Reflected Multi-Entropy in Holographic Duality

• The paper introduces reflected multi-entropy as a mixed-state generalization achieved through canonical purification, reducing to pure-state multi-entropy and standard reflected entropy in specific limits.
• It defines the measure with a UV-finite construction and a holographic dual represented by minimal surface webs, providing a robust formulation in AdS3/CFT2 contexts.
• The approach is validated through explicit zero- and finite-temperature checks and distinguishes itself from earlier multipartite reflected entropy constructions by its unique purification process.

Searching arXiv for papers on reflected multi-entropy and closely related reflected-entropy generalizations. Searching "reflected multi-entropy holographic dual canonical purification" Reflected multi-entropy is a mixed-state generalization of multi-entropy obtained through canonical purification. In the formulation introduced for tripartite systems, one starts from a mixed state $\rho_{ABC}$, forms the canonical purification $\ket{\sqrt{\rho_{ABC}}}$ in the doubled Hilbert space, and defines the reflected multi-entropy as the multi-entropy of the purified tripartite state,

$$S_R^{(\mathtt{q}=3)}(A;B;C)=S^{(\mathtt{q}=3)}(AA^*;BB^*;CC^*)_{\ket{\sqrt{\rho_{ABC}}}}.$$

The construction reduces to twice the pure-state multi-entropy when the input state is pure, reduces to ordinary reflected entropy in the bipartite limit, vanishes on factorized states, and was proposed together with a holographic dual in terms of a minimal surface web (Yuan et al., 2024). It belongs to a broader lineage of multipartite reflected-entropy constructions in holography, but it is conceptually distinct from earlier multipartite reflected entropies such as $\Delta_R$ and from cyclic-gluing candidates tied directly to multipartite entanglement wedge cross-sections (Chu et al., 2019, Bao et al., 2019).

1. Genealogy within the reflected-entropy program

Multipartite extensions of reflected-entropy ideas appeared before the specific notion of reflected multi-entropy. One line of work proposed a multipartite analogue of reflected entropy by cyclically gluing multiple copies of the bulk geometry and then canonically purifying the result. In that construction, the leading-order holographic behavior is

$S_R(A_1:\ldots:A_n)=I(n)\,E_W(A_1:\ldots:A_n),$

with candidate relations

$$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n \text{ even},\ 4E_W(A_1:\ldots:A_n), & n \text{ odd}, \end{cases}$$

for one candidate, and

$$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n>2,\ 4E_W(A_1:\ldots:A_n), & n=2, \end{cases}$$

for another (Bao et al., 2019). These proposals were designed so that the bulk dual minimal surface is built from multipartite entanglement wedge cross-section segments.

A second line of work generalized reflected entropy through repeated canonical purification. For three parties it introduced the symmetric quantity $\Delta_R(A:B:C)$, defined from an 8-copy purified state, and conjectured the holographic identification

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

with $\Delta_W$ the multipartite entanglement wedge cross section. In the same framework, $\ket{\sqrt{\rho_{ABC}}}$0 obeys

$\ket{\sqrt{\rho_{ABC}}}$1

for pure tripartite states and satisfies bounds such as

$\ket{\sqrt{\rho_{ABC}}}$2

(Chu et al., 2019).

Reflected multi-entropy differs from both of these earlier constructions. Rather than defining a new multipartite entropy directly from a multipartite purification hierarchy or from bulk gluing alone, it imports the already-defined pure-state multi-entropy into the mixed-state setting by canonical purification. This gives it a direct reduction to multi-entropy on pure states and to reflected entropy at $\ket{\sqrt{\rho_{ABC}}}$3 (Yuan et al., 2024).

2. Definition through canonical purification and multi-entropy

The basic tripartite setup begins with a pure state on a larger Hilbert space,

$\ket{\sqrt{\rho_{ABC}}}$4

and traces out the auxiliary degrees of freedom $\ket{\sqrt{\rho_{ABC}}}$5 to obtain

$\ket{\sqrt{\rho_{ABC}}}$6

Canonical purification doubles the Hilbert space and produces

$\ket{\sqrt{\rho_{ABC}}}$7

after which the reflected multi-entropy is defined as the multi-entropy of the purified state (Yuan et al., 2024).

