Entanglement Wedge Cross Sections

• Entanglement Wedge Cross Sections (EWCS) are defined as the minimal codimension-2 surfaces bisecting the entanglement wedge, capturing both quantum and classical mixed-state correlations.
• They are dual to key information measures such as entanglement of purification, reflected entropy, and logarithmic negativity, linking geometric constructs to information theory.
• EWCS is highly sensitive to bulk excitations and phase transitions, providing insights into multipartite entanglement and the interplay between quantum and classical correlations in holography.

Entanglement wedge cross sections (EWCS) are geometric quantities in holographic duality that probe the structure of mixed-state correlations in quantum field theory. As a minimal-area codimension-2 surface bisecting the entanglement wedge of two boundary regions, the EWCS computes a variety of important mixed-state information-theoretic measures—including entanglement of purification, reflected entropy, and, under specific prescriptions, logarithmic negativity—at leading order in the bulk semiclassical limit. This central object in the AdS/CFT correspondence is sensitive to both quantum and classical correlations, reacts distinctively to bulk excitations, and reflects a multipartite entanglement structure fundamentally richer than mutual information can capture.

1. Geometric Definition and Holographic Prescription

Given two boundary subregions $A$ and $B$ in a holographic CFT, the entanglement wedge $W_{AB}$ is the bulk domain of dependence bounded by $A \cup B$ and the minimal Ryu–Takayanagi (RT) surface $\gamma_{AB}$. The entanglement wedge cross section, denoted $E_W(A,B)$ or EWCS($A:B$), is the area (divided by $4G_N$) of the minimal codimension-2 bulk surface $\Sigma_{A:B} \subset W_{AB}$ that splits the wedge into regions homologous to $A$ and $B$ respectively:

$$E_W(A,B) = \min_{\Sigma_{A:B} \subset W_{AB}} \frac{\mathrm{Area}(\Sigma_{A:B})}{4G_N}\,.$$

For two parallel strips on the boundary separated by distance $h$ and width $\ell$ in pure AdS$_{d+1}$, the minimal cross-section is realized by a surface at constant $x=0$ connecting the turning points $r_d$ (for $h$) and $r_u$ (for $2\ell+h$), with explicit functionals provided in terms of the metric, e.g.,

$$E_W = \frac{R^{d-1}L^{d-2}}{4G_N}\int_{r_d}^{r_u} \frac{dr}{r^{d-1}\sqrt{f(r)}}$$

for Einstein gravity bulk with $f(r)=1$ in vacuum AdS (Sahraei et al., 2021, Velni et al., 2019).

If the mutual information $I(A:B)$ vanishes or the wedge disconnects, $E_W=0$; $E_W$ is continuous except for discontinuous transitions at wedge disconnections.

2. Dual Boundary Information Measures

EWCS is widely conjectured to be dual to several important mixed-state correlation measures:

• Entanglement of purification $E_P(A,B)$: $E_W(A,B)=E_P(A,B)$ for all connected-wedge configurations (Sahraei et al., 2021, Velni et al., 2019).
• Reflected entropy $S_R(A,B)$: $S_R(A,B)=2E_W(A,B)$, proven at leading order in AdS/CFT via both canonical purification in doubled Hilbert space and path-integral replica approaches (Akers et al., 2019, Tamaoka, 2018, Kusuki et al., 2019).
• Odd entanglement entropy (OEE): $\widehat S_o(A:B) = S(A \cup B) + E_W(A,B)$; OEE reduces to $E_W$ in CFTs with a geometric replica construction (Tamaoka, 2018).
• Logarithmic negativity $\mathcal{E}(A,B)$: Certain approaches relate $\mathcal{E}(A,B)$ to a rescaled $E_W(A,B)$, e.g., $\mathcal{E}(A,B) = \chi_d E_W(A,B)$ with $\chi_d$ a state-independent constant (Sahraei et al., 2021, Basak et al., 2020).

