Reflected Entropy in Quantum Systems

• Reflected entropy is a quantum measure defined via canonical purification that captures mixed-state and multipartite entanglement in bipartite systems.
• It satisfies inequalities such that for pure states it recovers doubled von Neumann entropy and connects to holographic entanglement wedge cross sections.
• It is computed using replica techniques and analytic continuation, providing key insights into black hole evaporation and quantum gravitational phenomena.

Reflected entropy is a quantum information-theoretic quantity designed to probe the correlation structure, particularly mixed-state and multipartite entanglement, in bipartite systems. It is defined via a canonical purification and exhibits deep connections with holography, entanglement wedge cross-sections, and entanglement island phenomena in quantum gravity. The following article details its rigorous definition, central properties, role in field theory and gravity, operational techniques, and outstanding subtleties.

1. Canonical Purification and Definition

Given a bipartite mixed state $\rho_{AB}$ on a finite or infinite-dimensional Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$, the reflected entropy $S_R(A:B)$ is constructed as follows:

• One forms the canonical purification $|\sqrt{\rho_{AB}}\rangle$ in an extended Hilbert space $$\mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_{A^*} \otimes \mathcal{H}_{B^*}$$. Explicitly,

$$|\sqrt{\rho_{AB}}\rangle = \sum_{ij} (\sqrt{\rho_{AB}})_{ij} \; |i\rangle_A |j\rangle_B \otimes |i\rangle_{A^*} |j\rangle_{B^*}$$

where $|i\rangle$, $|j\rangle$ are orthonormal bases and $\sqrt{\rho_{AB}}$ is the unique positive-definite square root.

• The partial trace over $B$, $B^*$ yields the reflected density matrix:

$$\rho_{AA^*} = \mathrm{Tr}_{BB^*} \left( |\sqrt{\rho_{AB}}\rangle\langle\sqrt{\rho_{AB}}| \right)$$

• The reflected entropy is defined as the von Neumann entropy of this subsystem:

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

In field theory and continuum QFTs, this construction is formalized via the split property and Tomita–Takesaki theory, with careful attention paid to type-III algebras (Bueno et al., 2020).

2. Fundamental Properties and Inequalities

Reflected entropy exhibits the following hallmark properties:

• Symmetry: $S_R(A:B) = S_R(B:A)$ by construction.
• Purity Recovery: For pure $\rho_{AB}$, $S_R(A:B) = 2 S(A) = 2 S(B)$.
• Bounds:

$\min\{2 S(A),\, 2 S(B)\} \geq S_R(A:B) \geq I(A:B)$

where $I(A:B) = S(A) + S(B) - S(AB)$ is the mutual information.

• Non-negativity: $S_R(A:B) \geq 0$.
• Polygamy: For tripartitions of a global pure state, $S_R(A:B) + S_R(A:C) \geq S_R(A:BC)$ (Basak et al., 2023).
• Monotonicity: In free and holographic field theories, $S_R(A:BC) \geq S_R(A:B)$ is observed (partial trace monotonicity) (Bueno et al., 2020, Bueno et al., 2020, Basak et al., 2023), but general monotonicity fails for arbitrary states, even classically (Hayden et al., 2023, Berthiere et al., 2023).
• Markov Gap: The difference $M(A:B) = S_R(A:B) - I(A:B) \geq 0$ characterizes irreducible tripartite entanglement, and takes universal (e.g., $\mathcal{O}(c)$) values in certain CFT and holographic settings (Basu et al., 2023, Berthiere et al., 2023, Afrasiar et al., 2022, Kusuki, 2022).

3. Replicated Construction and Calculation Techniques

The computation of $S_R(A:B)$ employs a two-parameter replica trick, generalizing the standard Rényi entropy:

• Rényi Reflected Entropy: Introduce integer parameters $m, n$, define

$$S_R^{(m,n)}(A:B) = \frac{1}{1-n} \ln \, \mathrm{Tr}\left[ \left( \rho_{AA^*}^{(m)} \right)^n \right]$$

where

$$|\psi^{(m)}\rangle = \frac{\rho_{AB}^{m/2}}{\sqrt{\mathrm{Tr} \rho_{AB}^m}} | \phi^+ \rangle, \qquad \rho_{AA^*}^{(m)} = \mathrm{Tr}_{BB^*} |\psi^{(m)}\rangle\langle\psi^{(m)}|$$

• Analytic Continuation: The true reflected entropy is recovered from the double limit:

$$S_R(A:B) = \lim_{m \to 1} \lim_{n \to 1} S_R^{(m,n)}(A:B)$$

• Path Integral—Twist Operator Construction: In $d=2$ CFTs (and BCFTs/WCFTs/GCFTs), reflected entropy is computed by multi-point correlations of appropriately replicated twist operators (Chen et al., 2022, Basu et al., 2023, Setare et al., 2022).
• Gaussian Theories: In free theory, explicit computation is possible via the symmetries of correlation matrices in the doubled system (Bueno et al., 2020, Bueno et al., 2020).

