Entanglement Wedge Cross Section in Holography

• Entanglement wedge cross section is defined as the minimal-area surface in the bulk, quantifying mixed-state entanglement in holographic setups.
• It establishes dualities with boundary measures like entanglement of purification and reflected entropy, providing a clear field-theoretic prescription.
• The framework uses replica tricks and twist operator techniques to compute phase transitions and multipartite entanglement in AdS/CFT models.

The entanglement wedge cross section (EWCS) is a geometric quantity in semiclassical gravity—specifically in the AdS/CFT correspondence—that generalizes notions of entanglement entropy for mixed states and provides rigorous connections between quantum information measures in conformal field theory (CFT) and minimal surfaces in the bulk holographic dual. Unlike the Ryu–Takayanagi (RT) formula, which computes the von Neumann entropy for pure states, the EWCS detects finer correlation structure in mixed states, with critical consequences for the understanding of multipartite entanglement in holographic many-body systems, quantum gravity, and beyond.

1. Definition and Geometric Construction

Let $A_1$ and $A_2$ be two disjoint subregions (or more generally, a bipartition $A_1 \cup A_2$ of some boundary region $A$) in a holographic CFT. The entanglement wedge $\mathcal{E}_{A}$ is the bulk region bounded by $A$ and the RT/HRT surface $\gamma_{A}$. The entanglement wedge cross section $E_W(A_1: A_2)$ is defined as the minimal-area codimension-2 surface $\Sigma_{A_1: A_2}^{\text{min}}$ homologous to the "interface" between $A_1$ and $A_2$ in $\mathcal{E}_{A}$, subject to the endpoints of $\Sigma_{A_1: A_2}^{\text{min}}$ lying on $\gamma_A$. The canonical formula is: $$E_W(A_1:A_2) = \frac{\text{Area}\,\big(\Sigma_{A_1: A_2}^{\text{min}}\big)}{4G_N}$$ where $G_N$ is Newton's constant in the bulk.

The EWCS generalizes to multipartite regions $A_1, \ldots, A_n$, in which case the minimal cross-section divides the wedge into $n$ parts, each anchored at a distinct $A_i$, and its area quantifies more complex entanglement structures (Bao et al., 2021).

2. Field-Theoretic Duals and Odd Entanglement Entropy

Various proposals link the EWCS to measures of correlations in the CFT. The essential result of (Tamaoka, 2018) introduces the odd entanglement entropy (OEE) $S_o$ defined for a bipartite density matrix $\rho_{A_1A_2}$ as: $$S_o(\rho_{A_1A_2}) = \lim_{n_o \to 1}\frac{1}{1 - n_o}\big[\operatorname{Tr}(\rho_{A_1A_2}^{T_{A_2}})^{n_o} -1\big]$$ where $\rho^{T_{A_2}}$ denotes the partial transpose of $\rho$ on $A_2$, and $n_o$ is analytically continued from odd integers. The OEE, constructed via the replica trick and sensitive to negative eigenvalues in the spectrum of $\rho^{T_{A_2}}$, encapsulates a Tsallis-like entropy. The central finding is that, for holographic CFTs,

$$E_W(\rho_{A_1A_2}) = S_o(\rho_{A_1A_2}) - S(\rho_{A_1A_2})$$

where $S$ denotes the von Neumann entropy.

This matching is verified in explicit calculations for two-intervals in AdS$_3$ and BTZ backgrounds, and the equality is conjectured to hold in higher dimensions. The result yields a precise field-theoretic prescription for the EWCS purely in terms of boundary data.

3. Dualities with Mixed-State Entanglement Measures

The EWCS is conjectured to be dual to several boundary measures of mixed-state correlation:

• Entanglement of Purification ($E_P$): The minimal entanglement entropy across all possible purifications; the duality proposal is $E_P = E_W$ (Velni et al., 2019).
• Reflected Entropy ($S_R$): Twice the EWCS, $S_R = 2E_W$, where $S_R$ is defined using the canonical purification (Kusuki et al., 2019, Akers et al., 2019).
• Logarithmic Negativity: Proposed to be proportional to a backreacted EWCS or a sum of “order-half” entropies (Basak et al., 2020).
• Odd Entanglement Entropy: As above.

These dualities extend the geometrical Ryu–Takayanagi paradigm to richer, non-pure quantum correlations, making the EWCS a central probe of mixed state entanglement in holographic systems.

