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Quantum Frontiers

A universal bound on entanglement velocity

Logan Nye

Published Apr 28, 2025• 0 citations
Publisher

Abstract

Abstract We establish a universal bound on entanglement dynamics, proving that for any quantum system with energy E, the rate of entanglement entropy change is limited by $\left |\frac{dS}{dt}\right | \leq \frac{2E}{\pi \hbar}$ | d S d t | ≤ 2 E π ħ . This bound, which can be saturated by specific protocols, emerges from a framework treating entanglement rate as a physical observable and connects directly to quantum circuit complexity. When extended to quantum field theory, we demonstrate that the maximum entanglement propagation velocity equals the speed of light, revealing a fundamental connection between quantum information constraints and relativistic causality. We propose experimental tests on quantum computing platforms and explore implications for quantum computing efficiency and spacetime structure. Our results suggest that information-theoretic principles may be more fundamental than spacetime concepts, unifying quantum information, computational complexity, and relativistic physics.

DOI
10.1007/s44214-025-00080-w

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Abstract We establish a universal bound on entanglement dynamics, proving that for any quantum system with energy E, the rate of entanglement entropy change is limited by ∣dSdt∣≤2Eπℏ\left |\frac{dS}{dt}\right | \leq \frac{2E}{\pi \hbar}​dtdS​​≤πℏ2E​ | d S d t | ≤ 2 E π ħ . This bound, which can be saturated by specific protocols, emerges from a framework treating entanglement rate as a physical observable and connects directly to quantum circuit complexity. When extended to quantum field theory, we demonstrate that the maximum entanglement propagation velocity equals the speed of light, revealing a fundamental connection between quantum information constraints and relativistic causality. We propose experimental tests on quantum computing platforms and explore implications for quantum computing efficiency and spacetime structure. Our results suggest that information-theoretic principles may be more fundamental than spacetime concepts, unifying quantum information, computational complexity, and relativistic physics.
We investigate causality constraints on the time evolution of entanglement entropy after a global quench in relativistic theories. We first provide a general proof that the so-called tsunami velocity is bounded by the speed of light. We then generalize the free particle streaming model of [1] to general dimensions and to an arbitrary entanglement pattern of the initial state. In more than two spacetime dimensions the spread of entanglement in these models is highly sensitive to the initial entanglement pattern, but we are able to prove an upper bound on the normalized rate of growth of entanglement entropy, and hence the tsunami velocity. The bound is smaller than what one gets for quenches in holographic theories, which highlights the importance of interactions in the spread of entanglement in many-body systems. We propose an interacting model which we believe provides an upper bound on the spread of entanglement for interacting relativistic theories. In two spacetime dimensions with multiple intervals, this model and its variations are able to reproduce intricate results exhibited by holographic theories for a significant part of the parameter space. For higher dimensions, the model bounds the tsunami velocity at the speed of light. Finally, we construct a geometric model for entanglement propagation based on a tensor network construction for global quenches.
We establish quantum circuit complexity as a fundamental physical observable and prove that it satisfies an uncertainty relation with energy, analogous to Heisenberg’s canonical uncertainty principle. Through rigorous operator theory, we demonstrate that the complexity operator meets all mathematical requirements for a legitimate quantum observable, including self-adjointness, gauge invariance, and proper spectral decomposition. This enables us to derive a fundamental bound that constrains how quickly complexity can increase in physical systems given available energy resources. We provide complete mathematical proofs of these results and demonstrate their far-reaching implications across quantum computation, black hole physics, and computational complexity theory. In particular, we show that this uncertainty relation imposes fundamental speed limits on quantum circuits, explains maximal complexity growth in black holes, and suggests that physical constraints may enforce an effective separation between complexity classes independent of their mathematical relationships. We outline explicit experimental protocols for testing these predictions using current quantum computing platforms and discuss the profound implications for our understanding of the relationship between computational complexity and fundamental physics. Our results indicate that computational requirements may be as basic to physics as energy conservation, suggesting a deep connection between the structure of physical law and fundamental limits on computation.