4. Holographic Duality and Entanglement Wedge Cross Sections

A central insight from AdS/CFT is the geometric dual:

• Bulk Dual: For holographic (large-$c$, semiclassical) states, reflected entropy is twice the area of the entanglement wedge cross section (EWCS) S_R(A:B) = 2 \, \mathrm{Area}(\mathrm{EWCS})/(4 G_N).
• Quantum Corrections: When bulk quantum fields are included, $S_R$ receives subleading contributions from the reflected entropy of bulk quantum matter, and the formula generalizes to a "quantum extremal cross-section" prescription (Ling et al., 2021, Li et al., 2020).

Quantity	Holographic formula
Entanglement entropy $S$	Minimal (quantum) extremal surface area/4G$_N$
Reflected entropy $S_R$	2 × EWCS area/4G$_N$ (+ bulk $S_R^{(\mathrm{bulk})}$)

• Phase Transitions & Page Curves: Reflected entropy exhibits phase transitions ("Page curves") across dominance swaps between connected and disconnected EWCS saddles (Akers et al., 2022, Li et al., 2020, Chen et al., 2024), capturing entanglement island transitions in evaporating black holes and related models.
• Random Tensor Networks: In RTNs, $S_R$ is similarly dual to twice the minimal cut ("EWCS") in the network, and the reflected spectrum is organized into superselection sectors associated to different topological indices (Akers et al., 2021, Akers et al., 2022).

5. Evaporating Black Holes and Non-Isometric Holography

Black hole evaporation presents unique challenges for holography due to non-isometric bulk/boundary maps:

• Non-Isometric Map: In realistic (post-Page time) evaporation, the holographic map from the effective (bulk EFT) degrees of freedom to the boundary fails to be isometric. This is modeled by random Haar unitaries and post-selected maximal entanglement (Chen et al., 2024).
• Two-Sided Model: Consider the semi-classical state canonically purified into a two-sided ("doubled") black hole+reservoir system, then map to the boundary using a non-isometric, Haar-random unitary. The reflected entropy between relevant boundary and radiation subsystems is computed via the canonical purification formalism and Haar averaging.
• Phase Structure: Analytic calculations reproduce Page-curve-like transitions:
  • $S_R$ between the two black holes (or two radiation systems) saturates its upper bound $2\min\{\log|B|,\log|R|\}$ before Page time and collapses after. Transitions are discontinuous, paralleling QES and island transitions.
  • $S_R$ between disconnected radiation pieces exhibits a jump at the transition $|R_1||R_2|=|B|$, again saturating lower or upper bounds in distinct phases.
  • Results are consistent with predictions from the QES/island formula and provide a nontrivial test of random-unitary holographic models of black hole information flow (Chen et al., 2024).

6. Nonrelativistic and Exotic Theories: Limitations and Markov Gap

• Nonrelativistic Theories: In certain systems such as $z=2$ Lifshitz field theories, reflected entropy exhibits a sharp violation of expected monotonicity under partial trace. Analytic calculations show that, for adjacent intervals, $S_R$ fails to be nonincreasing under B-enlargement, marking the first explicit free-theory violation of this property (Berthiere et al., 2023).
• Universal Markov Gap: In both free and holographic CFTs, including BCFT and theories with boundaries, the Markov gap $M = S_R - I$ is generically positive and, in many cases, universal (e.g., $M = (c/3)\log 2$ per EWCS endpoint), highlighting irreducible multipartite entanglement structures (Basu et al., 2023, Afrasiar et al., 2022, Kusuki, 2022).
• Failure as General Correlation Measure: There exist explicit classical and quantum finite-dimensional counterexamples where $S_R(A:BC) < S_R(A:B)$ (Hayden et al., 2023). Thus, unlike mutual information, reflected entropy is not a general correlation monotone.

