Entanglement Wedge Cross Section

• Entanglement Wedge Cross Section (EWCS) is a geometric measure in holographic duality that partitions the entanglement wedge to quantify mixed state correlations.
• It is conjectured to be dual to boundary information measures like entanglement of purification, reflected entropy, and odd entropy, linking bulk geometry with quantum information.
• EWCS computations using replica techniques and conformal block analysis provide insights into thermal and dynamical behaviors as well as higher-dimensional and higher-curvature effects in holography.

The entanglement wedge cross section (EWCS) is a geometric quantity in holographic duality that quantifies correlations in mixed states, serving as a generalization of the Ryu–Takayanagi minimal surface prescription for entanglement entropy. While the Ryu–Takayanagi formula computes entanglement entropy for pure states by identifying a minimal bulk surface homologous to the boundary region, the EWCS corresponds to the minimal area (or length, in three bulk dimensions) cross section that partitions the entanglement wedge of a bipartite mixed state. EWCS has emerged as a central object in the holographic duality program due to its conjectured equivalence to various quantum information measures for mixed states, such as the entanglement of purification, reflected entropy, logarithmic negativity, and odd entropy. Its explicit computation and physical interpretation have deepened the connection between geometry and quantum information in AdS/CFT and related frameworks.

1. Definition and Fundamental Properties

The EWCS is defined within the bulk entanglement wedge ℰ for a bipartitioned boundary mixed state $\rho_{A_1A_2}$. Given ℰ as the bulk region bounded by the union $A_1\cup A_2$ along with the associated Ryu–Takayanagi (RT) minimal surfaces, the EWCS, denoted $E_W(\rho_{A_1A_2})$, is computed as the minimal area cross section $\Sigma_{A_1A_2}^{\min}$ that separates $A_1$ from $A_2$ inside ℰ:

$$E_W(\rho_{A_1A_2}) = \frac{\mathrm{Area}(\Sigma_{A_1A_2}^{\min})}{4G_N},$$

where $G_N$ is Newton's constant. In three bulk dimensions, "area" reduces to length.

The EWCS is subject to fundamental inequalities:

• $$\frac{1}{2} I(A_1:A_2) \leq E_W(\rho_{A_1A_2}) \leq \min\{ S_{A_1}, S_{A_2} \}$$, where $I(A_1:A_2)$ is the mutual information and $S_{A_i}$ is the entanglement entropy of subsystem $A_i$.

In the context of the AdS/CFT duality, the EWCS reduces to the boundary entanglement entropy in the case of pure states, i.e., $E_W = S_{A_1} = S_{A_2}$. For product states, it is additive and coincides with the von Neumann entropy.

2. Holographic Duals and Quantum Information Measures

EWCS is central to several conjectured bulk–boundary correspondences:

• Entanglement of Purification (E$_P$): $E_P(\rho_{A_1A_2})$ is defined as the minimal entanglement entropy across a bipartition over all purifications of $\rho_{A_1A_2}$, with $E_W$ conjectured to be its precise holographic dual.
• Reflected Entropy ($S_R$): For a canonical (GNS) purification of $\rho_{A_1A_2}$, the reflected entropy is $S_R = S_{A_1A_1'}$ where $A_1'$ is the mirror copy. The leading semiclassical relation is $S_R = 2 E_W$.
• Odd Entanglement Entropy (OEE): The OEE exploits a replica trick with partial transposition and odd replication index, leading to $E_W = S_o - S$, where $S_o$ is the OEE and $S$ is the von Neumann entropy.

These identifications have been supported by direct computations in two-dimensional holographic CFTs—AdS$_3$ and planar BTZ black holes—where EWCS can be exactly reproduced as boundary information-theoretic quantities. For example, for two intervals with cross ratio $x$ and central charge $c$,

$$E_W(\rho_{A_1A_2}) = \frac{c}{6} \log\left( \frac{1+\sqrt{x}}{1-\sqrt{x}} \right),$$

directly matching both holographic calculations and appropriate boundary quantities. In thermal backgrounds,

$$E_W = \frac{c}{3} \log\left( \frac{\beta}{\pi} \sinh\left( \frac{\pi \ell}{\beta} \right) \right)$$

in the appropriate regime for an interval of width $\ell$ and inverse temperature $\beta$.

3. Replica Techniques and Explicit Computation

EWCS calculations often employ the replica trick in boundary CFTs:

• For OEE, one computes the Tsallis entropy for the partially transposed density matrix $\rho^{T_{A_2}}$ using odd replicas $n_o$:

$$S_o^{(n_o)}(\rho_{A_1A_2}) = \frac{1}{1-n_o} \left[ \mathrm{Tr} \left( (\rho_{A_1A_2}^{T_{A_2}})^{n_o} \right) - 1 \right].$$

Analytic continuation to $n_o \to 1$ yields the OEE.

• For reflected entropy, purification is achieved by doubling the Hilbert space, and the (replicated) von Neumann entropy of the relevant block is computed using swap operators and analytic continuation.

