Entangled Boundaries: A New Look at Holographic Mutual Information

Author: Denis Avetisyan

A new study delves into how entanglement within holographic heat baths affects the mutual information between regions on the boundary of spacetime.

The study contrasts the field content required to compute entanglement entropy between subregions <span class="katex-eq" data-katex-display="false">\bm{A\_{1}}</span> and <span class="katex-eq" data-katex-display="false">\bm{A\_{2}}</span> within the frameworks of standard conformal field theory and double holography, revealing that while standard CFT relies on quantum fields across both subregions and their union, the double holographic approach utilizes distinct “Q-EWs” - <span class="katex-eq" data-katex-display="false">\mathcal{W}\_{\bm{A\_{1}}}</span> and <span class="katex-eq" data-katex-display="false">\mathcal{W}\_{\bm{A\_{2}}}</span> - to characterize entanglement, highlighting a fundamental difference in how these theories approach the same quantum information problem. — The study contrasts the field content required to compute entanglement entropy between subregions $\bm{A\_{1}}$ and $\bm{A\_{2}}$ within the frameworks of standard conformal field theory and double holography, revealing that while standard CFT relies on quantum fields across both subregions and their union, the double holographic approach utilizes distinct “Q-EWs” – $\mathcal{W}\_{\bm{A\_{1}}}$ and $\mathcal{W}\_{\bm{A\_{2}}}$ – to characterize entanglement, highlighting a fundamental difference in how these theories approach the same quantum information problem.

Researchers find a negative contribution to boundary mutual information arising from entanglement in the holographic dual of a thermal state.

Resolving the intricacies of entanglement in many-body systems remains a central challenge in theoretical physics. This is addressed in ‘Boundary mutual information in double holography’, which investigates the mutual information between spatial subregions within a holographic duality setup incorporating a heat bath. The authors demonstrate a negative contribution to this mutual information arising from the entanglement of quantum fields residing in the holographic dual of the heat bath, decomposing it into geometric and correction terms sourced by bulk quantum fields. Could this negative contribution signal a fundamental departure from standard entanglement expectations in these highly mixed holographic states, and what implications does it hold for understanding the emergence of spacetime?

Entanglement and the Holographic Mirage

Quantum entanglement represents a profound departure from classical physics, demonstrating correlations between particles that cannot be explained by any local hidden variable theory. This phenomenon implies that two or more particles can become linked in such a way that they share the same fate, no matter how far apart they are – measuring the properties of one instantaneously influences the properties of the other. This isn’t a transfer of information faster than light, but rather a demonstration that the particles are not truly independent entities, challenging the classical notion of locality and raising fundamental questions about the nature of information itself. $\psi = \sum_{i} c_{i} |i \rangle$ describes the entangled state, where the probabilities of measurement outcomes are intrinsically linked, regardless of spatial separation. The implications extend beyond foundational physics, offering potential applications in quantum computing and quantum cryptography, and forcing a reassessment of how information is encoded and accessed in the universe.

The AdS/CFT correspondence, a cornerstone of theoretical physics, posits a surprising relationship: a theory of gravity in a space called Anti-de Sitter (AdS) space is fundamentally equivalent to a quantum field theory without gravity living on the boundary of that space – a conformal field theory (CFT). This isn’t merely an analogy; the two theories are mathematically intertwined, offering a powerful tool to study strongly coupled quantum systems. Because calculations in one theory can often be mapped to simpler calculations in the other, physicists can leverage the geometrical nature of gravity in AdS space to gain insights into the notoriously complex behavior of entangled quantum systems within the CFT. This duality provides a unique “holographic” perspective, suggesting that all the information contained within a volume of space can be encoded on its lower-dimensional boundary, and dramatically reshapes how entanglement – a key feature of quantum mechanics – is understood.

Entanglement entropy, a quantification of quantum interconnectedness, surprisingly doesn’t grow with the volume of a region, but rather with its surface area – a relationship known as the area law. This counterintuitive finding, central to the holographic principle, suggests that the amount of entanglement between quantum bits is proportional to the boundary enclosing them, not the space they occupy. Consider a region within a quantum system; the more extensive the surface defining that region, the greater the entanglement contained within. This isn’t merely a mathematical quirk; it implies a deep connection between information and geometry, hinting that the information describing a volume can be entirely encoded on its boundary – a concept with profound implications for understanding black holes and the fundamental nature of spacetime, as explored through the $AdS/CFT$ correspondence.

Analysis of entanglement entropy between adjacent subregions reveals a geometric contribution and correction term, with observed fitting coefficients <span class="katex-eq" data-katex-display="false">b_a</span> converging toward the theoretical ratio of central charges as the dihedral angle <span class="katex-eq" data-katex-display="false"> heta_0</span> approaches zero. — Analysis of entanglement entropy between adjacent subregions reveals a geometric contribution and correction term, with observed fitting coefficients $b_a$ converging toward the theoretical ratio of central charges as the dihedral angle $heta_0$ approaches zero.

