⚛️ high-energy theory

Approximate locality, black hole complementarity and overlapping qubits

This paper presents a toy model of an evaporating black hole using overlapping qubits and approximately local degrees of freedom, which naturally realizes black hole complementarity by preventing exact Hilbert space factorization and successfully reproduces the Page curve through entropy overlap.

Original authors: ChunJun Cao, Gong Cheng, Alexander Jahn, Thomas Koutsikos

Published 2026-08-12

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: ChunJun Cao, Gong Cheng, Alexander Jahn, Thomas Koutsikos

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as the ultimate magic show, where the most baffling trick involves a black hole. For decades, physicists have been arguing over what happens when a black hole eats something and then slowly evaporates away. The trouble starts with a rule called "unitarity," which is basically the universe's promise that information is never truly lost, only scrambled. If a black hole disappears, the information about what fell in should be hidden in the leftover radiation. But here's the catch: if you try to look at the radiation outside and the stuff still inside the black hole at the same time, it looks like the same information exists in two places at once. In the world of quantum mechanics, this is a big no-no, like a magician pulling the same rabbit out of two different hats simultaneously. This paradox has led to wild theories, including the idea that the black hole's surface is a fiery wall of energy (a "firewall") that destroys anything trying to cross it, or that the laws of physics break down in ways we can't yet explain. The question isn't just about black holes; it's about whether our understanding of how space, time, and information fit together is fundamentally broken.

Now, a team of researchers has proposed a new way to look at this puzzle using a concept called "overlapping qubits." Think of a qubit as a tiny, fundamental bit of information, like a pixel in a cosmic image. Usually, we imagine these pixels as distinct, separate tiles on a floor. But in this new model, the tiles aren't separate at all; they are like overlapping transparencies stacked on top of each other. If you have a stack of clear sheets, you can draw a picture on the top one and a different picture on the bottom one, but because they overlap, the lines from the bottom one bleed through slightly. The authors suggest that the "inside" of a black hole and the "outside" radiation aren't two separate rooms with a door between them. Instead, they are two different ways of looking at the exact same set of overlapping transparencies.

In their paper, the team built a simplified computer model to test this idea. They didn't prove that black holes work this way in real life, but they showed that if you assume the universe uses these overlapping bits of information, the math suddenly makes sense. They found that you can have the same information appear in two places without actually breaking the rules of quantum mechanics, because the "inside" and "outside" are just different representations of the same underlying algebra. It's like having two different maps of the same city: one map shows the streets, and the other shows the subway lines. They look different, but they are describing the exact same territory. Because of this overlap, the model allows the information to escape the black hole naturally, recovering a specific pattern of entropy growth known as the "Page curve," which is what we expect to see if information is preserved. However, the authors are careful to note that this is a toy model—a simplified simulation using random mathematical vectors and fermions. They suggest that while this approach avoids the "cloning" paradox and explains how information can be recovered, it relies on the idea that the strict separation between space and time is only an approximation. In their simulations, the "firewall" problem disappears because the interior and exterior are never truly separate to begin with, but this is a theoretical construction that still needs to be tested against more complex, realistic physical theories.

Technical Summary: Approximate Locality, Black Hole Complementarity, and Overlapping Qubits

Problem Statement
The paper addresses the black hole information paradox, specifically the tension between unitary quantum mechanics, the validity of semi-classical effective field theory (EFT), and the "no-drama" condition at the horizon. A naive application of unitary evolution to an evaporating black hole suggests that information is encoded simultaneously in the interior and the exterior Hawking radiation, seemingly violating the no-cloning theorem. While black hole complementarity proposes that no single observer can verify this cloning, the underlying mechanism remains unclear. Furthermore, existing proposals often rely on strong non-locality or ad-hoc modifications to local quantum field theory (QFT), lacking a natural theoretical framework for "weak" or approximate non-locality. The authors seek a framework that allows for the recovery of the Page curve (indicating unitary evaporation) without violating fundamental principles, while explaining how an observer might perceive cloning due to approximate locality.

Methodology
The authors construct a kinematic toy model based on the concept of overlapping qubits (and fermions), originally formulated by Chao, Reichardt, Sutherland, and Vidick (CRSV). The core methodology involves three distinct descriptions of the system:

