Multi-Entropy in Quantum Information

• Multi-entropy is a replica-based generalization of entanglement entropy that quantifies genuine multipartite entanglement in quantum states.
• Replica constructions define multi-entropy through symmetric contractions of replicated density matrices, naturally reducing to Rényi entropy in bipartite cases.
• Multi-entropy plays a key role in holography, quantum criticality, and network science, offering insights into quantum geometries and complex dynamics.

Searching arXiv for recent multi-entropy papers and closely related variants. Multi-entropy is a replica-based generalization of entanglement entropy to multipartite quantum states. In the quantum-information and holography literature, it is defined from symmetric contractions of replicated density matrices or wavefunction amplitudes, is invariant under local unitaries, reduces to the ordinary Rényi entropy when the number of parties is two, and has been developed as a probe of genuinely multipartite entanglement, mixed-state correlations, holographic geometry, quantum criticality, topological order, and black hole evaporation (Gadde et al., 2022, Iizuka et al., 11 Feb 2025, Berthière et al., 30 Aug 2025).

1. Formal definitions and replica constructions

The literature uses several closely related normalizations. For a $q$-partite pure state $|\psi\rangle$ on $$\mathcal H_{A_1}\otimes\cdots\otimes \mathcal H_{A_q}$$, one commonly used definition is

$$S^{(q)}_n(A_1:\cdots:A_q) := \frac{1}{1-n}\,\frac{1}{n^{q-2}} \log\!\left[ \frac{Z^{(q)}_n}{(Z^{(q)}_1)^{n^{q-1}}} \right],$$

with

$$Z^{(q)}_n := \langle\psi|^{\otimes n^{q-1}} \Sigma_1(g_1)\cdots\Sigma_q(g_q) |\psi\rangle^{\otimes n^{q-1}},$$

where the permutations $g_k$ act as cyclic shifts on replica labels arranged on a $(q-1)$-dimensional hypercube of side $n$, and $g_q=e$ is the identity. An earlier normalization is written in terms of a symmetric invariant $$I_n^{(m)}=\mathrm{Tr}\!\left[\rho^{\otimes n^{m-1}}P\right]$$ and

$|\psi\rangle$0

In both conventions, the $|\psi\rangle$1 case reduces to the ordinary Rényi entropy, and the construction is symmetric under exchange of parties; the cited literature also records local-unitary invariance and additivity on product states (Gadde et al., 2022, Iizuka et al., 11 Feb 2025).

For tripartite pure states, the replica action is particularly explicit. One may arrange the $|\psi\rangle$2 replicas on an $|\psi\rangle$3 grid and define

$|\psi\rangle$4

with

$|\psi\rangle$5

Here $|\psi\rangle$6 cycles vertically within each column, $|\psi\rangle$7 cycles horizontally within each row, and $|\psi\rangle$8 is the identity. For $|\psi\rangle$9, the resulting quantity is already closely related to Rényi reflected entropy and CCNR negativities (Berthière et al., 30 Aug 2025).

The same replica language extends beyond purely algebraic definitions to field theory. In two-dimensional conformal field theory, multi-entropy is computed from twist operators whose monodromies lie in finite groups such as $$\mathcal H_{A_1}\otimes\cdots\otimes \mathcal H_{A_q}$$0 rather than the single cyclic group used for ordinary Rényi entropy. The corresponding twist fields have identical cycle structure and conformal dimension

$$\mathcal H_{A_1}\otimes\cdots\otimes \mathcal H_{A_q}$$1

and the replica geometry is a branched cover whose genus is fixed by the monodromy data through the Riemann-Hurwitz relation (Harper et al., 2024).

2. Genuine multi-entropy and adjacent constructions

A central refinement is the genuine multi-entropy, designed to remove lower-arity contributions. For three parties, the standard definition is

$$\mathcal H_{A_1}\otimes\cdots\otimes \mathcal H_{A_q}$$2

or, in the notation of the Lifshitz analysis,

$$\mathcal H_{A_1}\otimes\cdots\otimes \mathcal H_{A_q}$$3

The subtraction is intended to remove averaged bipartite contributions and UV-sensitive terms, so that the resulting quantity vanishes on states whose entanglement order is less than $$\mathcal H_{A_1}\otimes\cdots\otimes \mathcal H_{A_q}$$4; for $$\mathcal H_{A_1}\otimes\cdots\otimes \mathcal H_{A_q}$$5 this construction is unique, while for $$\mathcal H_{A_1}\otimes\cdots\otimes \mathcal H_{A_q}$$6 the literature records a family of genuine combinations satisfying linear constraints such as $$\mathcal H_{A_1}\otimes\cdots\otimes \mathcal H_{A_q}$$7 (Iizuka et al., 11 Feb 2025, Berthière et al., 30 Aug 2025).

