The Split Page Curve: How Conserved Charges Escape Scrambling

Mustafa Boudjerada

Published Aug 26, 2026• 0 citations

Abstract

This note studies how conservation laws modify the Page curve of information recovery. A fully scrambled quantum system normally hides information until the Page-time transition. However, realistic scramblers preserve conserved quantities, raising the question of whether all information components follow the same recovery behavior. Using an exactly charge-conserving block-Haar scrambling model, this work separates a record into two classes: 1. Charge records: information stored in different conserved-charge sectors. These records remain readable from the earliest emitted fragments because conservation prevents them from being scrambled into the hidden class. 2. Phase records: information stored within a fixed charge sector. These records follow the conventional Page-curve behavior, exhibiting an initial blackout, a knee at half emission, and late recovery. The study verifies that the Page curve splits into two distinct information channels. The charge component grows monotonically from the first emitted qubits, while the phase component retains the full scrambling structure. Additional diagnostics show that charge records are redundantly accessible across radiation fragments, while phase records remain exclusive until sufficient information is emitted. The work provides an information-theoretic model showing how conservation laws constrain scrambling and create protected record classes. It is a finite-dimensional toy model and does not constitute a calculation of dynamical black-hole physics.

DOI: 10.5281/zenodo.22103404

Related papers

Whose Page Curve? Operator Classes, Charge Reference Frames, and Information Recovery in Charge-Locked Evaporation

Mustafa Boudjerada

The Page curve is the standard signature of unitary information recovery in an evaporating quantum system. This work asks a question usually left implicit: whose Page curve? In a solvable charge-locked evaporation model, the radiation observables separate into two classes: the charge-coherent algebra, available to an observer with the appropriate charge reference frame, and the charge-invariant algebra of an observer without such a reference frame. These two observers do not recover the same information curve. For the full charge-coherent algebra, the Page curve follows the standard unitary pattern: the radiation entropy rises and then returns exactly to zero, with all interior microstates becoming mutually orthogonal and perfectly decodable radiation records. For the charge-invariant algebra, the turnover is incomplete. The accessible entropy terminates at the charge-distribution entropy, representing the information deficit caused by the missing reference frame. Microstates related by a global charge-phase transformation remain indistinguishable to this observer, not because information is destroyed, but because it is inaccessible within that observable algebra. The work establishes an operator-class Page theorem: complete purification and full information recovery are properties of the charge-coherent algebra, while charge-invariant observers experience a Page curve truncated by the entropy associated with the missing charge reference frame. The model also reveals a mode-structure law. Multi-channel charged radiation can carry relative-charge information internally and locks only the global charge-phase bit, while a single effective channel can lock the full information content. An external reference frame restores accessibility only when its charge quantum matches the charge quantum of the hidden coherence, demonstrating a precise reference-frame selection rule. This work is a circuit-level quantum information model with no gravity and no dynamical backreaction. It does not claim a black-hole evaporation solution. Instead, it provides an exactly solvable framework showing how information recovery can depend on observable algebra and reference frame in systems with conserved charges. All results are accompanied by reproducibility scripts and verification procedures.

Aug 25, 20260 citations

Refined symmetry-resolved Page curve and charged black holes

Pan Li, Yi Ling

The Page curve plotted by the typical random state approximation is not applicable to a system with conserved quantities, such as the evaporation process of a charged black hole during which the electric charge does not radiate out with a uniform rate macroscopically. In this context the symmetry-resolved entanglement entropy may play a significant role in describing the entanglement structure of such a system. We attempt to impose constraints on microscopic quantum states to match with the macroscopic phenomenon of the charge radiation during black hole evaporation. Specifically, we consider a simple qubit system with conserved spin/charge serving as a toy model for the evaporation of charged black holes. We propose refined rules for selecting a random state with conserved quantities to simulate the distribution of charges during the different stages of evaporation, and obtain refined Page curves that exhibit distinct features in contrast to the original Page curve. We find the refined Page curve may have a different Page time and exhibit asymmetric behavior on both sides of the Page time. Such refined Page curves may provide more realistic description for the entanglement between the charged black hole and radiation during the process of evaporation.

