Manuscript Oai265-A@2 · Oai265-A.tex · lemma · c0a34a45

Lemma: What an entropy bound says about a Schmidt tail

Stage 3: PEPS existence, algorithms, and approximation budgets › Norm error and entropy error · lines 881–903 · proof included or linked

Oai265-A@2#lem:tail

Let $p_{1} \geq p_{2} \geq \dots$ be a Schmidt probability distribution. For $D \geq 1$ ,

j > D \sum p_{j} \leq \frac{S ( p )}{lo g ( D + 1 )} .

Keeping and renormalizing its first $D$ Schmidt components gives a vector approximation with squared error at most $2 \sum_{j > D} p_{j}$ .

Proof. For $j > D$ , $p_{j} \leq 1/ j \leq 1/ (D + 1)$ . Hence

S (p) \geq j > D \sum p_{j} lo g (D + 1) .

If the discarded weight is $τ$ , the optimal normalized truncation has overlap $1 - τ$ , so the squared vector distance is $2 (1 - 1 - τ) \leq 2 τ$ .

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