The vanishing property is explicit in benchmark states. For the tripartite triangle state

$$\mathcal H_{A_1}\otimes\cdots\otimes \mathcal H_{A_q}$$8

one has

$$\mathcal H_{A_1}\otimes\cdots\otimes \mathcal H_{A_q}$$9

hence $$S^{(q)}_n(A_1:\cdots:A_q) := \frac{1}{1-n}\,\frac{1}{n^{q-2}} \log\!\left[ \frac{Z^{(q)}_n}{(Z^{(q)}_1)^{n^{q-1}}} \right],$$0 for all $$S^{(q)}_n(A_1:\cdots:A_q) := \frac{1}{1-n}\,\frac{1}{n^{q-2}} \log\!\left[ \frac{Z^{(q)}_n}{(Z^{(q)}_1)^{n^{q-1}}} \right],$$1. By contrast, the cited literature reports $$S^{(q)}_n(A_1:\cdots:A_q) := \frac{1}{1-n}\,\frac{1}{n^{q-2}} \log\!\left[ \frac{Z^{(q)}_n}{(Z^{(q)}_1)^{n^{q-1}}} \right],$$2 in the von Neumann limit, $$S^{(q)}_n(A_1:\cdots:A_q) := \frac{1}{1-n}\,\frac{1}{n^{q-2}} \log\!\left[ \frac{Z^{(q)}_n}{(Z^{(q)}_1)^{n^{q-1}}} \right],$$3, and $$S^{(q)}_n(A_1:\cdots:A_q) := \frac{1}{1-n}\,\frac{1}{n^{q-2}} \log\!\left[ \frac{Z^{(q)}_n}{(Z^{(q)}_1)^{n^{q-1}}} \right],$$4; this separates genuine tripartite entanglement from purely pairwise constructions (Iizuka et al., 11 Feb 2025).

A second line of development relates multi-entropy to mixed-state quantities. For tripartitions at $$S^{(q)}_n(A_1:\cdots:A_q) := \frac{1}{1-n}\,\frac{1}{n^{q-2}} \log\!\left[ \frac{Z^{(q)}_n}{(Z^{(q)}_1)^{n^{q-1}}} \right],$$5,

$$S^{(q)}_n(A_1:\cdots:A_q) := \frac{1}{1-n}\,\frac{1}{n^{q-2}} \log\!\left[ \frac{Z^{(q)}_n}{(Z^{(q)}_1)^{n^{q-1}}} \right],$$6

where $$S^{(q)}_n(A_1:\cdots:A_q) := \frac{1}{1-n}\,\frac{1}{n^{q-2}} \log\!\left[ \frac{Z^{(q)}_n}{(Z^{(q)}_1)^{n^{q-1}}} \right],$$7 is the realignment of the reduced density matrix. The associated quantity satisfies

$$S^{(q)}_n(A_1:\cdots:A_q) := \frac{1}{1-n}\,\frac{1}{n^{q-2}} \log\!\left[ \frac{Z^{(q)}_n}{(Z^{(q)}_1)^{n^{q-1}}} \right],$$8

so $$S^{(q)}_n(A_1:\cdots:A_q) := \frac{1}{1-n}\,\frac{1}{n^{q-2}} \log\!\left[ \frac{Z^{(q)}_n}{(Z^{(q)}_1)^{n^{q-1}}} \right],$$9. More generally, reflected multi-entropy is defined by taking the multi-entropy of the canonical purification,

$$Z^{(q)}_n := \langle\psi|^{\otimes n^{q-1}} \Sigma_1(g_1)\cdots\Sigma_q(g_q) |\psi\rangle^{\otimes n^{q-1}},$$0

with the properties that it reduces to $$Z^{(q)}_n := \langle\psi|^{\otimes n^{q-1}} \Sigma_1(g_1)\cdots\Sigma_q(g_q) |\psi\rangle^{\otimes n^{q-1}},$$1 on pure states, reduces to reflected entropy at $$Z^{(q)}_n := \langle\psi|^{\otimes n^{q-1}} \Sigma_1(g_1)\cdots\Sigma_q(g_q) |\psi\rangle^{\otimes n^{q-1}},$$2, and vanishes on factorized states (Liu et al., 2024, Yuan et al., 2024).