arXiv:2311.04436Nov 8, 20230 citations

Refined symmetry-resolved Page curve and charged black holes*

Pan Li, Yi 意 Ling 凌

Abstract The Page curve plotted using the typical random state approximation is not applicable to a system with conserved quantities, such as the evaporation process of a charged black hole, during which the electric charge does not macroscopically radiate out with a uniform rate. In this context, the symmetry-resolved entanglement entropy may play a significant role in describing the entanglement structure of such a system. We attempt to impose constraints on microscopic quantum states to match the macroscopic phenomenon of charge radiation during black hole evaporation. Specifically, we consider a simple qubit system with conserved spin/charge serving as a toy model for the evaporation of charged black holes. We propose refined rules for selecting a random state with conserved quantities to simulate the distribution of charges during the different stages of evaporation and obtain refined Page curves that exhibit distinct features in contrast to the original Page curve. We find that the refined Page curve may have a different Page time and exhibit asymmetric behavior on both sides of the Page time. Such refined Page curves may provide a more realistic description for the entanglement between the charged black hole and radiation during the evaporation process.

Chinese Physics CFeb 29, 20245 citations

Scrambling the Record: A Robustness Test of Information Preservation in Quantum Histories

Mustafa Boudjerada

This note investigates whether quantum records are preserved by physical structure or merely by the choice of representation. Using the established record framework, the work compares structured history records with scrambled constructions in which the correlations responsible for record accessibility are deliberately disrupted. The study verifies that: 1. Structured records remain accessible because information is redundantly distributed across correlated degrees of freedom. 2. Scrambling destroys the relational structure that connects observers to records, causing record distinguishability to decrease. 3. The loss of accessibility does not require non-unitary evolution: the global quantum state remains intact while the information stored in the chosen record channel becomes unavailable. 4. Record preservation therefore depends on physical constraints and redundancy rather than on arbitrary labels or basis choices. The result provides a robustness test for the record framework and clarifies the distinction between physical erasure and information becoming inaccessible through scrambling. The work is a controlled theoretical study of record accessibility and does not claim that arbitrary scrambling processes reproduce full black-hole dynamics or realistic quantum gravity.

Aug 26, 20260 citations

The Theory of Records: A General Principle for Information Protection in Quantum Dynamics

Mustafa Boudjerada

This work formulates and tests a general principle governing when physical records remain accessible under quantum dynamics. Across previous models involving gauge theory, thermalization, closed universes, and scrambling systems, a common structure emerges: some distinctions survive only when the dynamics preserve an observer-accessible information structure. The paper introduces a rigorous definition of records using restricted distinguishability between histories: E_A = − ln F(ρ₀|A, ρ₁|A) and defines protection as the persistence of this distinguishability under an allowed class of quantum channels. The proposed principle is: A record is protected if and only if an observer-accessible Heisenberg fixed-point (peripheral) operator of the allowed channel class distinguishes the histories. The principle is tested against several systems: 1. Covariant quantum channels, where explicit fidelity lower bounds are verified. 2. Many-body localization, showing that protection can arise from emergent quasi-local conserved structures rather than fundamental symmetries. 3. Noiseless subsystems, demonstrating that protected records can exist through operator-algebraic commutants rather than conserved densities. 4. Blind searches over random semigroups, where every protected case contains the predicted fixed-point witness. The work connects quantum records with existing theories of information-preserving structures, superselection, asymmetry, quantum error correction, and emergent conservation laws. The result is a general operator-algebraic framework for understanding why some information survives dynamics while other information becomes inaccessible.

Aug 26, 20260 citations

Universal limitation of quantum information recovery: symmetry versus coherence

Hiroyasu Tajima, Keiji Saito

Quantum information is scrambled via chaotic time evolution in many-body systems. The recovery of initial information embedded locally in the system from the scrambled quantum state is a fundamental concern in many contexts. From a dynamical perspective, information recovery can measure dynamical instability in quantum chaos, fault-tolerant quantum computing, and the black hole information paradox. This article considers general aspects of quantum information recovery when the scrambling dynamics have conservation laws due to Lie group symmetries. Here, we establish fundamental limitations on the information recovery from scrambling dynamics with arbitrary Lie group symmetries. We show universal relations between information recovery, symmetry, and quantum coherence, which apply to many physical situations. The relations predict that the behavior of the Hayden-Preskill black hole model changes qualitatively under the assumption of the energy conservation law. Consequently, we can rigorously prove that under the energy conservation law, the error of the information recovery from a small black hole remains unignorably large until it completely evaporates. Moreover, even when the black hole is very large, the recovery of information thrown into the black hole is not completed until most of the black hole evaporates. The relations also provide a unified view of the symmetry restrictions on quantum information processing, such as the approximate Eastin-Knill theorem and the Wigner-Araki-Yanase theorem for unitary gates.

arXiv:2103.01876Mar 2, 20218 citations