⚛️ high-energy theory

The Capacity of Entanglement and Holographic Entropies at Finite Resources

This paper refines the Ryu-Takayanagi correspondence by introducing the capacity of entanglement as a dimension-free bound on entropy ambiguity and demonstrating that a single holographic geometry uniquely fixes all one-shot entropies, effectively treating large central charge as the asymptotic limit of many copies.

Original authors: Ning Bao, Jacob March

Published 2026-08-28

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CC BY 4.0

Original authors: Ning Bao, Jacob March

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

In the deepest corners of theoretical physics, where the laws of gravity meet the strange rules of quantum mechanics, scientists have long suspected that space itself is woven from information. A leading idea in this field suggests that the geometry of a hidden, higher-dimensional universe is encoded in the entanglement of particles on its boundary. This connection is so precise that a specific formula, known as the Ryu-Takayanagi relation, allows physicists to calculate the amount of quantum entanglement in a region of space simply by measuring the area of a minimal surface floating in the bulk. For decades, this has been treated as a perfect match: a geometric area equals a specific number called entropy. However, this equation was derived under idealized conditions, assuming an infinite number of copies of a system and a perfect, noise-free measurement. In the real world, experiments are limited by finite resources, and no preparation is ever perfectly precise. This raises a critical question: if we can only measure a single, imperfect copy of a quantum state, does the geometric area still tell us exactly what the entropy is, or does the fuzziness of our tools make the answer meaningless?

A team of researchers at Northeastern University and Brookhaven National Laboratory has now answered this question with a new level of precision. They found that while the entropy of a single, imperfectly prepared quantum state is not a single, sharp number, it is confined to a very narrow range that is still directly tied to the geometry. The key to this discovery was identifying a new property of quantum states called the "capacity of entanglement." Instead of relying on the size of the system, which can be infinitely large and unhelpful, the researchers used this capacity to measure how much the energy of the quantum state fluctuates. They proved that for the types of states described by holographic theories, this fluctuation is small enough that the entropy cannot wander far from the value predicted by the geometric area. Even with the inevitable noise of a real experiment, the entropy stays within a tight band around the geometric prediction, making the connection between geometry and information robust even when resources are limited.

The researchers also tackled a second, more subtle problem regarding how we define entropy for a single object. In standard information theory, the entropy of a system is often defined by looking at what happens when you have millions of identical copies of it. But a holographic geometry describes just one state, not a million. The team showed that for these specific holographic states, the entropy of a single copy behaves exactly as if it were part of a vast collection. This happens because the internal structure of the state is so highly organized that the fluctuations average out naturally, mimicking the behavior of many copies. This means that the geometric area does not just represent the entropy of a single state; it represents every possible way of measuring that entropy at once, whether one looks at the standard measure or other variations used in one-shot information theory. The geometry fixes the answer for all of them simultaneously.

To reach these conclusions, the authors had to replace old mathematical tools that relied on the dimension of the system with new ones that rely on the capacity of entanglement. The old tools suggested that as a system gets larger, the uncertainty in entropy grows so fast that the geometric prediction becomes useless. The new approach shows that for holographic states, the uncertainty grows much more slowly, scaling with the square root of the area rather than the volume. This distinction is crucial because it means the geometric prediction remains valid even as the system becomes very large. The researchers demonstrated that this holds true for any smooth variation of the state, provided the state does not undergo a sudden, dramatic change in its internal structure. If such a change were to occur, the tight link between geometry and entropy would break, but for the vast majority of holographic states, the connection remains unshaken.

The implications of this work extend to how we might test these theories in the future. Scientists hope to use these findings to verify whether a physical system has a holographic description by checking if its measured entropies fit within the predicted geometric bounds. The new results provide a clear criterion: if the measured entropies of a system deviate from the geometric prediction by more than a tiny, calculable amount, the system cannot be described by a simple semiclassical geometry. This offers a concrete way to distinguish between systems that are truly holographic and those that are not, even when working with imperfect data. The study does not claim to have solved every mystery of quantum gravity, but it has firmly established that the bridge between geometry and information is far sturdier than previously thought, holding firm even under the constraints of finite resources and single-copy measurements.