Fundamental Description: A finite-dimensional Hilbert space $\mathcal{H}_F$ of dimension $2^n$ (where $n$ is the black hole entropy in bits) spanned by fundamental degrees of freedom. The dynamics are assumed to be unitary.
Quasi-Local (Overlapping) Description: A set of $N(t)$ $N (t)$ operators (qubits or Majorana fermions) acting on $\mathcal{H}_F$ $H_{F}$ that mimic local EFT degrees of freedom. These operators are "overlapping," meaning they are not strictly independent. Specifically, operators acting on distinct "qubits" do not exactly commute (for qubits) or anti-commute (for fermions) but satisfy relations like $[O_i, O'_j] \sim \mathcal{O}(\epsilon)$ $[O_{i}, O_{j}^{'}] \sim O (ϵ)$ or $\{ \Psi_I, \Psi_J \} \sim 2\delta_{IJ} + 2\epsilon R_{IJ}$ ${Ψ_{I}, Ψ_{J}} \sim 2 δ_{I J} + 2 ϵ R_{I J}$ .
- The construction utilizes the Johnson-Lindenstrauss lemma to embed a large number of approximately orthogonal vectors (representing the overlapping operators) into a lower-dimensional vector space.
- The interior and exterior (radiation) degrees of freedom are defined as complementary sets of these overlapping operators.
Effective Description: A hypothetical description where the overlaps are ignored, treating the $N(t)$ operators as independent qubits in a much larger Hilbert space $\mathcal{H}_{eff}$ of dimension $2^{N(t)}$ .

The authors analyze the system using Majorana fermions for calculational tractability. They define three types of entropy corresponding to the three descriptions:

Fundamental Entropy: Algebraic entropy computed in the UV theory.
Effective Entropy: The von Neumann entropy of a state reconstructed by an observer who assumes the operators are independent (ignoring overlaps).
Quasi-Local Entropy: The entropy of a state reconstructed by an observer who detects the inconsistency (negativity) in the reconstructed density matrix and corrects it to find the physical state within the smaller fundamental Hilbert space.

Key Contributions and Results

Resolution of No-Cloning via Algebraic Equivalence:
The model demonstrates that the interior and exterior operators are not distinct subsystems but rather two different representations of the same fundamental algebra. After the Page time ( $t \sim n$ ), the set of exterior overlapping operators forms an overcomplete basis for the fundamental algebra. Consequently, any interior operator can be reconstructed as a linear combination of exterior operators. There is no cloning because the "interior" and "exterior" are simply different bases of the same underlying Hilbert space.
Recovery of the Page Curve:
- Effective Entropy (Hawking-like): An observer who ignores the overlaps (the standard observer) reconstructs a pseudo-state in the large effective Hilbert space. Due to the mismatch between the assumed dimension and the actual fundamental dimension, this state exhibits "negativity" (non-positive eigenvalues). If the observer corrects this by finding the nearest physical state (ignoring the identity block artifact), the resulting entropy grows linearly with time, mimicking Hawking's original prediction of ever-increasing entropy.
- Quasi-Local Entropy (Page-like): A more sophisticated observer who recognizes the compression (the existence of the identity block in the singular value decomposition of the overlap matrix) can isolate the physical subspace. The entropy of this corrected state follows a Page curve: it rises, peaks near the Page time, and then decreases, returning to zero as the black hole fully evaporates.
Complexity of Reconstruction:
The paper analyzes the complexity of reconstructing an interior mode from exterior radiation. While the reconstruction is theoretically possible after the Page time (due to the overcomplete basis), the authors argue that the circuit complexity remains high. Reconstructing a specific semiclassical mode requires a coherent superposition of many basis vectors, acting as a "complexity barrier" that prevents computationally bounded observers from verifying the cloning or accessing the interior information easily.
Generalization to General Operators:
The authors extend the fermionic results to general overlapping operators (including Pauli operators) using a random projection map. They show that the structure of the reconstructed state generally contains a tensor product of a physical state and an identity block, confirming that the compression mechanism is not specific to fermions.

Significance and Claims
The paper claims to provide a concrete, mathematically explicit framework for approximate locality and black hole complementarity. Its primary significance lies in:

Natural Emergence of Complementarity: It shows that complementarity arises naturally from the relaxation of exact tensor factorization. The "cloning" is an illusion created by the observer's assumption of exact locality (independence of subsystems) in a system where subsystems only approximately commute.
Unitary Evaporation without Firewalls: By preventing exact factorization, the model circumvents the assumptions required for the firewall paradox (specifically the strong subadditivity arguments applied to strictly factorized subsystems). The model preserves unitarity and the "no-drama" condition for low-weight operators, as the non-locality is weak ( $\epsilon$ ) and only detectable via high-complexity operations.
Holographic Compression: The model illustrates how a large number of overlapping degrees of freedom can be compressed into a smaller Hilbert space, analogous to the holographic principle, where volumetric degrees of freedom are encoded in area-scaling degrees of freedom.

The authors are modest about their claims, acknowledging that this is a kinematic toy model. They note that the current construction uses random vectors and does not yet incorporate specific spatial locality or causal structures of a full quantum gravity theory. They emphasize that while the model reproduces the Page curve and explains the mechanism of complementarity, a more refined embedding map is required to reproduce the low-point correlation functions of realistic local QFTs. The work serves as a proof-of-concept that approximate tensor factorization can resolve the information paradox without invoking radical new physics, but rather by re-evaluating the mathematical description of subsystems in quantum gravity.

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