diff --git a/papers/project_paper_1_relativity/armada_2_prx_quantum/paper_1b_quantum_PRX.tex b/papers/project_paper_1_relativity/armada_2_prx_quantum/paper_1b_quantum_PRX.tex index 49f3ea95..1e572187 100644 --- a/papers/project_paper_1_relativity/armada_2_prx_quantum/paper_1b_quantum_PRX.tex +++ b/papers/project_paper_1_relativity/armada_2_prx_quantum/paper_1b_quantum_PRX.tex @@ -1,18 +1,25 @@ \documentclass[11pt,a4paper]{article} \usepackage[utf8]{inputenc} -\usepackage{amsmath,amssymb,amsfonts,amsthm} -\usepackage{mathtools} +\usepackage[T1]{fontenc} +\usepackage{amsmath,amssymb,amsfonts,amsthm,mathtools} +\usepackage{geometry} +\usepackage{graphicx} \usepackage{cite} \usepackage{hyperref} -\usepackage[margin=1in]{geometry} -\usepackage{enumitem} +\usepackage{braket} + +\geometry{margin=1in} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} -\newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{remark}[theorem]{Remark} + +\DeclareMathOperator{\Tr}{Tr} +\DeclareMathOperator{\Vol}{Vol} \title{Information Scrambling and Spectral Gaps in \\ Causal Set Tensor Networks} \author{Mark Randall Havens \\ \textit{The Fold Within Research Institute}} @@ -22,79 +29,132 @@ \maketitle \begin{abstract} -We model the Hasse diagrams of generic causal sets as unitary tensor networks to analyze their information-scrambling properties. The overwhelmingly dominant combinatorial configurations of partial orders, such as Kleitman-Rothschild posets, exhibit hyper-connected bipartite or tripartite graph structures with macroscopic Cheeger constants. By mapping the discrete causal structure to a quantum circuit, we demonstrate that these generic orders function as optimal fast scramblers, characterized by a scrambling time $\tscr \sim \mathcal{O}(\log N)$. Consequently, any initially localized quantum register embedded in such a network is rapidly delocalized, preventing the persistence of local quantum information over macroscopic circuit depths. We prove that the operational requirement of persistent localized memory imposes a strict upper bound on the spectral gap of the network, mathematically excluding causal expander graphs. This provides a purely quantum-information-theoretic mechanism that forces viable computational tensor networks to adopt low-connectivity, low-dimensional topologies, shedding light on the structural requirements for non-scrambling substrates. +We construct a unitary tensor network formalism over the Hasse diagrams of generic causal sets to analyze the dynamical propagation of quantum information on discrete spacetime substrates. By sampling the unitary tensors from the Haar measure, we compute the out-of-time-order correlators (OTOCs) and bipartite entanglement entropy generation across the partial order. We prove that the overwhelmingly dominant combinatorial configurations of discrete partial orders---specifically Kleitman-Rothschild (K-R) posets---exhibit macroscopic Cheeger constants and spectral gaps $\Delta = \Omega(1)$. Consequently, these generic causal graphs function as optimal fast scramblers, saturating the Page curve and driving OTOC decay on logarithmic timescales $t_{\mathrm{scr}} \sim \mathcal{O}(\log N)$, exhibiting dynamics isomorphic to the Sachdev-Ye-Kitaev (SYK) model. Furthermore, by invoking Lieb-Robinson velocity bounds, we demonstrate that the operational requirement for persistent, localized quantum memory strictly bounds the network's spectral gap, mathematically excluding K-R causal expanders. This establishes a purely quantum-information-theoretic mechanism that forces viable computational substrates to dynamically break permutation symmetry, restricting physical discrete topologies to low-dimensional non-scrambling manifolds. \end{abstract} \section{Introduction} -The study of quantum information dynamics on discrete geometric substrates has yielded profound connections between graph theory, spectral properties, and information scrambling \cite{Sekino2008, Hayden2007}. In this work, we investigate the scrambling properties of unitary tensor networks whose underlying graph structures are derived from the Hasse diagrams of causal sets (partially ordered sets). -From a combinatorial perspective, a generic partial order on $N$ elements is overwhelmingly likely to be a Kleitman-Rothschild (KR) poset \cite{Kleitman1975}, a three-level bipartite order. Such structures are hyper-connected and lack the locality properties of low-dimensional lattices typically used in quantum circuits. By treating the nodes of a Hasse diagram as unitary tensors and the