The Ultrametric Hilbert Space Description of Quantum Measurements with Finite Precision

Andrew Khrennikov

Rowan Brad Quni-Gudzinas

Ultrametric spaces provide a natural geometry for hierarchically structured information. This paper synthesizes four independent research programs: Murtagh's data embedding in ultrametric spaces, Heydeman's holographic QEC on Bruhat-Tits trees, Palmer's p-adic quantum foundations, and Aniello's p-adic Hilbert space formalism. QNFO contributions: D=4 embedding theorem, adelic QEC, CMB analysis.

Masatoshi Namiki, Saverio Pascazio

Andrew Lesniewski

We revisit von Neumann's theory of infinite tensor products of Hilbert spaces. On the set $Γ$ of equivalence classes of $C_0$-sequences, which labels the incomplete tensor products inside the complete tensor product, we introduce a natural pseudo-ultrametric $d$: the distance between two classes is the convergence exponent of the series $\sum_j|\langle\varphi_j,ψ_j\rangle-1|$ formed from any pair of representatives. We show that $d$ is well defined on equivalence classes, satisfies the strong triangle inequality, and is complete. Distinct classes may lie at distance zero, so $d$ separates points only after passing to the quotient $\widetildeΓ$ of $Γ$ by the relation $d=0$; the pair $(\widetildeΓ,d)$ is then a complete ultrametric space. As an application, we show that a product unitary $\bigotimes_j U$ whose factor $U$ satisfies $\inf_{\|x\|=1}|\langle x,Ux\rangle-1|>0$ (in particular, a unitary on a finite dimensional space with $1\notinσ(U)$) displaces every class to the maximal distance $1$. Guided by the intended application -- a caricature of Everettian branching, in which the sectors of the infinite tensor product play the role of worlds -- we also develop a gauge-invariant variant $\tilde d$ of the metric, based on von Neumann's weak equivalence and matched to the quasi-equivalence of product states on the quasi-local algebra. The displacement of a class under a product unitary, measured by $\tilde d$, is class dependent and realizes every value in $[0,1]$. We interpret $\tilde d$ as a decoherence exponent: it measures the polynomial rate at which two branches of the universal state vector become operationally distinct as ever larger portions of the environment are monitored.

Mikio Namiki

QNFO Research

Ultrametric spaces provide a natural geometry for systems organized hierarchically. This paper synthesizes research connecting ultrametric geometry to quantum information theory: Murtagh's data embedding, Heydeman's holographic QEC on Bruhat-Tits trees, Palmer's p-adic quantum foundations, and Aniello's p-adic Hilbert space formalism. QNFO contributions: D=4 embedding theorem, adelic QEC, CMB analysis.