Reflected Entropy (RE) in Quantum Systems

• Reflected Entropy (RE) is a quantum measure defined via the canonical purification of a bipartite state, capturing intrinsic quantum correlations.
• Its holographic dual links RE to the entanglement wedge cross section, providing geometric insights into quantum many-body and gravitational systems.
• Generalizations in multipartite, nonrelativistic, and tensor network models reveal RE's limitations, including non-monotonic behavior, advancing our understanding of entanglement.

Reflected entropy (RE) is a quantum information-theoretic quantity defined for a bipartite system as the von Neumann entropy of the canonical purification of a mixed state. Originally introduced to diagnose correlations in bipartite mixed states, RE has since been generalized and extensively analyzed in quantum field theory, quantum many-body systems, random tensor networks, and holographic duality frameworks. The central significance of reflected entropy lies in its dual geometric interpretation—in holographic theories, RE is conjectured to be proportional to the area of the entanglement wedge cross section—and in the emergence of multipartite generalizations that probe irreducible many-body entanglement. However, recent developments highlight subtle issues regarding RE’s status as a correlation measure, especially concerning its monotonicity properties.

1. Definition and Canonical Purification

Given a bipartite density matrix $\rho_{AB}$ on $\mathcal{H}_A \otimes \mathcal{H}_B$, the reflected entropy $S_R(A:B)$ is computed as follows:

1. Canonical purification: Double the Hilbert space, promoting subsystems $A$ and $B$ to $A \otimes A^*$ and $B \otimes B^*$. Construct the canonical purification $|\sqrt{\rho_{AB}}\,\rangle$ as

$$|\sqrt{\rho_{AB}}\rangle = \sum_i \sqrt{\lambda_i} |i\rangle_{AB} |i\rangle_{A^*B^*}$$

where $\{\lambda_i\}$ is the eigenvalue spectrum of $\rho_{AB}$.

1. Reduced density matrix: Trace over $B, B^*$ to obtain $\rho_{AA^*}$.
2. Reflected entropy:

$$S_R(A:B) = -\text{Tr}_{AA^*}(\rho_{AA^*} \log \rho_{AA^*})$$

This construction generalizes to multipartite states in which the canonical purification is performed iteratively, with the reflected entropy replaced by higher-order entropic invariants that capture genuine multipartite correlations.

2. Holographic Dual: Entanglement Wedge Cross Section

The most influential advance in the paper of RE is its conjectured geometric dual for quantum systems with gravitational (AdS) duals. For bipartite systems in holographic (large-$c$) CFTs: $S_R(A:B) = 2 E_W(A:B)$ where $E_W(A:B)$ is the minimal area (length in AdS$_3$) cross section of the entanglement wedge associated with $A \cup B$ in the bulk dual geometry. For multipartite settings, the generalization involves minimal surfaces constructed from multipartite cross sections—e.g., for the tripartite case, introducing the invariant

$\Delta_R(A:B:C) = 2 \Delta_W(A:B:C)$

with $\Delta_W$ the multipartite entanglement wedge cross section, typically realized as the area of a bulk "triangle" bounded by the Ryu–Takayanagi surfaces of $A$, $B$, and $C$ (Chu et al., 2019).

The geometric duality is substantiated by replica trick computations in the large-$c$ limit: the purification and entropy computations reduce to evaluating multi-point twist operator correlators whose conformal blocks, in the semiclassical limit, exponentiate and match the variation of the corresponding minimal bulk surfaces.

3. Generalizations and Examples in Field Theory

• Multipartite reflected entropy: For a tripartite mixed state, a canonical 8-copy purification and an appropriately symmetrized partition define $\Delta_R(A:B:C)$, invariant under permutations and sensitive to irreducible three-body entanglement (Chu et al., 2019).
• Free fields and Gaussian systems: For free fermions and scalars, RE is computed via block correlators. The eigenvalues of the correlator matrix for the purified system yield the RE as a finite function, immune to the UV divergences plaguing single-interval entanglement entropy (Bueno et al., 2020, Bueno et al., 2020). For example, for fermions:

$$R_{\text{ferm.}} = -\sum_m [\nu_m \log \nu_m + (1-\nu_m) \log (1-\nu_m)]$$

where $\{\nu_m\}$ are eigenvalues of the AA$^*$ block.

• Monotonicity: In free field theory, lattice calculations indicate that RE is monotonic under the inclusion of more degrees of freedom ($R(A,BC) \geq R(A,B)$); however, exceptions in non-relativistic Lifshitz models and explicit counterexamples show this monotonicity does not hold universally (Bueno et al., 2020, Hayden et al., 2023, Berthiere et al., 2023).

