Symmetry and localisation in causally constrained quantum operator dynamics

TL;DR

This work addresses how causality constraints emerge in many-body quantum operator dynamics by introducing wall unitaries that enforce a bounded light cone in time-periodic brickwork circuits. It develops a rigorous C*-algebraic framework to identify invariant sub-algebras and the commutant structure that splits operator space into locally decoupled sectors, enabling an entanglement area law and robust localisation without relying on integrals of motion. By employing representation theory and normalisers, the authors classify Abelian and non-Abelian wall structures, derive conditions for local conserved quantities, and analyze fragmentation, measurement stability, and spectral form factors in random wall ensembles. The results offer a general, analytically tractable description of locally constrained quantum dynamics with potential applications in benchmarking quantum simulations and understanding ergodicity breaking beyond conventional paradigms.

Abstract

This paper explores the connection between causality and many-body dynamics by studying the algebraic structure of tri-partite unitaries ('walls') which permanently arrest local operator spreading in their time-periodic evolution. We show that the resulting causally independent subsystems arise from the invariance of an embedded sub-algebra in the system (ie. a generalised symmetry) that leads to the splitting of operator space into commuting sub-algebras. The commutant structure of the invariant algebra is then used to construct local conserved quantities. Using representation theory of finite matrix algebras, the general form of wall gates is derived as unitary automorphisms. Taking causal independence as a minimal model for non-ergodic dynamics, we study its effect on probes of many-body quantum chaos. We prove an entanglement area-law due to local constraints and we study its stability against projective measurements. In a random ensemble exhibiting causal independence, we compare spectral correlations with the universal (chaotic) ensemble using the spectral form factor. Our results offer a rigorous understanding of locally constrained quantum dynamics from a quantum information perspective.

Figures (8)

• Figure 1: We study the time-periodic evolution of local traceless operators in brickwork unitaries. A wall unitary tri-partitions the circuit and permanently obstructs the spreading of local operators. This leads to a bounded light cone (shown in red) and causal decoupling across spatial regions.
• Figure 2: Localisation of left-evolving operators in tensor network notation. Green operators correspond to traceless operators where dotted lines signify arbitrary superpositions. Elements of the full matrix algebra $m \in \mathcal{M}_L$ must localise under the wall evolution. The operator evolution is closed as a local sub-algebra $\mathcal{M}_L \otimes \mathcal{A}_C$ under \ref{['thm:left_right_closure']}.
• Figure 3: Stable embedding of a tri-partite wall unitary into an arbitrary circuit environment under \ref{['thm:left_right_equiv']}. The existence of a bounded light cone from the left-environment implies that of from the right inducing a splitting of operator space.
• Figure 4: Algebraic structure of conserved operators. A bounded light cone splits operator space with the algebra $\mathcal{C}$ of $C$-local conserved operators at the intersection of commutants from left-localised and right-localised operators. Local conserved charges can be independent from left-right coupling (ie. uncoupled subsystems) which are not elements of the localised algebras.
• Figure 5: Extending the wall condition to time-dependent unitary sequences. On a), the sequence consistes of elements of the normaliser (see \ref{['def:normaliser']}) group of $\overline{\mathcal{M}_L}$. On b), we dress this sequence through gauge transformations, inducing a sequence of isomorphic sub-algebra transformations, changing the global operator basis at every timestep.

Theorems & Definitions (86)

• Definition 2.1: Walls
• Theorem 2.1
• proof
• Definition 2.2
• Corollary 2.2
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5