Entanglement Wedge Cross Sections in Holography

• Entanglement Wedge Cross Sections (EWCS) is a geometric measure in holography that quantifies mixed-state correlations by minimizing the cross-sectional area within the entanglement wedge.
• EWCS is computed by partitioning the bulk domain into regions homologous to boundary subregions and is conjectured to be dual to measures like entanglement of purification and reflected entropy.
• EWCS obeys key inequalities and phase transitions, offering insights into the interplay of classical and quantum correlations across various holographic and dynamical settings.

The entanglement wedge cross section (EWCS) is a geometric quantity in holography encoding mixed-state correlations between boundary subregions. For boundary regions $A$ and $B$, the EWCS is defined as the minimal area of a codimension-2 surface inside the entanglement wedge—a bulk domain bounded by $A\cup B$ and their associated Ryu-Takayanagi (RT) surface—which splits the wedge into two parts, one homologous to $A$, the other to $B$. EWCS has been conjectured as the gravity dual of several boundary mixed-state correlation measures, notably entanglement of purification, reflected entropy, logarithmic negativity, and odd entanglement entropy.

1. Holographic Definition and Computation

For regions $A$ and $B$ on a boundary CFT spatial slice, let $\Gamma_{A\cup B}$ be the minimal-area RT surface homologous to $A\cup B$. The entanglement wedge $\mathcal{W}_{A\cup B}$ is the bulk region bounded by $A\cup B\cup\Gamma_{A\cup B}$. The EWCS is then

$$E_W(A\!:\!B) = \min_{\Sigma \subset \mathcal{W}_{A\cup B}} \frac{\mathrm{Area}(\Sigma)}{4 G_N}$$

where $\Sigma$ is a surface splitting $\mathcal{W}_{A\cup B}$ into two contiguous regions homologous to $A$ and $B$ respectively. If the entanglement wedge is disconnected (for sufficiently separated regions), $E_W(A:B) = 0$ (Sahraei et al., 2021, Umemoto, 2019).

For parallel strips of width $\ell$, separated by distance $h$ in $d$ spatial dimensions, the vacuum AdS result is

$$E_W^{\text{AdS}}(\ell,h) = \frac{R^{d-1} L^{d-2}}{4(d-2) G_N} \left[ \frac{1}{h^{d-2}} - \frac{1}{(2\ell + h)^{d-2}} \right]$$

computed by solving for the RT surface turning points and minimizing the area functional over candidate cross sections (Sahraei et al., 2021, Velni et al., 2019).

2. Boundary Measures Dual to EWCS

EWCS serves as a geometric dual to a class of boundary measures for mixed-state correlations:

• Entanglement of Purification ($E_P$): Minimum entanglement entropy over all purifications of the mixed state. Conjectured holographic dual: $E_P(A:B) = E_W(A:B)$ (Kudler-Flam et al., 2018, Velni et al., 2019).
• Reflected Entropy ($S_R$): Entanglement entropy of the canonical "reflected" purification; $S_R(A:B) = 2 E_W(A:B)$ (Kusuki et al., 2019, Akers et al., 2019).
• Logarithmic Negativity ($\mathcal{E}$): Quantum entanglement measure, with holographic dual typically $\mathcal{E}(A:B) = \chi_d E_W(A:B)$, where $\chi_d$ depends on spacetime dimension (Kudler-Flam et al., 2018, Basu et al., 2021).
• Odd Entanglement Entropy ($S_o$): Replica-trick defined via partial transpose; leading large-$N$ term is $S_o(A:B) - S(A \cup B) = E_W(A:B)$ (Tamaoka, 2018).

Each such measure inherits inequalities and phase transitions (e.g., vanishing for disconnected wedges) directly from EWCS (Umemoto, 2019, Bao et al., 2021).

3. Inequalities and Monogamy Relations

EWCS satisfies several key inequalities:

• Purity bounds: $$\frac{1}{2} I(A:B) \le E_W(A:B) \le \min(S(A), S(B))$$, where $I(A:B)$ is mutual information (Jain et al., 2022, Umemoto, 2019).
• Monotonicity/inclusion: $E_W(A:BC) \ge E_W(A:B)$.
• Weak monogamy: $$E_W(A:BC) + \frac{1}{2} I(A:BC) \ge E_W(A:B) + E_W(A:C)$$ always holds (Jain et al., 2022).
• Squared monogamy: $(E_W(A:BC))^2 \ge (E_W(A:B))^2 + (E_W(A:C))^2$.
• Multipartite extensions: By replicated geometry, multipartite EWCSs satisfy subadditivity, strong subadditivity, and monogamy inequalities analogous to those for entanglement entropy (Bao et al., 2021).

