arXiv:2605.26617 · Physical review. D/Physical review. D.

On a mixed-state extension of the holographic signal inequality

Joydeep Naskar

hep-thquant-phSubmitted May 26, 2026• Announced May 27, 2026• 0 citations• Revised Jun 3, 2026 (v2)

Abstract

A novel inequality involving the residual entropy and genuine multi-entropy was proposed in \cite{Balasubramanian:2025hxg} for tripartite holographic pure states, using which it was argued, that purely GHZ-like tripartite entanglement is not allowed in holography. In this work, we generalize this holographic signal inequality to mixed states. In a minimal extension, we compute the reflected genuine multi-entropy following \cite{Yuan:2024yfg} and find a class of holographic geometries that violate this minimally extended inequality due to vanishing Markov gap. We can symmetrize this prescription, where instead of computing the residual entropy on the given mixed state $ρ_{ABC}$, we compute it on its canonical purification. The inequality is restored on the canonically purified state, as expected. Finally, we conjecture a new inequality for tripartite holographic states and give supporting evidence.

Comments: v1 5 pages, 6 figures; v2 6 pages 7 figures

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arXiv:2605.26617 · Physical review. D/Physical review. D.

On a mixed-state extension of the holographic signal inequality

Joydeep Naskar

hep-thquant-phSubmitted May 26, 2026• Announced May 27, 2026• 0 citations• Revised Jun 3, 2026 (v2)

arXiv HTML PDF

Abstract

ρ_{ABC}

, we compute it on its canonical purification. The inequality is restored on the canonically purified state, as expected. Finally, we conjecture a new inequality for tripartite holographic states and give supporting evidence.

Comments: v1 5 pages, 6 figures; v2 6 pages 7 figures

Related papers

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Ning Bao, Illan F. Halpern

A bstract We study the conjectured holographic duality between entanglement of purification and the entanglement wedge cross-section. We generalize both quantities and prove several information theoretic inequalities involving them. These include upper bounds on conditional mutual information and tripartite information, as well as a lower bound for tripartite information. These inequalities are proven both holographically and for general quantum states. In addition, we use the cyclic entropy inequalities to derive a new holographic inequality for the entanglement wedge cross-section, and provide numerical evidence that the corresponding inequality for the entanglement of purification may be true in general. Finally, we use intuition from bit threads to extend the conjecture to holographic duals of suboptimal purifications.A bstract We study the conjectured holographic duality between entanglement of purification and the entanglement wedge cross-section. We generalize both quantities and prove several information theoretic inequalities involving them. These include upper bounds on conditional mutual information and tripartite information, as well as a lower bound for tripartite information. These inequalities are proven both holographically and for general quantum states. In addition, we use the cyclic entropy inequalities to derive a new holographic inequality for the entanglement wedge cross-section, and provide numerical evidence that the corresponding inequality for the entanglement of purification may be true in general. Finally, we use intuition from bit threads to extend the conjecture to holographic duals of suboptimal purifications.

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In this paper, we holographically quantify the entanglement and complexity for mixed states by following the prescription of purification. The bulk theory we consider in this work is a hyperscaling violating solution, characterized by two parameters, hyperscaling violating exponent $\ensuremath{\theta}$ and dynamical exponent $z$ . This geometry is due to a nonrelativistic strongly coupled theory with hidden Fermi surfaces. We first compute the holographic analogy of entanglement of purification, denoted as the minimal area of the entanglement wedge cross section and observe the effects of $z$ and $\ensuremath{\theta}$. Then in order to probe the mixed state complexity we compute the mutual complexity for the Banados Teitelboim Zanelli (BTZ) black hole and the hyperscaling violating geometry by incorporating the holographic subregion complexity conjecture. We carry this out for two disjoint subsystems separated by a distance and also when the subsystems are adjacent with subsystems making up the full system. Furthermore, various aspects of holographic entanglement entropy such as the entanglement Smarr relation, the Fisher information metric, and the butterfly velocity have also been discussed.In this paper, we holographically quantify the entanglement and complexity for mixed states by following the prescription of purification. The bulk theory we consider in this work is a hyperscaling violating solution, characterized by two parameters, hyperscaling violating exponent $\ensuremath{\theta}$ and dynamical exponent $z$. This geometry is due to a nonrelativistic strongly coupled theory with hidden Fermi surfaces. We first compute the holographic analogy of entanglement of purification, denoted as the minimal area of the entanglement wedge cross section and observe the effects of $z$ and $\ensuremath{\theta}$. Then in order to probe the mixed state complexity we compute the mutual complexity for the Banados Teitelboim Zanelli (BTZ) black hole and the hyperscaling violating geometry by incorporating the holographic subregion complexity conjecture. We carry this out for two disjoint subsystems separated by a distance and also when the subsystems are adjacent with subsystems making up the full system. Furthermore, various aspects of holographic entanglement entropy such as the entanglement Smarr relation, the Fisher information metric, and the butterfly velocity have also been discussed.

