= Typical subspace theorem
{title2=$\operatorname{Tr}(\pi^{\otimes n}\Pi_{n,\delta})\to1$}
For each fixed $\delta>0$, the <quantum typical subspace> projector selects eigenvalues of $\pi^{\otimes n}$ between $2^{-n(S(\pi)+\delta)}$ and $2^{-n(S(\pi)-\delta)}$. Its probability tends to one by the <weak law of large numbers> for the spectral information variable. If its probability is at least $1-\epsilon$, its dimension is between $(1-\epsilon)2^{n(S(\pi)-\delta)}$ and $2^{n(S(\pi)+\delta)}$. These statements give the size and concentration estimates behind <Schumacher compression>.
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