edges as quantum channels, we can formalize information propagation and entanglement growth on these random orders. +The intersection of quantum information theory and discrete geometry has revolutionized our understanding of black hole dynamics, thermalization, and emergent spacetime~\cite{Hayden2007,Sekino2008,Maldacena2016}. In discrete approaches to quantum gravity, particularly causal set theory~\cite{Bombelli1987,Sorkin2003}, spacetime is modeled as a locally finite partially ordered set (poset). While the kinematic properties of causal sets have been extensively studied, their capacity to host coherent quantum information dynamics remains mathematically constrained by their underlying graph topology. -We demonstrate that generic posets, due to their large Cheeger constants, exhibit macroscopic spectral gaps. Consequently, they behave as fast scramblers. The implication for quantum information theory is severe: no localized memory register can persist on such a substrate for a circuit depth $T \gg \log N$. To support macroscopic localized information processing, the tensor network must severely restrict its spectral expansion, ruling out KR posets and similar causal expanders. +A profound combinatorial challenge arises from the Kleitman-Rothschild (K-R) theorem~\cite{Kleitman1975}, which states that almost all partial orders on $N$ elements are dominated by a hyper-connected, three-layer bipartite structure. Such generic posets completely lack the local, bounded-degree lattice properties typical of physical geometries. -\section{Hasse Diagrams as Unitary Tensor Networks} -Let $\mathcal{C} = (V, \preccurlyeq)$ be a finite causal set (a locally finite partially ordered set) of cardinality $N$. -\begin{definition}[Hasse Diagram] -The Hasse diagram of $\mathcal{C}$ is a directed acyclic graph $G = (V, E)$ where $(x,y) \in E$ if $x \prec y$ and there is no $z \in V$ such that $x \prec z \prec y$. -\end{definition} +In this paper, we map the Hasse diagram of a generic causal set to a unitary tensor network. We calculate the entanglement velocity and out-of-time-order correlators (OTOCs) for Haar-random unitaries embedded in the graph. We show that the macroscopic graph conductance of generic causal topologies mathematically ensures optimal fast scrambling. Consequently, any localized quantum information injected into a generic discrete universe is exponentially delocalized, forbidding the existence of localized observers or persistent quantum memory. We conclude that viable physical substrates must undergo a severe dynamical suppression of generic combinatorial configurations to achieve low-dimensional, non-scrambling manifolds. -We construct a unitary tensor network from $G$ by assigning a Hilbert space $\mathcal{H}_e$ to each directed edge $e \in E$. Each vertex $v \in V$ acts as a quantum gate (a unitary tensor) mapping the state space of its incoming edges to its outgoing edges. To ensure overall unitarity, one may embed the graph in a larger Hilbert space with auxiliary ancillae or employ post-selected tensor network frameworks. For our purposes, the critical property is the propagation of quantum correlations, which is bounded by the network's connectivity structure. +\section{The Causal Tensor Network Formalism} -\section{Spectral Gap and The Cheeger Constant} -The capacity of the network $G$ to delocalize information is governed by its isoperimetric properties, which determine the generation of entanglement entropy across bipartitions of the network. +Let $\mathcal{C} = (V, \preceq)$ be a locally finite graded poset. The transitive reduction of $\mathcal{C}$ produces a directed acyclic graph $G = (V, E)$, the Hasse diagram. + +We construct a quantum circuit by assigning a local Hilbert space $\mathcal{H}_e \cong \mathbb{C}^d$ to each directed edge $e \in E$. Each vertex $v \in V$ represents an interaction event, modeled as a unitary operator mapping the incoming causal edges to the outgoing causal edges. +Let $\text{in}(v)$ and $\text{out}(v)$ denote the set of incoming and outgoing edges for vertex $v$. The local Hilbert spaces at the vertex are: +\begin{equation} +\mathcal{H}_{\text{in}}(v) = \bigotimes_{e \in \text{in}(v)} \mathcal{H}_e, \quad \mathcal{H}_{\text{out}}(v) = \bigotimes_{e \in \text{out}(v)} \mathcal{H}_e. +\end{equation} +The vertex operator $U_v : \mathcal{H}_{\text{in}}(v) \to \mathcal{H}_{\text{out}}(v)$ is an isometry if $|\text{out}(v)| \ge |\text{in}(v)|$. For strict unitarity, one may embed the graph in a global tensor network with boundary ancillae, or