Ultimately, this research clarifies the operational meaning of the holographic principle. It confirms that the area of a minimal surface is not just a theoretical abstraction for infinite systems, but a concrete, measurable quantity that dictates the behavior of real, finite quantum states. By showing that the entropy of a single state is tightly bound to its geometric dual, the work bridges the gap between the idealized mathematics of string theory and the practical realities of quantum information. It suggests that the universe's geometric structure is encoded in its quantum information with a precision that survives the noise of measurement, offering a new foundation for understanding how space and time emerge from the quantum world. The findings leave open the challenge of how to perform these measurements in practice, but they provide the theoretical assurance that such a task is, in principle, possible.

Technical Summary: The Capacity of Entanglement and Holographic Entropies at Finite Resources

Problem Statement
The Ryu-Takayanagi (RT) formula equates the area of a minimal surface in the bulk to the von Neumann entropy ( $S_{vN}$ ) of a boundary subregion. However, this identification faces two fundamental ambiguities when applied to finite resources and single states:

Continuity and Dimensionality: The operational meaning of $S_{vN}$ (as a compression or dilution rate) is defined asymptotically for many copies. For a single state prepared with finite fidelity (within a trace-norm ball $B_\epsilon(\rho)$ ), the standard Fannes-Audenaert continuity bound depends on the Hilbert space dimension $d$ . In a continuum Conformal Field Theory (CFT), $d$ diverges as the UV cutoff is removed, rendering $S_{vN}$ ill-posed as a target quantity for any finite-precision experiment.
Single-State vs. Asymptotic: Experiments measure integer Rényi moments ( $\text{Tr}\rho^\alpha$ ) which yield bare Rényi entropies $S_\alpha$ . For holographic states, $S_\alpha$ differs from $S_{vN}$ by an extensive amount (e.g., $S_2 = \frac{3}{4}S_{vN}$ for a vacuum interval). Furthermore, the "smooth" entropies (one-shot information theoretic quantities defined as extrema over the preparation ball) are typically distinct from $S_{vN}$ . The question is whether a single holographic geometry can fix all one-shot entropies simultaneously, or if the distinction between single-copy and asymptotic limits remains significant.

Methodology
The authors employ a combination of spectral analysis, modular Hamiltonian theory, and one-shot information theory, avoiding asymptotic limits in favor of finite-resource bounds.

Modular Energy Moments: The analysis centers on the modular energy $E = -\log \lambda$ (where $\lambda$ are eigenvalues of the density matrix). The mean of $E$ is $S_{vN}$ , and its variance is the capacity of entanglement ( $C$ ).
Spectral Truncation: The authors utilize the concentration of the modular energy spectrum. For holographic states, the distribution of $E$ is shown to be sub-Gaussian with a width scaling as $\sqrt{C}$ .
Continuity Bounds: Instead of counting eigenvalues (which introduces dimension $d$ ), the authors derive continuity bounds for entropy differences based on the trace distance and the capacities of the states involved.
Smooth Entropy Analysis: The paper analyzes the "flatness" of the smooth Rényi spectrum by constructing specific states within the preparation ball that maximize or minimize entropy, utilizing a "projected-state" construction to bound the rank and eigenvalues of states in the ball.

Key Contributions and Results

1. A Dimension-Free Continuity Bound (Theorem 3)
The paper replaces the dimensional factor in the Fannes-Audenaert inequality with the capacity of entanglement.