7. Extensions, Computable Bounds, and Operational Significance

• Gaussan Systems: For free scalars and fermions, explicit correlation-matrix formulas mean $S_R$ can be efficiently calculated numerically (via doubling and diagonalizing the extended correlation matrix) (Bueno et al., 2020, Bueno et al., 2020).
• Field Theories with Symmetries: The reflected entropy can be adapted to orbifold and subalgebra settings, with subtracted contributions from symmetry-sector (twist) entropies (Bueno et al., 2020).
• Topological Theories: In Chern–Simons TQFTs, $S_R$ matches the mutual information and encodes both quantum and classical (Shannon) topological data (Berthiere et al., 2020).
• Operational Proxies: In systems where the mutual information is zero, $S_R$ can still detect multipartite or topologically protected correlations invisible to ordinary bipartite entanglement measures.

8. Outstanding Issues and Current Directions

• Order-of-Limits and Analytic Continuation: Computing reflected entropy via the replicated path integral or in random tensor networks requires special care regarding the order of analytic continuation. Naive $m \to 1$ limits can violate known bounds and fail to reproduce the correct phase structure; careful saddle-point analysis is essential (Akers et al., 2021, Akers et al., 2022).
• Non-universal Monotonicity: Beyond free and holographic theories, monotonicity and even nonnegativity of the Markov gap can fail, so the universal validity of $S_R$ as a correlation measure is not guaranteed (Hayden et al., 2023, Berthiere et al., 2023).
• Multipartite Generalizations: Reflected entropy intrinsically probes multipartite and nonclassical entanglement. Positive Markov gaps, sectoral structure in random tensor networks, and area versus Shannon decompositions in TQFTs all indicate sensitivity to multipartite and topological entanglement regimes (Akers et al., 2022, Berthiere et al., 2020).
• Connection to Holographic Complexity and Black Hole Information: Fine-grained features of $S_R$—such as phase transitions, smoothing via area fluctuations, and multipartite "island" formulae—provide sensitive diagnostics of information flow during black hole evaporation and, in double-holography, quantum-gravitational dualities (Akers et al., 2022, Li et al., 2020, Chen et al., 2024).

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References:

• "Reflected entropy in an evaporating black hole through non-isometric map" (Chen et al., 2024)
• "Reflected entropy and Markov gap in non-inertial frames" (Basak et al., 2023)
• "Reflected entropy and Markov gap in Lifshitz theories" (Berthiere et al., 2023)
• "Reflected entropy in BCFTs on a black hole background" (Basu et al., 2023)
• "Reflected entropy is not a correlation measure" (Hayden et al., 2023)
• "The Page Curve for Reflected Entropy" (Akers et al., 2022)
• "Islands for Reflected Entropy" (Chandrasekaran et al., 2020)
• "Reflected Entropy for an Evaporating Black Hole" (Li et al., 2020)
• "Reflected Entropy in Double Holography" (Ling et al., 2021)
• "Reflected entropy, symmetries and free fermions" (Bueno et al., 2020)
• "Reflected entropy in random tensor networks" (Akers et al., 2021)
• "Reflected entropy in random tensor networks II: a topological index from the canonical purification" (Akers et al., 2022)
• "Reflected Entropy for Communicating Black Holes II: Planck Braneworlds" (Afrasiar et al., 2023)
• "Reflected Entropy for Communicating Black Holes I: Karch-Randall Braneworlds" (Afrasiar et al., 2022)
• "Reflected entropy for free scalars" (Bueno et al., 2020)
• "Reflected Entropy in Boundary/Interface Conformal Field Theory" (Kusuki, 2022)
• "Holographic Study of Reflected Entropy in Anisotropic Theories" (Vasli et al., 2022)
• "Reflected Entropy in AdS$_3$/WCFT" (Chen et al., 2022)
• "The reflected entropy in the GMMG/GCFT flat holography" (Setare et al., 2022)
• "Topological reflected entropy in Chern-Simons theories" (Berthiere et al., 2020)

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Summary Table: Fundamental Features of Reflected Entropy

Feature	Holds Universally?	References
$S_R(A:B) \geq I(A:B)$	Yes	(Bueno et al., 2020, Bueno et al., 2020)
Monotonicity (partial trace)	No (counterexamples exist)	(Hayden et al., 2023, Berthiere et al., 2023)
$S_R=2\,$EWCS (holography)	For holographic states	(Akers et al., 2022, Chandrasekaran et al., 2020)
Nonnegativity	Yes	(Bueno et al., 2020)
Detects multipartite entangl.	Yes	(Berthiere et al., 2023, Basu et al., 2023)
Sensitivity to islands	Yes (gravity, QES)	(Chandrasekaran et al., 2020, Akers et al., 2022)

Reflected entropy therefore constitutes a refined, multipartite correlation measure, underpinned by robust geometric dualities and directly sensitive to the subtle features of quantum and gravitational information flow. However, its status as a universal measure of correlations is limited by explicit counterexamples and the structure of nonrelativistic theories.