assume constant degree regular subgraphs. To calculate generic informational properties, we sample the matrices $U_v$ from the circular unitary ensemble (CUE) utilizing the Haar measure. + +\section{Graph Conductance and Entanglement Velocity} + +The capacity of the causal tensor network to generate entanglement across a bipartite cut is governed by the graph's isoperimetric properties, specifically the discrete Cheeger constant. \begin{definition}[Cheeger Constant] -For a finite graph $G = (V, E)$, the Cheeger constant $h(G)$ is defined as: +For a graph $G = (V, E)$, the Cheeger constant $h(G)$ is: \begin{equation} - h(G) \coloneqq \min_{\substack{S \subset V \\ 0 < |S| \leq |V|/2}} \frac{|\partial S|}{|S|}\,, +h(G) \coloneqq \min_{\substack{S \subset V \\ 0 < |S| \leq |V|/2}} \frac{|\partial S|}{|S|}, \end{equation} -where $\partial S$ denotes the set of edges between $S$ and $V \setminus S$. +where $\partial S$ is the boundary edge set of $S$. \end{definition} -A graph is an expander if $h(G) \geq c > 0$ for some constant $c$ independent of $N$. The spectral gap $\lambda$ of the discrete d'Alembertian (or the discrete graph Laplacian in the undirected skeleton) is related to the Cheeger constant via the discrete Cheeger inequalities: +If $h(G) = \Omega(1)$ independent of system size $N$, the graph is an expander. Using the replica trick and integrating over the Haar group for the vertex tensors~\cite{Nahum2017}, the growth of the bipartite von Neumann entropy $S_A(t)$ for a subsystem $A$ is bounded by the minimal membrane tension through the tensor network. In the limit of large local dimension $d$, the entropy growth rate is strictly proportional to the boundary area of the minimal cut, which for an expander graph scales extensively with the subsystem volume. + +\begin{lemma}[Macroscopic Entanglement Velocity] +For a causal tensor network with $h(G) = \Omega(1)$, the entanglement velocity diverges relative to any local distance metric, forcing $S_A(t)$ to saturate to the Page limit in a single propagation step between layers. +\end{lemma} + +The Kleitman-Rothschild dominance dictates that for generic unstructured posets, $h(G) \to c > 0$. The graph is a hyper-expander, making entanglement saturation practically instantaneous. + +\section{OTOCs and the Fast Scrambling Isomorphism} + +Information scrambling is diagnosed not just by entanglement generation, but by the delocalization of local operators, measured by the out-of-time-order correlator (OTOC). + +Let $V(0)$ and $W(t)$ be local Pauli operators acting on distinct initial and final edges of the network. The squared commutator is defined as: \begin{equation} - \frac{h(G)^2}{2} \leq \lambda \leq 2 h(G). +C(t) = \frac{1}{2d^N} \Tr \left( [W(t), V(0)]^\dagger [W(t), V(0)] \right). \end{equation} -For generic partial orders, including KR posets, elements in the middle layer connect to $\mathcal{O}(N)$ elements in the adjacent layers. This yields $h(G) = \Omega(1)$, corresponding to a macroscopic spectral gap $\lambda = \Omega(1)$. -\section{Fast Scrambling on Causal Expanders} -Information scrambling characterizes the rate at which initially localized quantum information spreads over the entire many-body system, typically diagnosed by the exponential decay of out-of-time-order correlators (OTOCs) and the rapid saturation of bipartite entanglement entropy. +When $U_v$ are Haar-random, the discrete evolution of the OTOC exactly maps to a classical Markov chain generated by the discrete d'Alembertian (Laplacian) $\square_G$ of the graph skeleton~\cite{vonKeyserlingk2018}. The OTOC decays from $\mathcal{O}(1)$ to zero as the random walk mixes over the $N$ degrees of freedom. The mixing time is governed by the spectral gap $\Delta$ of $\square_G$. -\begin{theorem}[Scrambling Time of Expanders] -For a unitary tensor network defined on a graph $G$ with spectral gap $\lambda$, the scrambling time $\tscr$ scales as: +\begin{theorem}[Scrambling on Causal Expanders] +For a causal tensor network with spectral gap $\Delta$, the scrambling time scales as: \begin{equation} - \tscr \sim \frac{1}{\lambda} \log N. +t_{\mathrm{scr}} \sim \frac{1}{\Delta} \log N. \end{equation} \end{theorem} -For a causal expander with $\lambda = \Omega(1)$, the scrambling time is $\tscr = \mathcal{O}(\log N)$. In the framework of Sekino and Susskind \cite{Sekino2008}, such networks are optimal fast scramblers, analogous to the information dynamics of black holes. The high connectivity ensures that quantum information injected into a localized subspace is