4. Computational Framework: Replica Trick, Conformal Blocks, and Explicit Formulas

The EWCS admits a precise computational framework in two-dimensional holographic CFTs. The key technical ingredient is the replica trick applied to the partially transposed density matrix, leading to correlation functions of twist operators with altered branch structure: $$\operatorname{Tr}[(\rho^{T_{A_2}})^{n_o}] = \langle \sigma_{n_o}(u_1)\,\bar{\sigma}_{n_o}(v_1)\,\bar{\sigma}_{n_o}(u_2)\,\sigma_{n_o}(v_2) \rangle$$ Analytic continuation from odd-integer $n_o$ is essential to ensure the correct treatment of negative spectrum contributions. At large central charge ($c \gg 1$), the block decomposition is tractable due to dominance by the vacuum or lowest-dimension operator. The OEE is thus found to match the geometric cross section: $$E_W(\rho_{A_1A_2}) = \frac{1}{4G_N}\log\left(\frac{1+\sqrt{x}}{1-\sqrt{x}}\right)$$ for the vacuum (AdS$_3$) with $x$ the conformal cross ratio; and

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

for thermal (planar BTZ) backgrounds.

The approach generalizes to higher genus and operator insertions, with extensions covering time-dependent setups such as quenches (Kusuki et al., 2019, Velni et al., 2020).

5. Physical Consequences and Phase Structure

The EWCS exhibits sharp phase transitions associated with the connectivity of entanglement wedges. For disjoint intervals, the EWCS is nonzero if and only if the union’s wedge is connected; as intervals are separated, a geometric phase transition causes the EWCS to vanish, reflecting loss of mutual correlations. At high temperature or for large intervals the EWCS obeys an area law, even as the entanglement entropy transitions to a volume law due to thermal contributions (Velni et al., 2019).

Bulk corrections, e.g., higher-curvature (Gauss–Bonnet) terms or nontrivial boundary geometries (branes in BCFT), nontrivially affect the critical points at which EWCS vanishes or transitions (Li et al., 2021, 2206.13417).

For multipartite splittings, inequalities are derived between different EWCS surfaces, extending known holographic entropy cone constraints to cross-section measures. Notably, standard monogamy/polygamy-type inequalities do not unambiguously constrain the EWCS for all geometries; instead, weaker or squared versions are sometimes obeyed (Jain et al., 2022).

6. Role in Multipartite Entanglement and Tripartite Structure

The connection of EWCS to the multipartite structure of entanglement is of foundational importance. The gap between reflected entropy or entanglement of purification and mutual information—parametrically at order $1/G_N$—precludes a simple bipartite entanglement structure in holographic states. Rigorous analysis shows that O($1/G_N$) tripartite entanglement is necessary to realize the holographic values for EWCS, ruling out “mostly-bipartite” conjectures and impacting the construction of tensor network duals (Akers et al., 2019).

The construction of replicated geometries allows for the mapping of multipartite EWCS surfaces to Ryu–Takayanagi surfaces in extended bulk spacetimes, providing a geometrically unified framework for entropy inequalities (Bao et al., 2021).

7. Applications and Extensions

The EWCS is a sensitive diagnostic in systems with:

• Nontrivial scaling (Lifshitz, hyperscaling violation): The scaling of EWCS with temperature and subsystem size depends nontrivially on the dynamical exponent $z$ and hyperscaling parameter $\theta$, with explicit analytic dependence in both early and linear growth regimes after quench and during thermalization (Velni et al., 2023, Velni et al., 2019).
• Non-local field theories (noncommutative SYM, Aether/axion gravity): The behavior and scaling of the EWCS encapsulate nonlocality, Lorentz violation, and disorder, displaying non-monotonic or cutoff–sensitive features not visible in von Neumann entropy or mutual information (Chowdhury et al., 2021, Cheng et al., 2021, Chen et al., 2021).
• Dynamics and thermalization: The EWCS manifests universal scaling regimes (early quadratic/linear growth, saturation, and sometimes plateaus), matching and extending the “entanglement tsunami” paradigm to mixed-state measures (Velni et al., 2020, Velni et al., 2023, Boruch, 2020).

Furthermore, the relation between odd entanglement entropy, reflected entropy, and the EWCS grounded in explicit field-theoretic and holographic constructions provides a nontrivial map from quantum information diagnostics to geometric bulk data, with implications for both quantum gravity and strongly-coupled many-body systems.

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In summary, the entanglement wedge cross section is a robust, precisely defined geometric quantity encoding the mixed-state correlation structure of quantum field theory states with semiclassical gravity duals. It plays a crucial role in generalized entropy formulas, multipartite entanglement algebra, and the holographic dictionary, providing an indispensable bridge between information-theoretic measures and geometric realization in AdS/CFT and related dualities.