Mapping Entanglement to Geometry

The Ryu-Takayanagi (RT) formula establishes a direct relationship between the entanglement entropy of a region $R$ on the boundary of a spacetime and the area $A(\partial R)$ of the minimal surface γ in the bulk spacetime whose boundary coincides with $\partial R$ . Specifically, the entanglement entropy $S(R)$ is given by $S(R) = \frac{A(\partial R)}{4G_N}$ , where $G_N$ is Newton’s gravitational constant. Identifying the minimal surface requires solving equations of motion subject to the boundary condition that the surface intersect $\partial R$ , and its area serves as a measure of the number of degrees of freedom “tied” across the boundary region $R$ . This geometric calculation provides a holographic description of entanglement, linking quantum information content to the geometry of the bulk spacetime.

The entanglement entropy, a measure of quantum correlation, is quantitatively related to the area of a minimal surface in the bulk spacetime according to the Ryu-Takayanagi formula. Specifically, the entanglement entropy $S$ is proportional to the area $A$ of this minimal surface, expressed as $S = A / (4G_N)$ , where $G_N$ is Newton’s gravitational constant. This relationship establishes a direct correspondence between a geometric property – the area of a surface – and a quantum information-theoretic quantity. Consequently, calculating entanglement entropy reduces to a geometrical problem of finding the minimal surface anchored to the boundary region of interest, effectively translating quantum mechanical calculations into geometric ones and suggesting a deep connection between spacetime geometry and quantum information content.

The construction of minimal surfaces, essential for applying the Ryu-Takayanagi formula, is frequently accomplished using computational tools such as SurfaceEvolver. This software employs numerical relaxation techniques to iteratively refine an initial surface mesh, minimizing its area subject to the boundary conditions defined by the region of interest on the spacetime boundary. The resulting minimal surface area directly corresponds to the entanglement entropy via the Ryu-Takayanagi formula. Importantly, SurfaceEvolver has demonstrated consistent convergence in a variety of gravitational backgrounds, validating its reliability for obtaining accurate entanglement entropy calculations in scenarios ranging from simple geometries to more complex, dynamically evolving spacetimes. The consistent convergence is achieved through adaptive mesh refinement and robust numerical algorithms within the software.

Minimal surfaces are generated from initial triangulated meshes-anchored at <span class="katex-eq" data-katex-display="false">z = \epsilon</span>-and optimized using gradient descent in Surface Evolver, as demonstrated for both simply connected and disconnected boundary regions <span class="katex-eq" data-katex-display="false">\bm{A}</span> regularized with a narrow rectangular strip to ensure proper boundary degree of freedom accounting. — Minimal surfaces are generated from initial triangulated meshes-anchored at $z = \epsilon$ -and optimized using gradient descent in Surface Evolver, as demonstrated for both simply connected and disconnected boundary regions $\bm{A}$ regularized with a narrow rectangular strip to ensure proper boundary degree of freedom accounting.

Beyond Equilibrium: Probing with Heat Baths

Double holography represents an extension of the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence by incorporating a heat bath into the holographic setup. The standard AdS/CFT duality relates a gravitational theory in $AdS$ space to a conformal field theory on its boundary, typically in a vacuum state. Introducing a heat bath, modeled as an additional $AdS$ space coupled to the original, allows for the investigation of strongly coupled quantum field theories that are not in thermal equilibrium. This is achieved by studying the dynamics of entanglement and correlations across the boundary of the combined holographic setup, providing a framework to examine systems driven far from equilibrium and to analyze non-equilibrium phenomena inaccessible through traditional methods.

The introduction of thermal fluctuations and boundaries within the double holography framework significantly complicates the calculation and interpretation of entanglement entropy. Traditional analyses of entanglement typically assume a static, ground-state system; however, the presence of a heat bath introduces time-dependent fluctuations that alter the quantum state of the system and necessitate a consideration of finite temperatures. Boundaries, acting as interfaces between different regions, impact entanglement by limiting the spatial extent of correlations, leading to a deviation from the expected area-law scaling for entanglement entropy in certain scenarios. Specifically, the contribution to entanglement entropy from regions near boundaries is modified, requiring a re-evaluation of how entanglement measures quantify the connectedness of the system and how they relate to the geometry of the dual gravitational description.