These identifications hold for a wide class of connected-wedge, large-$N$ holographic states and have exact realizations in AdS$_3$/CFT$_2$, where EWCS is the length of a minimal geodesic anchored on $\gamma_{AB}$.

3. Response to Bulk Excitations and Operator Condensates

First-order corrections to EWCS under small perturbations of the AdS background display sensitivity to the type of bulk field excitation:

• Gravitational/Energy Excitations: For isotropic stress-energy perturbations or small thermal states, one finds

$$\delta E_W(T) = \mathcal{C} L^{d-2} \ell(\ell+h) T_{00}$$

with $\mathcal{C}<0$; thus, increasing the bulk energy density decreases EWCS, reflecting correlation loss under thermalization (Sahraei et al., 2021).

• Gauge Fields (Current Density): In the presence of a bulk gauge field with nonzero charge density, the correction is positive, i.e., increases EWCS, indicating that charge density strengthens correlations between $A$ and $B$.
• Scalar Condensation: For scalar operator condensates of dimension $\Delta$, the sign of the correction flips depending on $\Delta$, with sufficiently relevant operators $(\Delta < d/2 + \pi/4)$ yielding a negative contribution (correlation decrease) (Sahraei et al., 2021).
• Matter-Coupling Parameters: In axion models and aether gravity, the behavior is further enriched; EWCS can be non-monotonic in momentum relaxation or Lorentz-violation parameters, exhibiting U-shaped or turning-point behaviors absent in HEE or MI, exposing the sensitivity of EWCS to deep bulk geometry (Cheng et al., 2021, Chen et al., 2021).

In all scenarios, EWCS is less affected by bulk thermal entropy than entanglement entropy and isolates nontrivial, genuinely quantum and subleading classical correlation channels (Yang et al., 2023).

4. Inequalities, Multipartite Extensions, and Phase Structure

EWCS satisfies a suite of geometric and information-theoretic inequalities:

• Bounds: $\frac12 I(A,B) \leq E_W(A,B) \leq \min\{S_A, S_B\}$ (Velni et al., 2019, Jain et al., 2022).
• Monotonicity: $E_W(A:BC) \geq E_W(A:B)$; enlarging one region cannot decrease EWCS.
• Tripartite Inequalities: While naive “polygamy” or “monogamy” inequalities for tripartite splits such as $E_W(A:B)+E_W(A:C) \geq E_W(A:BC)$ do not always hold (especially in dimensions $d>2$ or nontrivial geometries), weaker and squared monogamy-type bounds (e.g., $(E_W(A:BC))^2 \geq (E_W(A:B))^2 + (E_W(A:C))^2$) do hold universally (Jain et al., 2022).

Multipartite generalizations include the triangle information EI$_\Delta(A:B|E)$, an EWCS analogue of conditional mutual information:

$$EI_\Delta(A:B|E) = EWCS(A:EB) + EWCS(B:EA) - EWCS(E:AB)$$

which is non-negative and tightly bounded above by the holographic entanglement of assistance. Phase diagrams for EI$_\Delta$ in AdS$_3$/CFT$_2$ display nontrivial transitions as cross-ratios are varied, with regions of strict positivity and bound saturation (Ju et al., 25 Dec 2025).

Inequality	Statement	Notes
Lower bound	$E_W(A:B) \geq \frac12 I(A,B)$	Valid in all holographic phases
Upper bound	$E_W(A:B) \leq \min\{S_A, S_B\}$	Geometric minimality property
Monotonicity	$E_W(A:BC) \geq E_W(A:B)$	Enlargement cannot decrease EWCS
Squared monogamy	$(E_W(A:BC))^2 \geq (E_W(A:B))^2 + (E_W(A:C))^2$	Holds for all examined bulk backgrounds
Weak monogamy	$$E_W(A:BC) + \tfrac12 I(A:BC) \geq E_W(A:B) + E_W(A:C)$$	Universal with MI supplement

These findings indicate that EWCS is sensitive to multipartite, genuinely higher-order entanglement beyond bipartite measures (Akers et al., 2019, Jain et al., 2022).