In AdS/BCFT, these inequalities hold in all extremal surface "phases" (surfaces ending on branes), verified algebraically via candidate-set and wedge-nesting arguments (2206.13417).

4. Physical Interpretation and Comparison to Other Correlation Measures

EWCS extracts the "bulk thickness" of the entanglement wedge—that is, it diagnoses the minimal cross-link between subregions not accessible via boundary-only entanglement entropy or mutual information. Unlike entanglement entropy, which exhibits a volume law in thermal states, EWCS remains an area law, retaining sensitivity to quantum correlations (Velni et al., 2019).

Perturbative results show that under small increases in energy density (thermalization), EWCS strictly decreases, reflecting loss of correlation (Sahraei et al., 2021). However, introducing charge density can increase EWCS, signifying enhanced correlations mediated by matter fields. The change in EWCS under bulk scalar condensation depends on operator dimension, but generically relevant operators decrease EWCS (Sahraei et al., 2021).

In dynamical settings (e.g., global or local quenches), EWCS displays regimes of early quadratic growth, linear intermediate growth (with velocity equal to entanglement velocity), and late saturation or disentanglement (Velni et al., 2020, Velni et al., 2023, Boruch, 2020), paralleling known behaviors for entanglement entropy but with sharper phase transitions and distinct scaling exponents in nonrelativistic backgrounds (scaling as $t^{1+1/z}$ for Lifshitz exponent $z$) (Velni et al., 2023).

EWCS can remain nonzero beyond the mutual information phase transition—i.e., when mutual information vanishes, EWCS still detects nontrivial mixed correlations due to a still-connected entanglement wedge (Sahraei et al., 2021).

5. Classical versus Quantum Correlations and the Role of Multipartite Entanglement

Unlike pure quantum entanglement measures (entanglement of formation, squashed entanglement, conditional mutual information), EWCS and the entanglement wedge mutual information (EWMI) can strictly exceed these, capturing both quantum and classical correlations geometrically encoded in the bulk (Umemoto, 2019).

At phase boundaries (e.g., the Araki-Lieb transition for adjacent intervals), EWCS can equal the entropy of one subsystem even when entanglement of formation is strictly smaller, highlighting its sensitivity to classical correlations (Umemoto, 2019). EWMI is defined via an optimization among purifications and also admits strong superadditivity. The chain of inequalities is

$$\frac{1}{2} I \leq \text{EWMI} \leq \text{EWCS} \leq \min(S(A),S(B))$$

demonstrating EWCS as the largest among natural mixed-state measures in holographic settings.

Crucially, for holographic states to realize the nontrivial difference $2E_W - I = O(1/G_N)$ between reflected entropy (twice the EWCS) and mutual information, the boundary state must possess genuine multipartite entanglement at leading order in the large-$N$, large-$c$ expansion—a condition not met by mostly-bipartite toy tensor networks (Akers et al., 2019).

6. Geometric, Field-Theoretic, and Dynamical Variants

Analytical computation of EWCS is available for AdS/CFT vacuum, thermal black holes, and various deformations (confinement, RG flows, axion models, massive gravity, aether gravity, Gauss-Bonnet corrections) (Velni et al., 2019, Cheng et al., 2021, Liu et al., 2021, Chen et al., 2021, Li et al., 2021).

• Nonrelativistic backgrounds: EWCS is sensitive to dynamical ($z$) and hyperscaling-violating ($\theta$) exponents, with larger $z$ enhancing, larger $\theta$ suppressing spatial correlations (Velni et al., 2019, Velni et al., 2023).
• Corner/singular geometries: EWCS receives universal logarithmic contributions proportional to the CFT central charge, mapping singular boundary features to bulk geometry (Velni et al., 2019).
• Flat space holography: EWCS is defined via minimization of spacelike geodesic length in the bulk, and has a proportional relation to entanglement negativity in Galilean CFTs (Basu et al., 2021).

7. Open Problems and Future Directions

The precise matching between EWCS and boundary measures such as entanglement of purification, reflected entropy, and odd entanglement entropy is well-established at leading semiclassical order, but questions remain about quantum corrections, finite $N$, higher dimensions, and covariant time-dependent generalizations (Tamaoka, 2018, Kudler-Flam et al., 2018).

Further investigation into multipartite EWCSs, more refined inequalities, and the role of genuine multipartite entanglement is ongoing, with replicated geometry providing a systematic framework for generalization and constraint derivation (Bao et al., 2021). Understanding how classical and quantum correlations separate in different holographic backgrounds, as well as the impact of matter fields, phase transitions, and exotic geometries, remains a significant research direction.