Quantum information theory has provided the formal framework for describing how information is stored, transmitted, and transformed in physical quantum systems [6, 7, 9]. Its entropic formulations underpin our understanding of quantum computation, communication, and cryptography. Yet this theory traditionally treats all quantum operations as freely available, ignoring computational restrictions. In practice, however, any manipulation of quantum information must be performed by devices of bounded complexity and runtime. Capturing such realistic constraints requires extending quantum information theory to include computational efficiency as a fundamental component.This work takes a first step toward building a computational version of quantum information theory, one that treats efficiency as part of the theory itself. The goal is to understand how the behavior of quantum information changes when the parties involved can only perform computationally efficient operations. This approach bridges the abstract, ideal setting of quantum information theory with the practical limitations of real quantum devices, offering a means to study information processing under realistic resource constraints.At the center of this work are two new quantities: the quantum computational min-entropy and the quantum computational max-entropy. These entropies extend standard quantum entropies by explicitly limiting the computational power of the observer or adversary. The quantum computational min-entropy captures how unpredictable a quantum system A remains to an observer holding system B, when that observer is restricted to quantum circuits of bounded size. Formally, for a bipartite state rho AB, we define(c) H-s (min)(A|B)rho := - log d(A) max (epsilon s) B > A' F ((parallel to(A)circle times epsilon(s))(rho(AB)),|Phi (AA')> Quantum information theory; Theory of computation -> Quantum complexity theory
Contemporary physics encounters fundamental limitations when describing physical change under spacetime constraints, particularly the invariance of light speed and the requirement of infinite energy for massive bodies approaching this limit. This work proposes an alternative conceptual framework, termed Moses Walkway, which reformulates physical change not as motion through spacetime, but as a direct transition between physical states within an underlying informational structure.Rather than attempting to exceed relativistic bounds, the framework introduces the notion of state-to-state updates governed by informational cost, allowing non-local transitions without invoking superluminal propagation or violations of energy conservation. In this view, spacetime is treated as an emergent descriptive layer rather than a fundamental substrate.A formal representation based on networked state transitions is outlined, and conceptual connections with quantum non-locality, tunneling phenomena, and information-theoretic approaches to physics are discussed. The Moses Walkway is presented not as a complete physical theory, but as a foundational and pre-geometric model intended to re-express long-standing physical constraints and to open new directions for theoretical investigation and computational simulation.
The speed limit on entanglement is defined as the maximal rate at which entanglement can be generated or degraded in a physical process. We derive the speed limits on entanglement, using the relative entropy of entanglement and trace-distance entanglement, for unitary and arbitrary quantum dynamics, where we assume that the dynamics of the closest separable state can be approximately described by the closest separable dynamics of the actual dynamics of the system. For the unitary dynamics of isolated bipartite systems which are described by pure states, the rate of entanglement production is bounded by the product of fluctuations of the system's driving Hamiltonian and the surprisal operator, with an additional term reflecting the time-dependent nature of the closest separable state. Removing restrictions on the purity of the input and on the unitarity of the evolution, the two terms in the bound get suitably altered. Furthermore, we find a lower bound on the time required to generate or degrade a certain amount of entanglement by arbitrary quantum dynamics. We demonstrate the tightness of our speed limits on entanglement by considering quantum processes of practical interest.
We show that in any relativistic system, entanglement entropy obeys a speed limit set by the entanglement in thermal equilibrium. The bound is derived from inequalities on relative entropy with respect to a thermal reference state. Thus the thermal state constrains far-from-equilibrium entanglement dynamics whether or not the system actually equilibrates, in a manner reminiscent of fluctuation theorems in classical statistical mechanics. A similar shape-dependent bound constrains the full nonlinear time evolution, supporting a simple physical picture for entanglement propagation that has previously been motivated by holographic calculations in conformal field theory. We discuss general quantum field theories in any spacetime dimension, but also derive some results of independent interest for thermal relative entropy in 1+1d CFT.