Infeasibility Certificates and the Critical Line: LP Bounds from Quantum AME States to the Riemann Hypothesis

Abstract

We speculate that the linear programming technique used by Zeng and Zhang to bound absolutely maximally entangled (AME) quantum states—specifically their method of truncated MacWilliams identities and explicit infeasibility certificates—bears a formal analogy to potential linear programming approaches to the Riemann Hypothesis via the Weil explicit formula.

Overview

The source paper establishes explicit bounds for the existence of absolutely maximally entangled (AME) states with arbitrary defect using a truncated MacWilliams linear-programming system. These quantum states are characterized by the spectral property that their reduced density matrices are maximally mixed (uniform spectrum). The proof constructs an explicit infeasibility certificate—a non-negative dual vector λ satisfying orthogonality conditions that force a contradiction when the number of parties n exceeds a threshold of order 2(l+1)q².

The Proposed Analogy

We explore a structural correspondence between this quantum coding theory and the Hilbert–Pólya spectral interpretation of the Riemann Hypothesis. The analogy identifies:

• The weight enumerator coefficients A_i (non-negative by definition) with the Weil functional positivity constraints in the explicit formula.
• The MacWilliams transform (Krawtchouk polynomials) with the Mellin/Fourier transform in the explicit formula relating primes to zeros.
• The infeasibility certificate λ with a test function f in the explicit formula that would isolate a hypothetical zero off the critical line.

The strength of this analogy is rated as a formal analogy: the linear programming duality structure is identical, but the underlying spaces are discrete and finite (quantum codes) versus continuous and infinite (zeta zeros). No isomorphism of the objects exists.

Key Structural Parallel

Both systems employ a duality argument: the source paper proves that for a state of defect l, the vector of weight enumerator coefficients must satisfy a linear system M·A = b with A ≥ 0. When n is too large, a dual vector λ ≥ 0 exists such that λ^TM = 0 but λ^Tb < 0, proving infeasibility. Similarly, the Weil explicit formula can be viewed as a linear constraint on the zero distribution; a test function (dual variable) with positive transform can potentially rule out configurations with zeros off the critical line.

Assessment and Limitations

The essay proposes concrete experiments to test the convergence of the MacWilliams kernel to analytic kernels in the large-q limit. However, we explicitly identify the failure modes: the finite state space of the quantum system lacks an Euler product, and the "defect" parameter does not naturally extend to the complex plane. The analogy remains speculative, offering a methodological parallel rather than a proof strategy.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.