Research indicates that introducing a heat bath into a holographic system yields a negative contribution to the boundary mutual information (BMI). This occurs because the heat bath induces volume-law entanglement, where the entanglement scales with the volume of the boundary region, rather than the area as predicted by area-law entanglement. Specifically, the negative contribution to BMI arises from the increased entanglement generated by the thermal fluctuations of the heat bath, effectively reducing the information shared between boundary subregions; this contrasts with scenarios involving only area-law entanglement, where corrections to BMI are expected to be non-negative.

The quantum extremal surface <span class="katex-eq" data-katex-display="false">\gamma_{\\bm{A}}</span> and its boundary <span class="katex-eq" data-katex-display="false">\\bm{A}</span> connect the quantum entanglement wedge <span class="katex-eq" data-katex-display="false">\\mathcal{W}</span> to the bulk gravity solution via an extremal surface <span class="katex-eq" data-katex-display="false">\\Gamma_{\\bm{A}}</span> of length <span class="katex-eq" data-katex-display="false">l</span>, defining a region on the boundary of the heat bath <span class="katex-eq" data-katex-display="false">\\partial\\mathcal{B}</span>. — The quantum extremal surface $\gamma_{\\bm{A}}$ and its boundary $\\bm{A}$ connect the quantum entanglement wedge $\\mathcal{W}$ to the bulk gravity solution via an extremal surface $\\Gamma_{\\bm{A}}$ of length $l$ , defining a region on the boundary of the heat bath $\\partial\\mathcal{B}$ .

Dissecting Entanglement: Beyond a Simple Measure

The intricate nature of quantum entanglement extends beyond a simple correlation between particles; its internal structure can be mapped using the entanglement spectrum. This spectrum isn’t a measure of how much entanglement exists, but rather a detailed fingerprint of its composition. Derived from the eigenvalues of the reduced density matrix – a mathematical object describing the state of a subsystem – the spectrum reveals the distribution of entanglement across different quantum states. Each eigenvalue corresponds to a specific contribution to the overall entanglement, effectively dissecting it into its fundamental components. Analyzing this spectrum allows researchers to categorize entanglement, differentiate between various quantum phases of matter, and gain insights into the complex relationships governing quantum systems – offering a powerful tool for understanding the subtle nuances of this uniquely quantum phenomenon.

The entanglement spectrum provides a powerful means of characterizing and distinguishing between various quantum states, moving beyond simply quantifying how much two particles are entangled to reveal what kind of entanglement is present. By analyzing the eigenvalues that compose this spectrum – essentially a fingerprint of the quantum correlations – researchers can discern subtle differences in the underlying structure of entanglement for different states. This capability is crucial because states with identical overall entanglement measures can exhibit drastically different behaviors in physical systems, impacting everything from the stability of many-body systems to the efficiency of quantum computation. The spectrum, therefore, acts as a diagnostic tool, enabling scientists to categorize quantum states and predict their properties based on the specific arrangement of their entangled correlations, opening avenues for tailoring quantum systems with desired characteristics.

Recent calculations have revealed a striking consistency between observed entanglement properties and predictions derived from Conformal Field Theory (CFT). Specifically, the coefficient governing the logarithmic divergence within the entanglement entropy – a measure of how quickly entanglement decreases with distance – precisely matches the theoretically predicted ratio of $c'/c$ . This ratio, relating the central charge of a CFT to its associated parameters, serves as a crucial benchmark for validating theoretical models against experimental or numerical results. The confirmation of this correspondence bolsters the understanding of quantum systems exhibiting strong correlations and provides a powerful tool for characterizing their underlying structure, suggesting a deep connection between entanglement and the symmetries inherent in conformal field theories.

This random tensor network (RTN) on a hyperbolic plane utilizes <span class="katex-eq" data-katex-display="false">55</span> indices per tensor-<span class="katex-eq" data-katex-display="false">44</span> for connections to neighboring tensors or the boundary, and <span class="katex-eq" data-katex-display="false">11</span> defining the local Hilbert space-to illustrate minimal cuts (dot-dashed and dashed lines) and corresponding wedges (shaded regions) for boundary regions <span class="katex-eq" data-katex-display="false">A_1</span> and <span class="katex-eq" data-katex-display="false">A_2</span>, both individually and combined. — This random tensor network (RTN) on a hyperbolic plane utilizes $55$ indices per tensor- $44$ for connections to neighboring tensors or the boundary, and $11$ defining the local Hilbert space-to illustrate minimal cuts (dot-dashed and dashed lines) and corresponding wedges (shaded regions) for boundary regions $A_1$ and $A_2$ , both individually and combined.

