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

Example: Von Neumann control does not control α<1 without further assumptions

Stage 4: hypothesis tests on explicit models › Rényi and mutual-information corrections · lines 1135–1157

For $N \geq 3$ , let $δ_{N} = 1/ lo g N$ , and consider

p^{(N)} = (1 - δ_{N}, \frac{δ _{N}}{N}, \dots, \frac{δ _{N}}{N}) .

Then

S (p^{(N)}) = h_{2} (δ_{N}) + 1 \leq lo g 2 + 1.

For fixed $0 < α < 1$ ,

S_{α} (p^{(N)}) \geq lo g N - \frac{α}{1 - α} lo g lo g N ⟶ \infty.

These are Schmidt spectra of finite bipartite pure states. The example disproves a spectrum-only implication; it is not claimed to be a counterexample in the frozen gapped-local class.

Displayed formulas in this passage

unlabeled · line 1138
$p^{(N)} = (1 - δ_{N}, \frac{δ _{N}}{N}, \dots, \frac{δ _{N}}{N}) .$
unlabeled · line 1144
$S (p^{(N)}) = h_{2} (δ_{N}) + 1 \leq lo g 2 + 1.$
unlabeled · line 1148
$S_{α} (p^{(N)}) \geq lo g N - \frac{α}{1 - α} lo g lo g N ⟶ \infty.$

Example: Von Neumann control does not control α<1 without further assumptions

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