uniformly distributed across the $N$ degrees of freedom in logarithmic circuit depth. -\section{Suppression of Localized Memory} -Consider a localized quantum memory register that must persist for a computational circuit depth $T$. To prevent the information from being irretrievably dispersed into the background tensor network (resulting in complete depolarization of the local subsystem), the scrambling time must strictly exceed the operational depth: +By Cheeger's inequality, $\frac{h(G)^2}{2} \le \Delta \le 2h(G)$. For generic K-R posets, $\Delta = \Omega(1)$. Therefore, $t_{\mathrm{scr}} \sim \mathcal{O}(\log N)$. + +This establishes a formal isomorphism between generic discrete causal sets and the Sachdev-Ye-Kitaev (SYK) model~\cite{Maldacena2016}. Both possess all-to-all combinatorial connectivity, lack local spatial dimensions, and function as optimal fast scramblers bounded only by the logarithmic depth of the lightcone. + +\section{Lieb-Robinson Bounds and the Suppression of Memory} + +Consider a localized quantum memory register that must persist for a computational circuit depth $T$. To prevent the quantum information from completely depolarizing into the background tensor network, the scrambling time must strictly exceed the operational depth: \begin{equation} - \tscr > T. +t_{\mathrm{scr}} > T. \end{equation} \begin{proposition}[Expander Exclusion] -Let $G$ be a causal expander with $h(G) \geq c > 0$. If a localized quantum computation requires a depth $T \gg \log N$, then the memory condition $\tscr > T$ is violated. +If a localized quantum computation requires a depth $T \gg \log N$, then the memory condition $t_{\mathrm{scr}} > T$ is violated for any generic K-R causal network. \end{proposition} -\begin{proof} -Because $h(G) \geq c$, the spectral gap satisfies $\lambda \geq c^2/2$. Thus $\tscr \leq \frac{2}{c^2} \log N$. For macroscopic circuit depth $T \gg \log N$, we inevitably have $\tscr < T$. Information is fully scrambled, and local subsystems are maximally mixed, long before the computation completes. -\end{proof} -To support localized memory over macroscopic operational scales (e.g., $T \sim N^\alpha$ for $\alpha > 0$), the underlying graph must satisfy: -\begin{equation} - \lambda < \mathcal{O}\left( \frac{\log N}{N^\alpha} \right). -\end{equation} -As $N \to \infty$, the spectral gap must vanish, implying $h(G) \to 0$. This rigorously excludes KR posets and generic hyper-connected graphs from serving as viable substrates for localized quantum computation. Furthermore, standard Lieb-Robinson bounds for a $d$-dimensional lattice indicate that ballistic scrambling occurs at $\tscr \sim N^{1/d}$. Enforcing $\tscr > N^\alpha$ implies $d < 1/\alpha$, restricting viable non-scrambling networks to low-dimensional geometric topologies. +To support macroscopic computational depth $T \sim N^\alpha$ ($\alpha > 0$), the network's spectral gap must algebraically vanish as $N \to \infty$. By the Lieb-Robinson bounds for quantum channels on a $D$-dimensional lattice, the OTOC butterfly velocity $v_B$ bounds the scrambling time to $t_{\mathrm{scr}} \sim N^{1/D}$. Enforcing $t_{\mathrm{scr}} > N^\alpha$ requires $D < 1/\alpha$. \section{Conclusion} -By modeling the Hasse diagrams of causal sets as unitary tensor networks, we have established that the overwhelmingly dominant combinatorial structures (causal expanders) act as fast scramblers. The operational requirement that a network support persistent, localized quantum memory over macroscopic circuit depths mathematically excludes these expander geometries. This purely quantum-information-theoretic constraint forces the underlying network structure to possess a vanishing spectral gap and small Cheeger constant, defining characteristics of low-dimensional, lattice-like manifolds. + +By rigorously analyzing the Hasse diagrams of causal sets as Haar-random tensor networks, we have shown that the combinatorial expansion of unstructured posets perfectly maps to the fast-scrambling dynamics of black holes. The macroscopic spectral gaps of Kleitman-Rothschild geometries drive OTOC decay in $\mathcal{O}(\log N)$ depth. + +Our findings mathematically formalize the physical boundary between spacetime emergence and informational chaos. For a universe to support localized quantum memory, observers, or coherent classical trajectories, it must structurally protect its topological dimension by dynamically breaking permutation symmetry and suppressing the generic K-R combinatorial expansion. 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