RT Entropy Cone in Holographic Theories

Updated 1 September 2025

The RT entropy cone is a geometric and combinatorial structure that defines all possible entropy vectors in holographic quantum systems using the Ryu–Takayanagi formula.
It encodes both classical entropic principles like strong subadditivity and uniquely holographic constraints such as the monogamy of mutual information, verified through null reductions and graph models.
Extensions to dynamic settings with the HRT prescription and to generalized spacetime frameworks demonstrate the cone's robustness and universal role in holographic entanglement.

The RT entropy cone (Ryu–Takayanagi entropy cone) is a geometric and combinatorial structure central to the paper of entanglement constraints in holographic quantum systems, particularly those described by the AdS/CFT correspondence. It delineates the set of all possible entropy vectors for boundary regions whose entropies can be calculated using the Ryu–Takayanagi formula, i.e., as areas of minimal (or extremal) surfaces in an appropriate bulk gravitational spacetime. The RT entropy cone is defined by an infinite hierarchy of linear entropy inequalities, encoding both classical (Shannon-type) and uniquely holographic constraints, some of which are not satisfied by all quantum states but are universal in holographic theories.

1. Definition and Mathematical Structure of the RT Entropy Cone

Let $\mathcal{C}_N$ be the holographic or RT entropy cone for $N$ boundary regions. Each point in this cone is an entropy vector $S = \{S_I\}_{I \subseteq \{1,\dots,N\}, I \neq \emptyset}$ , where $S_I$ denotes the entanglement entropy of region $I$ . For holographic states, these vectors are computed with the Ryu–Takayanagi prescription: $S(I) = \frac{\operatorname{Area}(\gamma_I)}{4G_N}$ with $\gamma_I$ the codimension-2 minimal surface homologous to $I$ .

The RT cone is polyhedral in static spacetimes and is uniquely characterized by a finite set of homogeneous linear inequalities for each $N$ : $\sum_{J} Q_J S_J \geq 0 \qquad Q_J \in \mathbb{Z}$ Important examples include strong subadditivity (SSA) and the monogamy of mutual information (MMI): $S(AB) + S(AC) + S(BC) \geq S(A) + S(B) + S(C) + S(ABC)$ For $N=5$ , $\mathcal{C}_5$ is generated by 372 inequalities reducible to 8 symmetry orbits, each corresponding to a primitive information quantity defining a facet of the cone (Hernández-Cuenca, 2019).

Recently, a new characterization based on null reductions and a majorization test offers an efficient and robust diagnostic. Any candidate RT inequality, when “null reduced”—i.e., projected onto degrees of freedom involving a central party—must satisfy a majorization condition under perturbations of the bulk metric. If this is the case for all reductions, it is a valid RT inequality (Grimaldi et al., 29 Aug 2025). This test is both necessary and sufficient.

The structure of the RT entropy cone admits dual descriptions: the facet (inequality) description and the extreme ray (generator) description. The Farkas–Minkowski–Weyl theorem guarantees that any polyhedral cone allows both. For $\mathcal{C}_5$ , all 2267 extreme rays (in 19 orbits) are realized by explicit graph models (Hernández-Cuenca, 2019). The graph model representation is grounded in the min-cut/max-flow principle: a weighted undirected graph encodes bulk connectivity, and the entropy $S_I$ for a region $I$ is the minimal cut separating $I$ from its complement.

$S_I = \min_{\rm{cuts}} \sum_{\rm{edges} \in \rm{cut}} w(\rm{edge})$

This construction demonstrates both the combinatorial richness and the algorithmic tractability of the RT entropy cone in static settings.

3. Extension Beyond AdS/CFT and Generalized Settings

The RT entropy cone is not limited to AdS/CFT boundary regions. All known RT area inequalities extend to generalized static entanglement wedges in arbitrary spacetimes, provided a mutual independence condition is satisfied: each input bulk region $a_i$ must be outside the entanglement wedge of all other inputs, $a_i \subset [E(\cup_{j \neq i} a_j)]'$ . This guarantees the mutual independence of the semiclassical degrees of freedom associated with each region, essential for the validity of the cone inequalities (Bousso et al., 5 Feb 2025).

$\sum_l \alpha_l A[E(v_l)] \geq \sum_r \beta_r A[E(w_r)]$

where $E(\cdot)$ denotes the generalized entanglement wedge and $(v_l, w_r)$ are specified unions of input regions.

This extension points to a universal feature of holography: the emergent geometry and allowed entanglement structures obey the same constraints, regardless of the presence of asymptotic boundaries or the details of the gravitational dynamics.

4. Covariant, Time-Dependent, and Minimax Formulations

In dynamical (covariant) settings, entanglement entropy is defined via the HRT (Hubeny–Rangamani–Takayanagi) prescription: $S(I) = \frac{\operatorname{Area}(\gamma_I^\text{ext})}{4G_N}$ where $\gamma_I^\text{ext}$ is an extremal (not minimal) codimension-2 surface.

