Multipartite Reflected Entropy

• Multipartite reflected entropy is a measure that extends reflected entropy to systems with three or more subsystems using canonical purification and replica techniques.
• It has a holographic dual in AdS/CFT, represented by minimal surface webs that partition the entanglement wedge of multiple boundary regions.
• The measure distinguishes non-Markovian multipartite correlations, offering insights into quantum recoverability and the role of gravitational 'islands' in entanglement dynamics.

Multipartite reflected entropy is a quantum information-theoretic quantity designed to generalize reflected entropy to systems of three or more subsystems. It quantifies multipartite correlations in mixed quantum states and has an explicit holographic dual in AdS/CFT, capturing the minimal surface web spanning the entanglement wedge of multiple boundary regions. This measure, along with related quantities such as multipartite Markov gaps and genuine reflected multiparty entropies, is central to the analysis of irreducible multipartite entanglement and quantum recoverability in both field-theoretic and holographic settings (Yuan et al., 11 Oct 2024, Iizuka et al., 21 Jul 2025, Chu et al., 2019, Bao et al., 2019, Chandrasekaran et al., 2020).

1. Formal Definition and Canonical Purification

Given a mixed state $\rho_{A_1\cdots A_q}$ on $q$ spatial boundary regions, the multipartite reflected entropy utilizes the canonical purification procedure, extending the Dutta–Faulkner bipartite construction. The canonical purification of $\rho_{A_1\cdots A_q}$ is the pure state $|\sqrt{\rho_{A_1\cdots A_q}}\rangle$ in

$$(\mathcal{H}_{A_1} \otimes \mathcal{H}_{A_1}^*) \otimes \cdots \otimes (\mathcal{H}_{A_q} \otimes \mathcal{H}_{A_q}^*)$$

such that tracing out all the star copies yields the original $\rho$.

On this canonical purification, multipartite multi-entropy $S^{(q)}$ is defined using a replica construction with $n$ copies, and the reflected multi-entropy is

$$S_R^{(q)}(A_1 : \cdots : A_q) \equiv S^{(q)}(A_1A_1^* : \cdots : A_qA_q^*)_{|\sqrt{\rho}\rangle}.$$

For $q=2$ this reduces to the conventional reflected entropy, while for pure states, it reduces to twice the multi-entropy.

The multipartite reflected entropy can also be generalized to other purification and tracing sequences, yielding a family of entropic correlation measures including symmetric invariants such as $\Delta_R(A:B:C)$ for the tripartite case (Chu et al., 2019).

2. Replica-Trick and CFT Representation

Evaluation of $S_R^{(q)}$ proceeds via a two-fold replica trick: introducing replica indices $n$ and $m$ for the multi-entropy and purification steps, respectively. The core object is the ratio of partition functions:

$$S_R^{(q)} = \lim_{m \to 1} \lim_{n \to 1} \frac{1}{1-n} \log \left[\frac{Z_{n^{q-1}, m}}{(Z_{1, m})^{n^{q-1}}}\right]$$

where $Z_{n^{q-1}, m}$ is constructed as a $2q$-point correlation function of twist operators in a $CFT^{\otimes m n^{q-1}}$, each twist insertion representing an entangling interval endpoint.

In the tripartite case ($q=3$), explicit computation involves solving the monodromy problem for a six-point function of twist fields at large $c$. Accessory parameters in the stress tensor are fixed by requiring trivial monodromy around specific cycles, and the entropic quantity is then extracted from the semiclassical block (Yuan et al., 11 Oct 2024). At both zero and finite temperature, this replica-trick calculation yields exact agreement with the holographic minimal surface construction.

3. Holographic Dual: Minimal Surface Webs and Multiway Cuts

The AdS/CFT dual of multipartite reflected entropy is a minimal-area surface web (multiway cut) in the entanglement wedge of the boundary union $A_1 \cup \cdots \cup A_q$, anchored on the relevant Ryu–Takayanagi surface and partitioning the wedge into sub-wedges containing each $(A_i, A_i^*)$. The multipartite reflected entropy is given by

$$S_R^{(q)}(A_1:\cdots:A_q) = \frac{2}{4G_N} \min_{\mathcal{W}} \mathrm{Area}(\mathcal{W})$$

where the web $\mathcal{W}$ connects the RT surface of the full union to each of the regions, and the minimization ensures global connectedness subject to homology constraints (Yuan et al., 11 Oct 2024, Iizuka et al., 21 Jul 2025). For $q=3$ at zero temperature (AdS${}_3$/CFT${}_2$), this web is Y-shaped, with legs meeting at a junction on $\Gamma_{ABC}$. At finite temperature, the construction is generalized to the BTZ black hole geometry with appropriate modifications.

