New explicit constructions show exponential tightness of the product-versus-operator norm gap and noncommutative Bohnenblust-Hille constant, disprove the dimension-free quantum Fourier Entropy-Influence inequality, and prove the Aaronson-Ambainis bound for anti-commuting Pauli-string Hamiltonians.
Entropy versus influence for complex functions of modulus one 2
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abstract
This is a simplification of a previous version of this ArXiv note. We present an example of a function $f$ from $\{-1,1\}^n$ to the unit sphere in $\mathbb{C}$ with influence bounded by $1$ and entropy of $|\hat f|^2$ larger than $\frac12\log n$.
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Tightness of and counterexamples to several quantum estimates
New explicit constructions show exponential tightness of the product-versus-operator norm gap and noncommutative Bohnenblust-Hille constant, disprove the dimension-free quantum Fourier Entropy-Influence inequality, and prove the Aaronson-Ambainis bound for anti-commuting Pauli-string Hamiltonians.