The dihedral invariants form a further related family. Using $$Z^{(q)}_n := \langle\psi|^{\otimes n^{q-1}} \Sigma_1(g_1)\cdots\Sigma_q(g_q) |\psi\rangle^{\otimes n^{q-1}},$$3 replicas and the dihedral group $$Z^{(q)}_n := \langle\psi|^{\otimes n^{q-1}} \Sigma_1(g_1)\cdots\Sigma_q(g_q) |\psi\rangle^{\otimes n^{q-1}},$$4, one defines

$$Z^{(q)}_n := \langle\psi|^{\otimes n^{q-1}} \Sigma_1(g_1)\cdots\Sigma_q(g_q) |\psi\rangle^{\otimes n^{q-1}},$$5

For general tripartite pure states, the cited analysis shows

$$Z^{(q)}_n := \langle\psi|^{\otimes n^{q-1}} \Sigma_1(g_1)\cdots\Sigma_q(g_q) |\psi\rangle^{\otimes n^{q-1}},$$6

while the unnormalized dihedral partition function gives the Rényi CCNR negativity. The same work also proves that for $$Z^{(q)}_n := \langle\psi|^{\otimes n^{q-1}} \Sigma_1(g_1)\cdots\Sigma_q(g_q) |\psi\rangle^{\otimes n^{q-1}},$$7 the family $$Z^{(q)}_n := \langle\psi|^{\otimes n^{q-1}} \Sigma_1(g_1)\cdots\Sigma_q(g_q) |\psi\rangle^{\otimes n^{q-1}},$$8 is not monotonically non-increasing under partial trace, so in that range it is not a correlation measure (Berthière et al., 30 Aug 2025).

3. Exact results in critical, topological, and link-state systems

In $$Z^{(q)}_n := \langle\psi|^{\otimes n^{q-1}} \Sigma_1(g_1)\cdots\Sigma_q(g_q) |\psi\rangle^{\otimes n^{q-1}},$$9-dimensional critical systems, multi-entropy admits explicit conformal-field-theoretic forms. For three adjacent intervals,

$g_k$0

and the constant $g_k$1 is universal at fixed theory and replica index. In the massless free scalar, the zero mode modifies the expected constant and gives $g_k$2, with new lattice-supported predictions $g_k$3 and $g_k$4. In the transverse-field Ising model, finite-size scaling reproduces $g_k$5 with $g_k$6. The same work gives the dihedral excess

$g_k$7

for $g_k$8 CFTs with discrete spectra, while again noting zero-mode-induced deviations in the free boson (Harper et al., 12 Jun 2025).

Low-Rényi-index calculations in two-dimensional CFT also make the bulk geometry visible. For $g_k$9, $(q-1)$0, the connected three-point function takes the form

$(q-1)$1

so that

$(q-1)$2

The same analysis identifies a holographic corner term

$(q-1)$3

associated with trivalent geodesic intersections, and it records that the standard bosonization procedure for free fermions fails for multi-entropy twist operators, so a different treatment is required (Harper et al., 2024).

Lifshitz and Rokhsar-Kivelson ground states provide a distinct exactly solvable setting. For the $(q-1)$4 Lifshitz boson in $(q-1)$5 dimensions, the genuine tripartite multi-entropy admits an analytic continuation in $(q-1)$6 and the closed form

$(q-1)$7

with von Neumann limit

$(q-1)$8

In the studied geometries it is UV-finite, vanishes as one party shrinks to zero or as the separating region grows to infinity, reduces to a quantity proportional to $(q-1)$9 when the reduced state on $n$0 is separable, and satisfies $n$1 in the Lifshitz ground states examined there (Berthière et al., 30 Aug 2025).

Topological phases and Chern-Simons link states furnish two further exact settings. For $n$2-dimensional gapped ground states, the cited work shows

$n$3

and derives the lower bounds

$n$4

with $n$5 the central charge of ungappable edge degrees of freedom. For Abelian $n$6 three-component link states, the genuine tri-entropy has a closed form in terms of $n$7 combinations of the linking numbers, and at $n$8 it satisfies

$n$9

so it counts GHZ$g_q=e$0 triples in the stabilizer-state local-unitary normal form, whereas logarithmic negativity counts EPR pairs. In the same family, $g_q=e$1 (Liu et al., 2024, Yuan et al., 21 Oct 2025).

