# Average entropy of a subsystem.

@article{Page1993AverageEO,
title={Average entropy of a subsystem.},
author={Page},
journal={Physical review letters},
year={1993},
volume={71 9},
pages={
1291-1294
}
}
• Page
• Published 7 May 1993
• Mathematics
• Physical review letters
If a quantum system of Hilbert space dimension mn is in a random pure state, the average entropy of a subsystem of dimension m\ensuremath{\le}n is conjectured to be ${\mathit{S}}_{\mathit{m},\mathit{n}}$= ${\mathit{S}}_{\mathit{k}=\mathit{n}+1}^{\mathit{mn}}$ 1/k-m-1/2n and is shown to be \ensuremath{\simeq}lnm-m/2n for 1\ensuremath{\ll}m\ensuremath{\le}n. Thus there is less than one-half unit of information, on average, in the smaller subsystem of a total system in a random pure state.
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## References

SHOWING 1-10 OF 15 REFERENCES
Average quantal behavior and thermodynamic isolation
• Physics
• 1993
We study the thermodynamic concept of isolation. The causal motion of a system that models a thermodynamic “universe” but nevertheless couples to a surround is reconciled with an increase of
Black hole information
Hawking's 1974 calculation of thermal emission from a classical black hole led to his 1976 proposal that information may be lost from our universe as a pure quantum state collapses gravitationally
Mathematical Methods in Physics
Volume II J. S. R. Chisholm and Rosa M. Morris Amsterdam: North-Holland. 1964 Pp. xviii + 717. Price 72s. This book is a valuable addition to an ever-growing collection of mathematical texts written
Phys
• 19, 1028 (1978); cf. E. Lubkin and T. Lubkin, ”Average Quantal Behavior and Thermodynamic Isolation,” University of Wisconsin- Milwaukee preprint WISC-MILW-92-TH-12
• 1992
Ann. Phys
• Ann. Phys
• 1988
Black Hole Information” (University of Alberta report Alberta- Thy-23-93
• hep-th/9305040), to be published in Proceedings of the 5th Canadian Conference on General Relativity and Relativistic Astrophysics, University of Waterloo, 13–15 May, 1993, edited by R. B. Mann and R. G. McLenaghan
• 1994
Information in Black Hole Radiation ” ( University of Alberta report AlbertaThy2493 , hepth / 9306083 ) , submitted to
• Black Hole Information ” ( University of Alberta report AlbertaThy2393 , hepth / 9305040 ) , to be published in Proceedings of the 5 th Canadian Conference on General Relativity and Relativistic Astrophysics
• 1994
Black Hole InformationUniversity of Alberta report Alberta- Thy-23-93, hep-th/9305040), to be
• Proceedings of the 5th Canadian Conference on General Relativity and Relativistic Astrophysics
• 1993
Commun
• Math. Phys. 18, 160
• 1970
Commun. Math. Phys
• Commun. Math. Phys
• 1970