Recent advances demonstrate that the set of inequalities defining the static RT cone persists for the HRT cone under dynamic evolution. This is supported by the construction of covariant graph models via minimax/minimin prescriptions for time-sheets in the bulk (Grado-White et al., 14 Feb 2025). The central conjecture is that if the relevant “partial” minimal surfaces are maximal on their associated time-sheets (“cooperating” property), then the min-cut on the associated spacetime graph computes the HRT entropy, ensuring the equivalence of the RT and HRT cones.

In large region and late time limits, extensive contributions to entropy in dynamical, thermalizing spacetimes can be computed by a minimal membrane theory. This reduction to a codimension-1 minimization problem recovers the same geometric inclusion–exclusion arguments as in the static RT case, reaffirming that the entropy cone inequalities are robust against time evolution in the extensive scaling regime (Bao et al., 2018).

5. Universal Coefficients and Subleading Structure: Conical and Corner Contributions

Subleading universal contributions to entanglement entropy—especially those arising from conical (or “corner”) singularities in the entangling surface—are encoded via universal functions $a^{(d)}(\Omega)$ and coefficients $\sigma^{(d)}$ : $a^{(d)}(\Omega \to \pi/2) = 4 \sigma^{(d)} \left(\frac{\pi}{2} - \Omega\right)^2$ In holographic CFTs, $\sigma^{(d)}$ is universally related to the stress tensor central charge $C_T$ : $\sigma^{(d)} = \frac{\pi^{d-1} (d-1)(d-2)\Gamma(\frac{d-1}{2})^2}{8\Gamma(\frac{d}{2})^2 \Gamma(d+2)} \times \begin{cases} \pi, & \text{odd } d\ 1, & \text{even } d \end{cases}$ This analytic control over universal coefficients provides robust constraints within the RT entropy cone, serving as quantitative probes of non-area-law physics and constraining the structure of allowed entanglement for holographic theories (Bueno et al., 2015).

6. RT Cone versus Quantum and Stabilizer Entropy Cones

The RT entropy cone forms a proper subset of the full quantum entropy cone. Holographic (RT) states satisfy all classical inequalities and further obey genuinely holographic constraints—such as MMI—not satisfied by generic quantum states. Some families of non-holographic quantum states (e.g., $W_N^d$ or Dicke states) violate MMI and so occupy regions outside the RT cone (Schnitzer, 2022, Munizzi et al., 2023). Conversely, in special stabilizer and topological theories (e.g., SU(N) $_1$ Chern–Simons with odd prime $N$ ), the topological entropy cone coincides with the stabilizer cone but is strictly smaller than the RT cone (Schnitzer, 2020).

Explicit construction of the cone for stabilizer states reveals that, for four parties, the entropy cone is fully determined by strong subadditivity, weak monotonicity, and the Ingleton inequality—a linear rank constraint originally arising from matroid theory (Linden et al., 2013).

7. Majorization Characterization and Future Directions

A critical recent development is the majorization test: any RT inequality must, under null reduction, satisfy a majorization condition with respect to the set of “offsets” associated to HRT surfaces in light-cone configurations (Grimaldi et al., 29 Aug 2025). Formally, the majorization of two vectors $(y_1, ..., y_N) \succ (x_1, ..., x_N)$ indicates that, for any concave function $h$ , unbalanced contributions to entropy shifts are forbidden under perturbations preserving the null energy condition. This insight leads to an efficient, computationally tractable method for verifying holographic entropy inequalities and suggests a deep link between the Markov property, Markov states, and the underlying bulk/boundary geometry.

The extension of RT cone inequalities to generalized spacetimes with appropriate independence conditions, the equivalence of the RT and HRT cones via minimax constructions, and the ongoing search for a full set of primitive holographic inequalities continue to drive the field. Open directions include leveraging the majorization test to algorithmically uncover new inequalities at high $N$ , further exploring the geometry of the cone near the apex with non-homogeneous bounds (Christandl et al., 2023), and integrating combinatorial and geometric techniques for classifying entropy cones in holography and quantum information at large.

Inequality Name	Canonical Expression	Type
Subadditivity	$S_A + S_B \geq S_{AB}$	Uplifting of classical
Monogamy of Mutual Info	$S_{AB} + S_{AC} + S_{BC} \geq S_A + S_B + S_C + S_{ABC}$	Holographic (MMI)
QCycl	$S_{ABC} + S_{ABD} + S_{ACE} + S_{BDE} + S_{CDE} \geq S_{AB} + S_{AC} + S_{BD} + S_{CE} + ...$	Primitive five-party

The RT entropy cone thus encodes a set of combinatorial, geometric, and physical constraints that sharply delineate the possible patterns of entanglement in holographic field theories, and its mathematical structure continues to illuminate the interplay of geometry and information in quantum gravity.