Let O be a Hermitian operator on an n-qubit system satisfying O^2=I. Write H[O] for its Fourier entropy and I[O] for its influence. Quantum Fourier entropy-influence conjecture. There is a constant c independent of n such that H[O]≤ c I[O]. This conjecture would improve the preceding entropy-influence bound by removing its dependence on the dimension of the qubit system. Its resolution is not established in the supplied text.

Quantum Fourier entropy-influence conjecture for qubit systems

From papers

Let $O$ be a Hermitian operator on an $n$ -qubit system satisfying $O^2=I$ . Write $H[O]$ for its Fourier entropy and $I[O]$ for its influence.

Quantum Fourier entropy-influence conjecture. There is a constant $c$ independent of $n$ such that

H[O]\leq c I[O].

This conjecture would improve the preceding entropy-influence bound by removing its dependence on the dimension of the qubit system. Its resolution is not established in the supplied text.

Progress summary

Solved

Two August 2026 preprints claim to have disproved the conjecture, but neither claim has yet been independently verified.

The conjecture, proposed by Bu et al. in 2022, asks whether the Fourier entropy of every qubit-valued observable is bounded by a constant times its influence, independently of system size.

Known results

Bu et al. (2022) record the conjecture and prove only a dimension-dependent bound involving $\log n$ (and, more generally, $\log d$ ).
The same work notes that the conjecture would imply the classical Fourier entropy-influence conjecture.

August 2026 counterexample claims

One preprint constructs involutions with $H[\widehat{A_d}^{\,2}]=\Theta(d^2)$ and $I[A_d]=d$ , with the ratio growing at least logarithmically in the number of qubits.
A second gives $H[O_n]=\log_2 n+O(1)$ while $\operatorname{Inf}[O_n]<2$ , so the ratio diverges. Both present these as disproofs, but no independent verification is supplied.

Current status (as of August 2026): Two preprints claim the dimension-free conjecture is false via explicit counterexamples, but their mathematical correctness remains unverified.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

Kaifeng Bu, Roy J. Garcia, Arthur Jaffe, Dax Enshan Koh and Lu Li, “Complexity of quantum circuits via sensitivity, magic, and coherence”, arXiv:2204.12051 (2022).

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