When "islands" are present due to semiclassical gravity corrections, the area functional must include the contributions from generalized quantum extremal surfaces and intersections of island boundaries, extending the formula to dynamical spacetimes (Chandrasekaran et al., 2020).

4. Multipartite Markov Gap and Genuine Multipartite Entanglement

A central diagnostic enabled by multipartite reflected entropy is the Multipartite Markov Gap (MG). For $q$ parties, the Markov gap is defined as

$$MG^{M(q-1)}(A_1:\cdots:A_{q-1}) = S_{R\,(q-1)}(A_1:\cdots:A_{q-1}) - \left[ \sum_{i=1}^{q-1} S(A_i) - S(A_1\cdots A_{q-1}) \right]$$

where $S_{R\,(q-1)}$ is the reflected multi-entropy for the $(q-1)$-party reduced state. This quantity measures the non-Markovianity and quantifies irreducible multipartite entanglement: Markov gap vanishes if and only if the system obeys a quantum Markov condition (perfect recoverability via a quantum channel), and its holographic dual is the surface area deficit relative to the sum of RT surfaces (Iizuka et al., 21 Jul 2025).

To further single out only genuine $q$-party correlations (vanishing on states with entanglement distributed among fewer than $q$ parties), a "genuine" multipartite reflected entropy is constructed as a specific linear combination of multi-entropy and lower-partite reflected entropies. For $q=4$, this involves subtractions and additions of tripartite and bipartite entropic terms, and is designed to vanish on all partially separable states.

5. Key Properties and Inequalities

Multipartite reflected entropy exhibits symmetry under party permutations and is always nonnegative. Strong subadditivity and holographic surface rearrangement guarantee the inequality

$$S_R^{(q)} \geq \sum_{i=1}^q S(A_i) - S(A_1\cdots A_q)$$

and specifically for the Markov gap,

$MG^{M(q-1)} \ge 0.$

Upper and lower bounds relate to known quantities such as mutual information, standard entropic inequalities, and multipartite lower bounds (e.g., $D_3(A:B:C)$ for $q=3$) (Chu et al., 2019).

Polygamy-type inequalities hold, reflecting the distribution of correlations among subsets, e.g.

$$\Delta_R(A_1A_2:B:C) \le \Delta_R(A_1:B:C) + \Delta_R(A_2:B:C).$$

For bipartite and certain multipartite limits, these measures reduce to twice the entanglement wedge cross-section or to mutual information (Bao et al., 2019). For $d>2$ the geometric constructions generalize but require careful treatment of bulk topology.

6. Operational and Physical Interpretations

Multipartite reflected entropy is UV-finite and sensitive to genuinely multipartite quantum correlations in mixed states. It detects correlations not visible to simple combinations of bipartite entanglement entropy and mutual information, being sensitive to structures such as W-states over GHZ states in qubit examples (Chandrasekaran et al., 2020, Iizuka et al., 21 Jul 2025). The Markov gap bounds the maximum achievable fidelity of quantum recovery maps (e.g., rotated Petz channel recovery):

• $MG=0$: perfect recoverability of the canonical purification from any single marginal.
• $MG>0$: presence of irreducible multipartite entanglement and obstruction to Markov recovery.

Holographically, multipartite reflected entropy provides a unified geometric probe of bulk connectivity, the phase structure of multi-interval entanglement, and the emergence of multiboundary wormholes in time-symmetric AdS${}_3$ slices (Bao et al., 2019, Yuan et al., 11 Oct 2024). Its generalization in the presence of gravitational "islands" further refines the understanding of quantum extremal surfaces and the Page curve.

7. Connections, Extensions, and Open Directions

The multipartite reflected entropy, multipartite Markov gaps, and genuine multipartite entropic measures supply a broad framework for analyzing multipartite entanglement in both field theory and holography. Extensions include:

• Higher $q$, with minimal surface webs and generalized multiway cuts (Yuan et al., 11 Oct 2024, Iizuka et al., 21 Jul 2025).
• Covariant (time-dependent) generalizations using extremal rather than minimal surfaces.
• Island contributions for semiclassical gravity, via quantum extremal surfaces and replica wormholes (Chandrasekaran et al., 2020).
• Connections to operational tasks in quantum information, including bounds on correlation distillation and channel recoverability (Iizuka et al., 21 Jul 2025).
• Generalization to higher dimensions is conceptually straightforward for the surface web prescription but construction of smooth multiboundary wormholes remains open for $d>2$ (Bao et al., 2019).

These developments posit multipartite reflected entropy as a comprehensive and geometrically robust measure of genuine multipartite entanglement and non-Markovianity in complex quantum systems, both in abstract QFT and holographic contexts.