The construction was presented with three consistency conditions. For the pure-state limit,

$\ket{\sqrt{\rho_{ABC}}}$8

For the bipartite limit,

$\ket{\sqrt{\rho_{ABC}}}$9

For a factorized state,

$$S_R^{(\mathtt{q}=3)}(A;B;C)=S^{(\mathtt{q}=3)}(AA^*;BB^*;CC^*)_{\ket{\sqrt{\rho_{ABC}}}}.$$0

(Yuan et al., 2024).

The same source emphasizes a structural distinction from the original multi-entropy: unlike the original multi-entropy, reflected multi-entropy is described as UV finite / intrinsically well-defined because canonical purification ties the construction to the mixed state rather than to arbitrary endpoint-dependent divergences (Yuan et al., 2024). The paper also states lower bounds

$$S_R^{(\mathtt{q}=3)}(A;B;C)=S^{(\mathtt{q}=3)}(AA^*;BB^*;CC^*)_{\ket{\sqrt{\rho_{ABC}}}}.$$1

where $$S_R^{(\mathtt{q}=3)}(A;B;C)=S^{(\mathtt{q}=3)}(AA^*;BB^*;CC^*)_{\ket{\sqrt{\rho_{ABC}}}}.$$2 is the multipartite entanglement of purification, and, for holographic states,

$$S_R^{(\mathtt{q}=3)}(A;B;C)=S^{(\mathtt{q}=3)}(AA^*;BB^*;CC^*)_{\ket{\sqrt{\rho_{ABC}}}}.$$3

which places reflected multi-entropy above an earlier multipartite reflected-entropy quantity in the holographic hierarchy (Yuan et al., 2024).

A later formalization extended the definition to general $$S_R^{(\mathtt{q}=3)}(A;B;C)=S^{(\mathtt{q}=3)}(AA^*;BB^*;CC^*)_{\ket{\sqrt{\rho_{ABC}}}}.$$4. In that notation, one starts from a pure $$S_R^{(\mathtt{q}=3)}(A;B;C)=S^{(\mathtt{q}=3)}(AA^*;BB^*;CC^*)_{\ket{\sqrt{\rho_{ABC}}}}.$$5-partite state, traces out one subsystem to obtain a reduced $$S_R^{(\mathtt{q}=3)}(A;B;C)=S^{(\mathtt{q}=3)}(AA^*;BB^*;CC^*)_{\ket{\sqrt{\rho_{ABC}}}}.$$6-partite mixed state, canonically purifies $$S_R^{(\mathtt{q}=3)}(A;B;C)=S^{(\mathtt{q}=3)}(AA^*;BB^*;CC^*)_{\ket{\sqrt{\rho_{ABC}}}}.$$7, and applies the $$S_R^{(\mathtt{q}=3)}(A;B;C)=S^{(\mathtt{q}=3)}(AA^*;BB^*;CC^*)_{\ket{\sqrt{\rho_{ABC}}}}.$$8-partite multi-entropy to the purified state. The reflected multi-entropy is then

$$S_R^{(\mathtt{q}=3)}(A;B;C)=S^{(\mathtt{q}=3)}(AA^*;BB^*;CC^*)_{\ket{\sqrt{\rho_{ABC}}}}.$$9

with the bipartite case reducing to the usual reflected entropy (Iizuka et al., 21 Jul 2025).

3. Replica construction and tripartite field-theory realization

For pure multipartite states, multi-entropy is defined by replica permutations. In the tripartite case $\Delta_R$0, the relevant replica geometry is a square lattice of $\Delta_R$1 copies with $\Delta_R$2 symmetry. Reflected multi-entropy introduces an additional replica index $\Delta_R$3 from canonical purification and defines a reflected Rényi quantity by a partition function $\Delta_R$4, followed by the limits $\Delta_R$5 and $\Delta_R$6 (Yuan et al., 2024).

For $\Delta_R$7, the CFT object is a six-point function of twist operators,

$\Delta_R$8

with twist conformal weights

$\Delta_R$9

and dominant fusion-channel operators of weight

$S_R(A_1:\ldots:A_n)=I(n)\,E_W(A_1:\ldots:A_n),$0

For $S_R(A_1:\ldots:A_n)=I(n)\,E_W(A_1:\ldots:A_n),$1, this becomes

$S_R(A_1:\ldots:A_n)=I(n)\,E_W(A_1:\ldots:A_n),$2

(Yuan et al., 2024).