4. Behavior in Topological, Nonrelativistic, and Random Tensor Network Models

• Topological order (Chern–Simons theories): RE equals the mutual information in all (2+1)D Chern–Simons cases studied, with area-law contributions canceling and the universal part governed by the classical Shannon entropy in the topological sector distribution (Berthiere et al., 2020).
• Lifshitz theories: Analytic evaluation reveals a discrete, "thermal"-like reflected entanglement spectrum for disjoint intervals and a universal, positive "Markov gap" (RE – mutual information). Reflected entropy is non-monotonic in these theories, highlighting its sensitivity to multipartite entanglement structures inaccessible to bipartite measures (Berthiere et al., 2023).
• Random tensor networks: The reflected entanglement spectrum is non-flat and organizes into superselection sectors labeled by topological invariants, matching gravitational duals with higher-genus wormhole saddle contributions. Replica calculations recover the duality $S_R = 2(EW)/(4G_N)$ away from phase transitions and give nontrivial analytic continuations of RE through the Page transition (Akers et al., 2021, Akers et al., 2022).

5. Extensions: Holographic Islands, Nonrelativistic & Boundary/Interface CFT, and Anisotropic Theories

• Island formula for RE: In gravitational-subsystem-coupled QFTs, the reflected entropy acquires "island contributions" via

$$S_R(A:B) = S_R^{\text{eff}}(A \cup \text{Is}_R(A) : B \cup \text{Is}_R(B)) + \frac{\text{Area}[\partial \text{Is}_R(A) \cap \partial \text{Is}_R(B)]}{2 G_N}$$

where $\text{Is}_R(A)$, $\text{Is}_R(B)$ are portions of the conventional entanglement island (Chandrasekaran et al., 2020, Li et al., 2020).

• Boundary/Interface CFT (BCFT/ICFT): The concept of "left-right reflected entropy" (LRRE) is introduced, dual to twice the area of a minimal entanglement wedge cross section in AdS/BCFT. Markov gap ($S_R - I$) acquires a universal term and signals genuine multipartite entanglement across boundaries or interfaces (Kusuki, 2022, Afrasiar et al., 2022).
• Nonrelativistic holography (GCFT, WCFT): The correspondence $S_R = 2 E_W$ holds for GCFTs dual to GMMG gravity as well as for AdS$_3$/WCFT (Setare et al., 2022, Chen et al., 2022). In WCFT, RE is computed via non-Abelian orbifold twist correlators, and the dual minimal surface is generalized to a "pre-entanglement wedge cross section."
• Anisotropic backgrounds: The phase structure of RE, including critical separation dependence and orientation effects, is governed by the anisotropy of the boundary theory, reflecting in the minimal surface problem for the entanglement wedge cross section (Vasli et al., 2022).

6. Reflected Entropy as a Correlation Measure: Monotonicity, Markov Gap, and Limitations

• Monotonicity violation: Explicit counterexamples demonstrate that the family of Rényi reflected entropies $S_R^{(\alpha)}$ with $0 < \alpha < 2$ are not monotonic under partial trace. RE can increase when discarding degrees of freedom, even for states diagonal in product bases (i.e., classical distributions), thereby failing as a bona fide correlation measure (Hayden et al., 2023, Berthiere et al., 2023). For integer $\alpha \geq 2$, monotonicity may be restored.
• Markov gap: The difference $S_R(A:B) - I(A:B)$, known as the Markov gap, is positive and universal in many settings, notably for adjacent intervals in CFTs and Lifshitz field theories. A nonzero Markov gap signals irreducible tripartite entanglement and departure from perfect quantum Markov chains.
• Operational meaning: While RE was originally motivated as a correlation measure, its non-monotonicity in general cases disqualifies it as a strict measure of correlations. However, it remains valuable for diagnosing multipartite entanglement structure, irreducible correlations, and dual geometric properties in holography (Hayden et al., 2023).

7. Microscopic and Coarse-Grained Realizations: BCFT Tensor Networks, Modular Flow, and Geometry

Recent work achieves a microscopic CFT derivation of the $S_R$–$E_W$ correspondence using coarse-grained BCFT tensor networks. The canonical purification is constructed via cutting-and-gluing of CFT path integrals, with modular flow in the large-$c$ limit simplifying the analysis. Coarse-graining over heavy states (Cardy spectrum) replaces the microscopic OPE structure by Liouville theory partition functions with ZZ boundary conditions, encoding universal hyperbolic geometry on the Cauchy slice (Bao et al., 12 Sep 2025). Replica partition functions, after ensemble averaging, reproduce all candidate entanglement wedge cross sections, with the dominant pattern yielding the RE and EW duality without invoking a bulk-boundary dictionary as an axiom.

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In summary, reflected entropy is a powerful, technically rich probe of multipartite entanglement, with a robust dual holographic geometric interpretation through the entanglement wedge cross section and nontrivial links to irreducible correlations. However, it does not satisfy all the properties required of a correlation measure, most notably monotonicity under partial trace, especially outside the pure or holographic settings. In CFT, tensor network, and holographic contexts, RE continues to provide deep insights into the structure of entanglement, bulk emergence, and the quantum information–geometry correspondence.