Result: For two states $\rho$ and $\sigma$ with trace distance $\delta = \|\rho - \sigma\|_1$ , the difference in von Neumann entropy is bounded by:
$|S_{vN}(\rho) - S_{vN}(\sigma)| \leq \sqrt{\frac{\delta}{2-\delta}} \left(\sqrt{C_\rho} + \sqrt{C_\sigma}\right) - \log\left(1 - \frac{\delta}{2}\right)$
Significance: This bound is dimension-free. For holographic states where $C \sim S_{vN}$ , the ambiguity scales as $\mathcal{O}(\sqrt{\delta S_{vN}})$ , which is subextensive, provided the capacity $C_\sigma$ of the perturbed state is small compared with $S_{vN}^2$ times the trace norm. This contrasts with the standard bound which scales as $\delta \log d$ (extensive). The bound is saturated by specific pairs of states (e.g., a pure state mixed with a maximally mixed state), proving the coefficient cannot be improved.
Condition: The subextensive scaling holds provided the states are "holographic" (single dominant replica saddle). If two saddles compete (e.g., near an entanglement transition), $C$ can scale as $S_{vN}^2$ , and the ambiguity becomes extensive again, rendering the von Neumann entropy of a prepared state ill-posed once more.

2. Flatness of the Smooth Rényi Spectrum (Theorem 5 and Corollary 6)
The paper establishes that for a single holographic state, all smooth Rényi entropies at fixed $\alpha > 1$ collapse onto the von Neumann entropy.

Result: For a holographic state with a single dominant replica saddle, the smooth Rényi entropy $S^\epsilon_\alpha(\rho)$ satisfies:
$S_{vN}(\rho) - W_\epsilon \leq S^\epsilon_\alpha(\rho) \leq S_{vN}(\rho) + W_\epsilon + \mathcal{O}(1)$
where the half-width $W_\epsilon \sim \sqrt{C} \sim \sqrt{S_{vN}}$ .
Mechanism: The large central charge $c$ (or $1/G_N$ ) plays the role of the number of copies $n$ in the asymptotic equipartition property. The modular energy concentrates with a relative width of $1/\sqrt{S_{vN}}$ , causing the spectrum to appear flat to any one-shot entropy measurement.
Implication: The minimal surface area fixes every one-shot entropy of the state simultaneously. The geometry does not distinguish between different Rényi indices or min/max entropies at leading order in $1/G_N$ .

3. Limitations of Preparation Noise (Section 4.3)
The authors demonstrate that generic preparation noise (modeled as mixing with a contaminant state) cannot reproduce the smoothing effect required to reach the optimal smooth entropy.

Result: Unless the contaminant is macroscopically distinct (separated by a constant trace distance), the mixing noise only shifts the entropy by $\mathcal{O}(1)$ or logarithmic factors, failing to suppress the extensive difference between bare Rényi entropies and $S_{vN}$ .
Consequence: The "smoothing" required to align measured moments with the RT area must be treated as an optimization problem over the ball, not as a natural consequence of experimental noise.

Significance and Claims
The paper claims to resolve the ambiguity of how sharply a semiclassical geometry fixes an entropy at finite resources:

Resolution of Ambiguity: The capacity of entanglement, rather than the Hilbert space dimension, governs the precision of the entropy-area correspondence. This makes the von Neumann entropy a well-posed quantity for single holographic states, provided the capacity is small compared with $S_{vN}^2$ times the trace norm and a single replica saddle dominates. If two saddles compete, the entropy becomes ill-posed.
One-Shot Holography: The results justify using the RT area as a proxy for all smooth one-shot entropies in holographic theories, effectively extending the asymptotic equipartition property to single copies via the large central charge limit.
Entropy Cone Constraints: The flatness result implies that a holographic state's smooth entropy vector lies within $\mathcal{O}(\sqrt{S_{vN}})$ of the holographic entropy cone. Consequently, an experimental observation of a vector violating the cone by an extensive margin would definitively rule out a holographic dual.
Failure Modes: The paper explicitly notes that these results fail at entanglement transitions where the modular spectrum becomes bimodal (two competing saddles). In such cases, the capacity diverges relative to the entropy, the continuity bound becomes extensive, and the flatness of the spectrum breaks down.

The authors maintain that while the theoretical bounds are established, the practical application to finite-copy estimation and certification remains an open challenge, particularly regarding the estimation of capacity from limited data and the handling of multi-component regions where saddle exchanges occur.

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