4. Holographic geometry and convex structures

The earliest holographic proposal identifies multi-entropy with the area of a minimal bulk “soap-film” web. In an $g_q=e$2-partite holographic CFT state, the leading-order prescription is

$g_q=e$3

where $g_q=e$4 is the globally minimal web anchored on the entangling boundaries and containing sub-webs homologous to each region. In static slices this web obeys Plateau-type rules: in $g_q=e$5, three sheets meet at $g_q=e$6 and four such lines meet at angle $g_q=e$7. The same work proposes a covariant extremal-web version and a quantum-corrected functional

$g_q=e$8

to be extremized by a quantum extremal soap-film (Gadde et al., 2022).

Genuine multi-entropy sharpens this picture. Under the assumptions that the $g_q=e$9 multi-entropy has a holographic area prescription and that the boundary subregions are connected so a bulk multiway junction exists, the cited holographic analysis proves

$$I_n^{(m)}=\mathrm{Tr}\!\left[\rho^{\otimes n^{m-1}}P\right]$$0

with strict positivity when the bulk admits intersection surfaces. In these connected configurations, non-bipartite multi-entropies are $$I_n^{(m)}=\mathrm{Tr}\!\left[\rho^{\otimes n^{m-1}}P\right]$$1, which the same paper interprets as evidence that genuine multipartite entanglement is not small in holography (Iizuka et al., 11 Feb 2025).

Reflected multi-entropy has an analogous geometric dual. For holographic mixed states, canonical purification doubles the entanglement wedge along the RT surface, and the proposal is

$$I_n^{(m)}=\mathrm{Tr}\!\left[\rho^{\otimes n^{m-1}}P\right]$$2

where $$I_n^{(m)}=\mathrm{Tr}\!\left[\rho^{\otimes n^{m-1}}P\right]$$3 is a minimal codimension-2 surface web anchored at the RT surface of the union and containing sub-webs homologous to each subsystem. For the tripartite case in AdS$$I_n^{(m)}=\mathrm{Tr}\!\left[\rho^{\otimes n^{m-1}}P\right]$$4/CFT$$I_n^{(m)}=\mathrm{Tr}\!\left[\rho^{\otimes n^{m-1}}P\right]$$5, both zero-temperature and finite-temperature large-$$I_n^{(m)}=\mathrm{Tr}\!\left[\rho^{\otimes n^{m-1}}P\right]$$6 six-point twist calculations match the bulk web lengths exactly, including the regime change signaled by $$I_n^{(m)}=\mathrm{Tr}\!\left[\rho^{\otimes n^{m-1}}P\right]$$7 in the BTZ result (Yuan et al., 2024).

A more global holographic development is the holographic multi-entropy cone. In this framework, one adjoins partition-labeled multi-entropy coordinates to the usual bipartite entropy coordinates and obtains a cone $$I_n^{(m)}=\mathrm{Tr}\!\left[\rho^{\otimes n^{m-1}}P\right]$$8 in dimension $$I_n^{(m)}=\mathrm{Tr}\!\left[\rho^{\otimes n^{m-1}}P\right]$$9. The cited work proves that for fixed $|\psi\rangle$00 this cone is rational polyhedral, using a universal weighted-graph model and multicontraction maps as exact certificates of holographic multi-entropy inequalities. For $|\psi\rangle$01 it finds two facet orbits, including

$|\psi\rangle$02

and for $|\psi\rangle$03 it finds seven facet orbits, with monogamy of mutual information decomposing as

$|\psi\rangle$04

The same paper conjectures that HEC facet inequalities are convex combinations of HMEC facet inequalities and that primitive HMEC facets obey a balanced-but-not-too-balanced principle (Ju et al., 13 Jun 2026).

