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New Journal of Physics

Entanglement and analytical continuation: an intimate relation told by the Riemann zeta function

C. Feiler, W. P. Schleich

Published Jun 7, 2013• 20 citations
Publisher

Abstract

We propose measurements on a quantum system to realize the Riemann zeta function ζ . A single system, that is classical interference, suffices to create the Dirichlet representation of ζ . In contrast, we need measurements performed on two entangled quantum systems to extend ζ into the critical strip of complex space where the non-trivial zeros of ζ are located. As a consequence, we can view these zeros as a result of a Schrödinger cat which is by its very construction similar to, but in its details very different from, the superposition formed by two coherent states of identical amplitudes but opposite phases. This interpretation suggests that entanglement in quantum mechanics is the analogue of analytic continuation of complex analysis.

DOI
10.1088/1367-2630/15/6/063009

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We propose measurements on a quantum system to realize the Riemann zeta function ζ . A single system, that is classical interference, suffices to create the Dirichlet representation of ζ . In contrast, we need measurements performed on two entangled quantum systems to extend ζ into the critical strip of complex space where the non-trivial zeros of ζ are located. As a consequence, we can view these zeros as a result of a Schrödinger cat which is by its very construction similar to, but in its details very different from, the superposition formed by two coherent states of identical amplitudes but opposite phases. This interpretation suggests that entanglement in quantum mechanics is the analogue of analytic continuation of complex analysis.
The intimate connection between the zeros of the Riemann zeta function and the distribution of prime numbers makes this function famous beyond the borders of mathematics. Since the precise location of the non-trivial zeros (Riemann hypothesis) is still unconfirmed many efforts are directed to the investigation of the zeta function, not only with mathematical but also with physical methods. The physical approach in this thesis is inspired by the interaction of a cavity field with a two-level atom described by the Jaynes-Cummings-Paul model. However, in our case the interaction Hamiltonian depends logarithmically on the photon number operator. The time-evolved quantum states produce with appropriately chosen reference states the values of the Riemann zeta function. It is shown that in the complex half-plane with real parts larger than one, the cavity field suffices to create the zeta function whereas the analytical continuation of the zeta function can only be realized with entangled states. Moreover, the continuous Newton method is used to illustrate the behavior of the Riemann zeta function in the complex plane. This method depicts a complex function F by lines of constant phase which start at the poles of the function and lead into its zeros. In the case of the Riemann zeta function, this method would reveal if the Riemann hypothesis was violated.
The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender-Brody-M\"uller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a unique Hermitian Hamiltonian that maps to the zeros of the analytic continuation of the Riemann zeta function, and provide an analytical expression for the eigenvalues of the results. The holomorphicity of the resulting eigenvalues is examined. Moreover, a second quantization of the resulting Schr\"odinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the eigenvalues it is shown that the real part of every nontrivial zero of the Riemann zeta function converges at σ=1/2\sigma=1/2σ=1/2.
The one-atom maser of Herbert Walther has ushered in a new era in quantum optics and paved the way to quantum information science. In this talk we focus on three quantum phenomena motivated by the seminal work of Herbert Walther. We concentrate on the Quantum FEL, cold atoms in space and the connection between analytic continuation in complex analysis and entanglement in quantum mechanics illuminated by the Riemann zeta function. The key ingredient of the three examples is the elementary interaction between light and matter that is also at the very heart of the one-atom maser.
The Riemann Hypothesis, a long-standing and stubbornly unsolved problem in number theory, has captivated mathematicians for over 150 years. Current computational methods struggle to definitively prove or disprove it, highlighting a significant gap in our understanding of the zeta function. This paper proposes a novel approach to tackle this problem utilizing quantum-enhanced measurement techniques and entangled variables. We outline a new algorithm for exploring the Riemann Hypothesis, leveraging the correlations between entangled particles to identify subtle patterns within the zeta function. The core mechanism centers on creating entangled particles to probe the structure of the zeta function, potentially revealing previously missed connections. This investigation leverages a hybrid quantum-classical approach, combining the precision of quantum mechanics with the analytical power of classical computation. The goal is to develop a framework that can potentially lead to new insights into the Riemann Hypothesis and provide a more robust theoretical foundation.
We analyze quantum-mechanical effects that emerge from the compression of the extended zeta function ζ φ (s; ε), which carries an additional phase φ and a Hurwitz-ε shift, onto the classical Riemann zeta function. The 4D structure (s, φ; ε) projects to the 2D s-plane under a natural projection, and this loss of fibre information yields analogues of wavefunction collapse, uncertainty relations, double-slit interference, spin doublets, and wormhole-type correlations. Relative Lorentz-like effects and a thermodynamic indication at the balance line ℜ(s) = 1 2 are also derived. The framework shows how classical quantum phenomena can be interpreted geometrically via the extended zeta function.
Abstract This paper proposes that the ultimate limits of quantum evolution are governed not only by geometric symmetries but also by analytic invariants originating in number theory. Within the framework of multicarrier phase geometry, we show that the continuity of the Phase–Cone Inequality, κ5=4 ⟶ κ6≃π+ζ(3),\kappa_{5} = 4 \;\longrightarrow\; \kappa_{6} \simeq \pi + \zeta(3),κ5=4⟶κ6≃π+ζ(3), marks a transition from an algebraic (elliptic) regime to an analytically continued (hyperbolic) one. In this transition, π\piπ encodes spatial periodicity—the geometric closure of interference—while ζ(3)\zeta(3)ζ(3), Apéry’s constant, represents analytic curvature arising from the hypergeometric continuation of the phase metric. Their coexistence defines a Geometry–Arithmetic Bridge, in which quantum motion reflects both algebraic closure and analytic continuation. This bridge implies that the quantum speed limit (QSL), typically formulated as a purely geometric constraint on energy dispersion, acquires a transcendental correction: vϕ(max⁡) = π+ζ(3) ΔEℏ.v_{\phi}^{(\max)} \;=\; \sqrt{\pi + \zeta(3)}\,\frac{\Delta E}{\hbar}.vϕ(max)=π+ζ(3)ℏΔE. The resulting 19% deviation from the algebraic limit (κ5=4\kappa_{5}=4κ5=4) may manifest as a measurable phase drift in six-path interferometers, offering a potential empirical link between the analytic structure of the Riemann zeta function and quantum interferometry. In this framework, π\piπ measures the geometric closure of space, ζ(3)\zeta(3)ζ(3) measures the openness of analytic continuation, and their union encodes a dual foundation of physical law: geometry in space and arithmetic in phase.