The large-$S_R(A_1:\ldots:A_n)=I(n)\,E_W(A_1:\ldots:A_n),$3 field-theory analysis uses a semiclassical Virasoro block

$S_R(A_1:\ldots:A_n)=I(n)\,E_W(A_1:\ldots:A_n),$4

and the monodromy equation

$S_R(A_1:\ldots:A_n)=I(n)\,E_W(A_1:\ldots:A_n),$5

with accessory parameters fixed by the asymptotic conditions at infinity and monodromy constraints

$S_R(A_1:\ldots:A_n)=I(n)\,E_W(A_1:\ldots:A_n),$6

Derivatives of the reflected multi-entropy are then extracted from

$S_R(A_1:\ldots:A_n)=I(n)\,E_W(A_1:\ldots:A_n),$7

(Yuan et al., 2024).

This six-point-twist realization is significant because it shows that reflected multi-entropy is not merely a formal transplant of canonical purification into a multipartite setting. It admits a concrete large-$S_R(A_1:\ldots:A_n)=I(n)\,E_W(A_1:\ldots:A_n),$8 CFT computation parallel to the bipartite reflected-entropy program, but with a genuinely tripartite replica structure (Yuan et al., 2024).

4. Holographic dual: minimal surface webs and explicit checks

The proposed holographic dual of reflected multi-entropy is a minimal surface web $S_R(A_1:\ldots:A_n)=I(n)\,E_W(A_1:\ldots:A_n),$9. For a tripartite mixed state $$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n \text{ even},\ 4E_W(A_1:\ldots:A_n), & n \text{ odd}, \end{cases}$$0, one begins with the entanglement wedge of $$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n \text{ even},\ 4E_W(A_1:\ldots:A_n), & n \text{ odd}, \end{cases}$$1, duplicates it, and glues the two copies along the RT surface $$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n \text{ even},\ 4E_W(A_1:\ldots:A_n), & n \text{ odd}, \end{cases}$$2. The resulting bulk geometry is taken to be the dual of the canonical purification $$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n \text{ even},\ 4E_W(A_1:\ldots:A_n), & n \text{ odd}, \end{cases}$$3. The reflected multi-entropy is then dual to a minimal codimension-2 web satisfying two conditions: $$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n \text{ even},\ 4E_W(A_1:\ldots:A_n), & n \text{ odd}, \end{cases}$$4 is anchored at $$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n \text{ even},\ 4E_W(A_1:\ldots:A_n), & n \text{ odd}, \end{cases}$$5, and $$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n \text{ even},\ 4E_W(A_1:\ldots:A_n), & n \text{ odd}, \end{cases}$$6 contains sub-webs homologous to all subsystems $$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n \text{ even},\ 4E_W(A_1:\ldots:A_n), & n \text{ odd}, \end{cases}$$7. The proposed formula is

$$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n \text{ even},\ 4E_W(A_1:\ldots:A_n), & n \text{ odd}, \end{cases}$$8

where $$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n \text{ even},\ 4E_W(A_1:\ldots:A_n), & n \text{ odd}, \end{cases}$$9 is the length or area of the web in one copy of the geometry (Yuan et al., 2024).

At zero temperature, the explicit check uses three intervals in the Poincaré half-plane,

$$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n>2,\ 4E_W(A_1:\ldots:A_n), & n=2, \end{cases}$$0

with symmetry parameters

$$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n>2,\ 4E_W(A_1:\ldots:A_n), & n=2, \end{cases}$$1

Using the bulk geodesic distance

$$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n>2,\ 4E_W(A_1:\ldots:A_n), & n=2, \end{cases}$$2

the web length is

$$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n>2,\ 4E_W(A_1:\ldots:A_n), & n=2, \end{cases}$$3

The numerical derivatives of the CFT result match those from

$$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n>2,\ 4E_W(A_1:\ldots:A_n), & n=2, \end{cases}$$4

(Yuan et al., 2024).