5. Dynamical, random-state, and symmetry-resolved behavior

In black hole applications, multi-entropy generalizes the Page curve by partitioning the Hawking radiation into finer subsystems. For a $|\psi\rangle$05-partite model with $|\psi\rangle$06 equal radiation factors of dimension $|\psi\rangle$07 and a black-hole factor of dimension $|\psi\rangle$08, with $|\psi\rangle$09 fixed, the black hole multi-entropy curve grows at early time, peaks at the multi-entropy time

$|\psi\rangle$10

which is later than the Page time $|\psi\rangle$11 for $|\psi\rangle$12, and then decreases. Unlike the bipartite Page curve, it does not decrease to zero at complete evaporation; the cited asymptotic value is

$|\psi\rangle$13

interpreted there as “secret entanglement” among Hawking particles (Iizuka et al., 2024).

A related random-state analysis of tripartite genuine multi-entropy concludes that, in an evaporating-black-hole model with $|\psi\rangle$14, the genuine quantity is essentially zero at early times, becomes nonzero only after the Page time, reaches a maximum at the multi-entropy time, and then decreases to zero as $|\psi\rangle$15. In the same model,

$|\psi\rangle$16

up to $|\psi\rangle$17 corrections, where $|\psi\rangle$18 is the generalized Rényi Markov gap. The same paper argues that genuine multi-entropy and the Markov gap diagnose different multipartite structures, since the former is fully symmetric in the three parties whereas the latter is a two-party quantity after tracing out the third (Iizuka et al., 11 Feb 2025).

Heavy local quenches in holographic $|\psi\rangle$19 CFTs exhibit a different dynamical mechanism. For adjacent intervals, the genuine tri-entropy is

$|\psi\rangle$20

A perturbative bulk analysis shows that the first-order small-mass correction cancels identically at all times,

$|\psi\rangle$21

because the vacuum trivalent network is extremal and the kernel identities match the bipartite subtraction. In the fully back-reacted geometry, however, a nonzero vacuum-subtracted genuine tri-entropy arises from a mismatch between the winding selected by the trivalent geodesic network and the windings selected independently by the pairwise geodesics. In the sharp-quench limit, its time dependence becomes a piecewise logarithm of rational functions of time, independent of the heavy operator dimension (Fujiki et al., 10 Jun 2026).

Symmetry resolution adds another layer. For Haar-random states with a global $|\psi\rangle$22 charge, the thermodynamic-limit multi-entropy keeps the same piecewise-linear dependence on subsystem sizes as in the fully Haar-random case, but every slope is rescaled by

$|\psi\rangle$23

For example, the tripartite genuine multi-entropy obeys

$|\psi\rangle$24

The same study finds that for random graph states $|\psi\rangle$25 for all graph states, while

$|\psi\rangle$26

so the $|\psi\rangle$27 quantity isolates GHZ content rather than generic Haar-like multipartite structure (Iizuka et al., 2 Nov 2025).

6. Terminological extensions beyond multipartite entanglement

Although the dominant recent usage of multi-entropy on arXiv is the replica-based multipartite entanglement quantity described above, the term also appears in distinct settings. In network science, a “multiscale entropy framework” called “Multi-Entropy” extends entropy-based structural analysis by spectral graph reduction. It tracks compression-based and link-prediction entropies across coarsening levels, identifies stable, increasing, and hybrid regimes, and improves predictability estimates relative to a single-scale model, with the cited regression improving from

$|\psi\rangle$28

when multiscale features are included (Brzovic et al., 13 Oct 2025).

In network intrusion detection, the related but distinct “Multi-Level Distributional Entropy” framework constructs entropy features from pre-aggregated flow statistics at three levels: within-flow Gaussian differential entropy, cross-directional Jensen-Shannon divergence, and TCP flag-pattern Shannon entropy. Across four datasets, the paper reports entropy-only weighted F1 in the range $|\psi\rangle$29-$|\psi\rangle$30 and emphasizes that high AUC or F1 can still conceal low detection rate under temporal shift or held-out attack families (Bouke et al., 29 Jun 2026).

In materials informatics, “multi-entropy” appears in the context of configurational stabilization of high-entropy compounds. The cited oxide study combines mixing-enthalpy descriptors with ideal configurational entropy,

$|\psi\rangle$31

and Monte Carlo simulations to rank candidate multi-component oxides and to predict temperature-dependent single-phase stability and secondary-phase fractions (Pitike et al., 2020).

These distinct uses share an information-theoretic vocabulary but not a common formalism. In current quantum-information, field-theoretic, and holographic research, multi-entropy refers specifically to the multipartite replica constructions, their genuine and reflected variants, and the associated geometric, algebraic, and dynamical structures developed across the works cited above.