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Average entropy of a subsystem. (English) Zbl 0972.81504

Summary: If a quantum system of Hilbert space dimension mn is in a random pure state, the average entropy of a subsystem of dimension \(m\leq n\) is conjectured to be \(S_{m,n}= S^{mn}_{k=n+1} 1/k-m-1/2n\) and is shown to be \(\sim=\text{ln}m-m/2n\) for \(1 \ll m\leq n\). 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.

MSC:

81P05 General and philosophical questions in quantum theory
82B03 Foundations of equilibrium statistical mechanics
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References:

[1] H. Araki, Commun. Math. Phys. 18 pp 160– (1970)
[2] E. Lubkin, J. Math. Phys. 19 pp 1028– (1978) · Zbl 0389.94006
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