At finite temperature, the bulk dual is the BTZ black hole,

$$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n>2,\ 4E_W(A_1:\ldots:A_n), & n=2, \end{cases}$$5

with boundary temperature

$$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n>2,\ 4E_W(A_1:\ldots:A_n), & n=2, \end{cases}$$6

For intervals

$$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n>2,\ 4E_W(A_1:\ldots:A_n), & n=2, \end{cases}$$7

and $$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n>2,\ 4E_W(A_1:\ldots:A_n), & n=2, \end{cases}$$8, minimization gives

$$S_R(A_1:\ldots:A_n)= \begin{cases} 2E_W(A_1:\ldots:A_n), & n>2,\ 4E_W(A_1:\ldots:A_n), & n=2, \end{cases}$$9

and

$\Delta_R(A:B:C)$0

The derivative of the large-$\Delta_R(A:B:C)$1 CFT result again agrees precisely with the holographic derivative from $\Delta_R(A:B:C)$2 (Yuan et al., 2024).

5. Relation to multipartite reflected entropy, islands, and later extensions

Reflected multi-entropy coexists with, rather than replaces, earlier multipartite reflected entropies. In the 2024 construction it is explicitly compared with $\Delta_R(A:B:C)$3, the symmetric multipartite reflected entropy from earlier work, through the bound

$\Delta_R(A:B:C)$4

for holographic states (Yuan et al., 2024). The comparison is informative because $\Delta_R(A:B:C)$5 is built from repeated purification and special bipartitions of the final purified state, whereas reflected multi-entropy is obtained by applying the independent multi-entropy construction to the canonical purification (Chu et al., 2019, Yuan et al., 2024).

Island physics has also been linked to this broader program. The island formula proposed for ordinary reflected entropy states

$\Delta_R(A:B:C)$6

together with a covariant maximin form. That work explicitly notes that multipartite generalizations of reflected entropy already exist in the literature and expects the same island logic to extend to reflected multi-entropy: one should first include islands in the relevant entanglement wedge and then apply the multipartite reflected-entropy construction (Chandrasekaran et al., 2020). This is an expectation rather than a completed formula for reflected multi-entropy itself.

A subsequent development generalized the Markov-gap program to multipartite systems by using reflected multi-entropy. The resulting multipartite Markov gap was defined as reflected multi-entropy minus the natural entropic lower bound, and a further refinement called genuine reflected multi-entropy was introduced to vanish for states containing only lower-partite entanglement (Iizuka et al., 21 Jul 2025). In that work, reflected multi-entropy is interpreted holographically in terms of minimal multiway cuts, extending the bipartite relation between reflected entropy and entanglement wedge cross-sections (Iizuka et al., 21 Jul 2025).

6. Interpretive status, cautions, and open directions

The original reflected multi-entropy paper presents the quantity as a natural mixed-state multipartite measure with a clean holographic dual and explicit large-$\Delta_R(A:B:C)$7 checks (Yuan et al., 2024). At the same time, the broader reflected-entropy literature provides a cautionary backdrop. Ordinary reflected entropy and the Rényi reflected entropies $\Delta_R(A:B:C)$8 for $\Delta_R(A:B:C)$9 are not monotone under partial trace in general; explicit counterexamples exist already for classical probability distributions, and reflected entropy therefore fails to be a universal correlation measure in that regime (Hayden et al., 2023). No general monotonicity theorem for reflected multi-entropy is supplied in the cited reflected-multi-entropy sources. This suggests caution in extrapolating bipartite intuitions about “correlation measures” to the multipartite setting.

The 2024 construction also leaves several directions open. The authors explicitly mention multiple canonical purifications, time-dependent or covariant generalization in which the dual would be a minimal extremal surface web, and quantum corrections analogous to bulk quantum corrections in holographic entanglement entropy (Yuan et al., 2024). These are natural next steps because the present formulation is classical on the bulk side and is developed mainly in static settings.

Within the literature assembled so far, reflected multi-entropy is therefore best understood as a specific canonical-purification lift of pure-state multi-entropy to mixed multipartite states, supported by exact zero- and finite-temperature checks in AdS$\Delta_R(A:B:C)=2\,\Delta_W(A:B:C),$0/CFT$\Delta_R(A:B:C)=2\,\Delta_W(A:B:C),$1, structurally related to—but not identical with—earlier multipartite reflected entropies, and already being used as an input to multipartite Markov-gap and genuine multipartite-entanglement diagnostics (Yuan et al., 2024, Iizuka et al., 21 Jul 2025).