (1994)Petz, D.: Monotonicity of quantum relative entropy revisited. Rev. Math. Phys.

**15**, 79–91 (2003)Ruskai, M.B.: Inequalities for quantum entropy: A review with conditions with equality. J. Math. Phys.

**43**, 4358–4375 (2002)Schumacher, B.W., Nielsen, M.A.: Quantum data processing and error correction. Phys. Rev. A

**54**, 2629–2635 (1996)Schwartz, J.T.:

*W*^{*}*-algebras*. New York-London-Paris: Gordon and Breach Science Publishers, 1967Strătilă, Ş:

*Modular theory of operator algebras*. Tunbridge Wells: Abacus Press, 1981Strasser, H.:

*Mathematical theory of statistics. Statistical experiments and asymptotic decision theory*. Berlin: Walter de Gruyter, 1985

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

Communicated by M.B. Ruskai

Supported by the EU Research Training Network Quantum Probability with Applications to Physics, Information Theory and Biology and Center of Excellence SAS Physics of Information I/2/2005.

Supported by the Hungarian grant OTKA T032662

## Rights and permissions

## About this article

### Cite this article

Jenčová, A., Petz, D. Sufficiency in Quantum Statistical Inference.
*Commun. Math. Phys.* **263**, 259–276 (2006). https://doi.org/10.1007/s00220-005-1510-7

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s00220-005-1510-7