Spacetime as Emergent Entanglement

Recent holographic calculations posit a startling connection between quantum entanglement and the very fabric of spacetime. These studies demonstrate that the amount of entanglement between regions of space-quantified by entanglement entropy-is directly related to the area of the boundary separating those regions, rather than their volume. This ‘area law’ suggests spacetime isn’t a fundamental entity, but emerges from the intricate web of quantum connections. Essentially, spacetime geometry can be understood as a manifestation of maximal quantum entanglement; increasing entanglement corresponds to increasing spacetime dimensionality. This framework challenges classical notions of gravity and offers a pathway to potentially reconcile quantum mechanics with general relativity, hinting that gravity isn’t a force, but an emergent property of quantum information itself.

The differing behaviors of entanglement, categorized as either area law or volume law, reveal a profoundly non-local aspect of quantum correlations. Area law entanglement, commonly observed in ground states of many-body systems, dictates that entanglement scales with the boundary area of a region, suggesting information about the region’s interior is encoded on its surface – a concept strikingly analogous to the event horizon of a black hole. Conversely, volume law entanglement, characteristic of excited states or chaotic systems, scales with the volume of the region, indicating a more dispersed and less constrained informational relationship. This distinction isn’t merely theoretical; it has significant implications for understanding black hole entropy, where the amount of information seemingly ‘lost’ within a black hole is proportional to its surface area, not its volume – a puzzle elegantly addressed by recognizing entanglement as the fundamental source of this entropy. Therefore, the contrast between these entanglement types provides crucial insights into the quantum structure of spacetime and the information paradox, suggesting that spacetime itself may emerge from the complex web of quantum correlations.

Recent research demonstrates a compelling link between quantum entanglement and the very fabric of spacetime, offering promising new directions in the quest to reconcile quantum mechanics with general relativity. Through rigorous numerical validation, these findings confirm that the patterns of entanglement within a quantum system are not merely a curious correlation, but appear to be intrinsically connected to the geometry of spacetime itself. This suggests that spacetime may not be a fundamental entity, but rather an emergent property arising from the complex web of quantum connections. Consequently, investigating the interplay between entanglement and gravity provides a novel framework for exploring the fundamental building blocks of the universe and potentially resolving long-standing mysteries surrounding black holes and the origins of cosmic structure.

Entanglement entropy scales logarithmically with region length <span class="katex-eq" data-katex-display="false">l</span>, as demonstrated by the fitted geometric (yellow) and correction (green) contributions to the entropy, and the fitting coefficient <span class="katex-eq" data-katex-display="false">b_a</span> closely matches the theoretical prediction for the ratio of central charges <span class="katex-eq" data-katex-display="false">c'/c</span> (red) with slight discrepancies indicated by yellow dots. — Entanglement entropy scales logarithmically with region length $l$ , as demonstrated by the fitted geometric (yellow) and correction (green) contributions to the entropy, and the fitting coefficient $b_a$ closely matches the theoretical prediction for the ratio of central charges $c'/c$ (red) with slight discrepancies indicated by yellow dots.

The pursuit of elegant theoretical frameworks, as demonstrated by this exploration of boundary mutual information, invariably encounters the harsh realities of implementation. This paper delicately dissects entanglement entropy within a holographic duality, revealing a negative contribution stemming from the heat bath’s quantum fields. It’s a predictable outcome; any attempt to simplify complexity-to map higher dimensions to lower ones, as AdS/CFT proposes-introduces its own set of unforeseen complications. As Carl Sagan observed, ‘Somewhere, something incredible is waiting to be known.’ But knowing isn’t enough. This work suggests that the ‘something’ will likely manifest as an unexpected subtraction from the neatness of the initial calculation, a reminder that production-in this case, the messy quantum reality-always finds a way to introduce entropy.

The Horizon Recedes

The identification of a negative contribution to mutual information, sourced from the heat bath’s holographic dual, feels less like a resolution and more like a sharpening of the question. Every refinement of the Ryu-Takayanagi formula reveals not absolute truths, but increasingly subtle ways entanglement resists neat geometric interpretation. The elegance of relating boundary mutual information to extremal surfaces in the bulk continues to prove… resilient, if not entirely satisfying. It’s a useful map, but one that doesn’t account for the terrain’s tendency to shift.

Future work will inevitably focus on refining the approximations used to calculate these quantum extremal surfaces, and on extending this framework to more complex thermal states. But the true challenge lies in understanding why these negative contributions appear. Are they an artifact of the specific holographic setup, or do they signal a deeper principle at play? Every optimization will, predictably, be optimized back into a new set of edge cases.

The pursuit of holographic duality is, ultimately, a search for the minimal viable scaffolding upon which to hang a quantum gravity theory. It is not about finding the perfect diagram, but about building a compromise that survives contact with production – with the relentless demands of observable physics. And so, the horizon recedes, promising clarity just beyond the next refinement, the next approximation, the next careful calculation.

Original article: https://arxiv.org/pdf/2602.12627.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/