5. Critical Phenomena, Thermalization, and Dynamics

EWCS serves as a sensitive probe of phase transitions and dynamical processes:

• Thermal and Quantum Phase Transitions: Near second-order critical points (e.g., in holographic p-wave superconductors or massive gravity), EWCS, HEE, and MI all exhibit the same critical exponents, typically $E_W - E_{W,c} \sim (T - T_c)^\alpha$ with $\alpha$ determined by the universality class. For p-wave superconductors, the EWCS critical exponent equals twice that of the condensate (e.g., $\alpha_{E_W} = 1$ when $\langle J_x \rangle \sim (1-T/T_c)^{1/2}$) (Yang et al., 2023).
• Quench Dynamics: After global quenches modeled by AdS-Vaidya spacetimes, EWCS evolution displays three regimes: early-time quadratic onset, intermediate linear growth (with velocity matching entanglement tsunami speed), and late saturation, with possible logarithmic growth for extremal quenches (Velni et al., 2020, Velni et al., 2023, Boruch, 2020). The thermalization speed and plateau height depend on spacetime dimensionality, dynamical exponents, and coupling to higher-curvature or matter fields.

Dynamical studies confirm EWCS's utility for tracking the build-up (or decay) of mixed-state correlations—correlations often overlooked by simpler measures such as mutual information.

6. Physical Significance: Quantum vs Classical Correlations and Beyond-Bipartite Entanglement

EWCS captures both quantum and classical correlation content in holographic states. It can exceed quantum entanglement measures (e.g., entanglement of formation) in mixed states, indicating a non-vanishing classical component, and strictly reflects $\mathcal{O}(1/G_N)$ tripartite entanglement in holographic CFTs (Umemoto, 2019, Akers et al., 2019). In contrast to bipartite-optimized ansatzes, EWCS (and its multipartite generalizations) expose the necessity of genuinely multipartite entanglement and invalidate the “mostly-bipartite” assumption for holographic large-$N$ states.

Moreover, EWCS remains robust against contamination from UV or IR artifacts in the bulk geometry, enabling refined detection of both quantum order and classical correlation reorganization. It is sensitive to matter-induced deformations, geometric singularities (e.g., wedge or cone boundaries), higher-curvature corrections (e.g., Gauss-Bonnet, topological mass), and Lorentz-violating effects (Li et al., 2021, Wen et al., 2022, Chen et al., 2021).

7. Outlook and Open Directions

The study of entanglement wedge cross sections continues to illuminate the interplay of geometry, quantum information, and quantum gravity:

• Beyond Einstein Gravity: EWCS prescriptions have been extended to higher-derivative (Gauss-Bonnet, Chern-Simons) and Lorentz-violating (aether) theories, revealing distinctive responses to finite coupling, anomaly, and symmetry-breaking corrections (Li et al., 2021, Wen et al., 2022, Chen et al., 2021).
• Multipartite Extensions: New structures, inequalities, and information quantities generalizing EWCS, such as triangle information and entanglement-of-assistance bounds, enrich analytic control and physical interpretation (Ju et al., 25 Dec 2025).
• Dynamical and Boundary Effects: The area law and information-theoretic inequalities remain robust in covariant settings (HRT surfaces) and in boundary CFT dualities (AdS/BCFT), with additional rich phase structures induced by brane or boundary end-points (2206.13417).
• Open Problems: Key open questions include a direct operational interpretation of EWCS in boundary quantum information, monotonicity properties under quenches and RG flows, precise mapping to multipartite or operationally-defined entropy tasks, and systematic characterization of continuity and phase transitions in higher rank and higher-dimensional setups.

In summary, EWCS provides a unified and remarkably versatile holographic tool for diagnosing and quantifying mixed-state correlations—including those of multipartite, classical, and quantum origin—binding together geometric extremality in AdS with advanced concepts in quantum information theory and condensed matter physics (Sahraei et al., 2021, Ju et al., 25 Dec 2025, Yang et al., 2023, Jain et al., 2022, Akers et al., 2019).