The crucial step in holographic CFTs is the identification of the dominant conformal block in the light-cone channel, leading to the universal expressions above in the large $c$ limit. In thermal (BTZ) cases, a conformal map is used to transfer the calculation from the cylinder to the plane, followed by the standard four-point function computation.

For singular entangling regions (e.g., corners or creases in higher dimensions), the EWCS acquires universal logarithmic contributions whose coefficients are controlled by CFT data, notably the central charge. The area law holds for higher-dimensional singularities.

4. Thermal and Dynamical Behavior

EWCS exhibits distinct scaling and phase structure at finite temperature and out of equilibrium:

• Finite Temperature: In the low-temperature limit ($\ell T \ll 1$), thermal corrections decrease EWCS as quantum correlations are suppressed. In the high-temperature (large $\ell T$) regime, although entanglement entropy obeys a volume law, EWCS remains governed by an area law (Velni et al., 2019).
• Nonrelativistic Deformations: In models with Lifshitz exponent $z$ and hyperscaling violation $\theta$, EWCS increases monotonically with $z$ (enhanced spatial correlations) and decreases with $\theta$ (reduced effective dimensionality) (Velni et al., 2019).
• Dynamical Quenches: Following a global thermal quench, EWCS displays characteristic temporal regimes: early quadratic growth, intermediate linear growth at a rate matching the entanglement velocity (which depends on $z$ and $\theta$), and late-time saturation (Velni et al., 2023, Velni et al., 2020). In extremal (T=0) electromagnetic quenches, linear growth is replaced by a logarithmic regime.

5. Advanced Geometry, Corrections, and Extensions

• Phase Transitions and Criticality: The EWCS can record discontinuous vanishing and jumps (as in mutual information) when the minimal surface transitions between connected and disconnected phases. In confining geometries or holographic RG flows (e.g., massive ABJM backgrounds), EWCS and mutual information can be non-monotonic and display scale-dependent purification (Jokela et al., 2019).
• Higher-Derivative Gravity: Inclusion of Gauss–Bonnet terms (and, more generally, higher curvature corrections) affects the magnitude and the detailed thermalization pattern of EWCS, modifying the disentangling transitions and their temporal evolution (Li et al., 2021). In topological massive gravity (chiral CFTs), parity-violating Chern–Simons terms contribute an additional correction to EWCS, which must be computed with carefully prescribed normal frames at the endpoints of the cross section (Wen et al., 2022).
• Non-Lorentzian and Flat Holography: In nonrelativistic (e.g., axion, Aether) or asymptotically flat holographic duals, EWCS retains sensitivity to correlation structure, showing, for example, non-monotonic dependence on parameters such as Lorentz violation strength or axion coupling (Basu et al., 2021, Chen et al., 2021).
• Covariant Scenarios: In dynamical or time-dependent situations, EWCS is generalized to an extremal surface, maintaining the correspondence to boundary information-theoretic measures and matching the covariant generalization of reflected entropy and balanced partial entanglement (Wen et al., 2022).

6. Multipartite Structure and Inequalities

The monogamy/polygamy and additivity properties of EWCS in multipartite settings are intricate and reflect the underlying geometry:

• Inequalities in Tripartite Systems: While $E_W(A:BC) \geq E_W(A:B)$ universally holds, a simple polygamy inequality $E_W(A:B) + E_W(A:C) \geq E_W(A:BC)$ does not apply generically; the outcome depends on geometry, number of dimensions, and system configuration (Jain et al., 2022).
• Weak Monogamy and Lower Bounds: To restore robust inequalities, a weaker monogamy relation involving mutual information is invoked:

$$E_W(A:BC) + \frac{1}{2} I(A:BC) \geq E_W(A:B) + E_W(A:C),$$

and it is found that the squared EWCS is strictly monogamous:

$$\big(E_W(A:BC)\big)^2 \geq \big(E_W(A:B)\big)^2 + \big(E_W(A:C)\big)^2.$$

• Strong Multipartite Entanglement in Holography: Holographic states exhibit $O(1/G_N)$ tripartite entanglement, as signaled by EWCS differing from mutual information at leading order, which cannot be accounted for by "mostly bipartite" entanglement models (Akers et al., 2019).

7. Significance and Future Directions

EWCS represents a unifying geometric diagnostic for mixed state correlations in the holographic context. Its direct calculability from the dual density matrix (via OEE, reflected entropy, or other constructions) bridges the gap between quantum information theory and gravitational bulk geometry. It captures features not probed by entanglement entropy or mutual information—such as classical correlations and multipartite structure—while exhibiting robust scaling, area laws, and universality in singular and dynamical regimes. Ongoing research is extending these insights to more general boundary theories (e.g., with boundaries or defects (2206.13417)), higher-derivative corrections, nonrelativistic dualities, and explicit realizations in tensor network and bit-thread models.

EWCS's role as a boundary-accessible probe of the bulk geometry and its underlying entanglement structure continues to provide a powerful tool for elucidating spacetime emergence, information flow, and mixed state physics in quantum gravity and holography.