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Tripartite Form Universality in Holographic Entropy Inequalities

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To elucidate the meaning of holographic entropy inequalities (beyond subadditivity) which characterize the entanglement structure of geometric states in holography, arXiv:2309.06296 proposed the "tripartite form" for these inequalities, consisting of tripartite information and conditional tripartite information terms with unit coefficients. While this provides a compact and useful packaging of the inequalities, it is not a priori guaranteed that all inequalities can be recast in this form. Here we conjecture that they can, and present substantial evidence, by proving that the two known infinite families of holographic entropy inequalities found in arXiv:2309.15145 can indeed be written in the tripartite form. This is significant because such recasting is particularly nontrivial for these families. Apart from providing the explicit tripartite form expressions for every member of these two infinite families, we detail how we arrived at them, in the process deriving several useful identities which may serve as stepping stones to formulate further repackaging of the corresponding information quantities. We also illustrate the power of the tripartite form by proving a number of structural properties satisfied by any holographic entropy inequality.To elucidate the meaning of holographic entropy inequalities (beyond subadditivity) which characterize the entanglement structure of geometric states in holography, arXiv:2309.06296 proposed the "tripartite form" for these inequalities, consisting of tripartite information and conditional tripartite information terms with unit coefficients. While this provides a compact and useful packaging of the inequalities, it is not a priori guaranteed that all inequalities can be recast in this form. Here we conjecture that they can, and present substantial evidence, by proving that the two known infinite families of holographic entropy inequalities found in arXiv:2309.15145 can indeed be written in the tripartite form. This is significant because such recasting is particularly nontrivial for these families. Apart from providing the explicit tripartite form expressions for every member of these two infinite families, we detail how we arrived at them, in the process deriving several useful identities which may serve as stepping stones to formulate further repackaging of the corresponding information quantities. We also illustrate the power of the tripartite form by proving a number of structural properties satisfied by any holographic entropy inequality.

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Tripartite correlation signal from multipartite entanglement of purification

Ning Bao, Keiichiro Furuya, Joydeep Naskar

A bstract We propose a signal ${\Delta }_p^{(3)}$ ∆ p 3 for tripartite correlations in finite-dimensional quantum systems and ${\Delta }_w^{(3)}$ ∆ w 3 for holographic systems. We prove that ${\Delta }_p^{(3)}$ ∆ p 3 is non-negative for any tripartite entangled mixed states and vanishes for mixed product states. It is a correlation signal because it is generally nonzero for a separable state containing a classical mixture. Based on the conjecture, the equality between an entanglement wedge cross section E w and entanglement of purification E p , i.e., E w = E P in the semiclassical limit, we apply the tripartite correlation signal to study the structures of tripartite entanglement in AdS 3 /CFT 2 , especially for pure AdS 3 . We comment on a generalization to n -partite correlation signals ${\Delta}_p^{(n)}\left({A}_1:\cdots :{A}_n\right)$ Δ p n A 1 : ⋯ : A n .A bstract We propose a signal $$ {\Delta }_p^{(3)} $$ ∆ p 3 for tripartite correlations in finite-dimensional quantum systems and $$ {\Delta }_w^{(3)} $$ ∆ w 3 for holographic systems. We prove that $$ {\Delta }_p^{(3)} $$ ∆ p 3 is non-negative for any tripartite entangled mixed states and vanishes for mixed product states. It is a correlation signal because it is generally nonzero for a separable state containing a classical mixture. Based on the conjecture, the equality between an entanglement wedge cross section E w and entanglement of purification E p , i.e., E w = E P in the semiclassical limit, we apply the tripartite correlation signal to study the structures of tripartite entanglement in AdS 3 /CFT 2 , especially for pure AdS 3 . We comment on a generalization to n -partite correlation signals $$ {\Delta}_p^{(n)}\left({A}_1:\cdots :{A}_n\right) $$ Δ p n A 1 : ⋯ : A n .

arXiv:2509.08209Journal of High Energy PhysicsMay 20, 20262 citations

ρ_{ABC}

z

. This geometry is due to a nonrelativistic strongly coupled theory with hidden Fermi surfaces. We first compute the holographic analogy of entanglement of purification, denoted as the minimal area of the entanglement wedge cross section and observe the effects of

z

and $\ensuremath{\theta}$. Then in order to probe the mixed state complexity we compute the mutual complexity for the Banados Teitelboim Zanelli (BTZ) black hole and the hyperscaling violating geometry by incorporating the holographic subregion complexity conjecture. We carry this out for two disjoint subsystems separated by a distance and also when the subsystems are adjacent with subsystems making up the full system. Furthermore, various aspects of holographic entanglement entropy such as the entanglement Smarr relation, the Fisher information metric, and the butterfly velocity have also been discussed.

We show that in pure time-reflection-symmetric holographic states several known four-party entanglement signals vanish unless the triple information

I_3

is non-zero. In this sense, our results show that

I_3

is the strongest known signal of the presence of quadripartite entanglement. Additionally,

I_3

quantitatively bounds all four-party entanglement signals built from the multi-entropy. However, the residual entropy

Q_4

, also a measure of four-party entanglement, is not bounded by

I_3

, although

I_3=0

does imply

Q_4=0

for holographic states (except on a set of measure zero for which

Q_4

is ill-defined).

A bstract We propose a signal

{\Delta }_p^{(3)}

∆ p 3 for tripartite correlations in finite-dimensional quantum systems and

{\Delta }_w^{(3)}

∆ w 3 for holographic systems. We prove that

{\Delta }_p^{(3)}

∆ p 3 is non-negative for any tripartite entangled mixed states and vanishes for mixed product states. It is a correlation signal because it is generally nonzero for a separable state containing a classical mixture. Based on the conjecture, the equality between an entanglement wedge cross section E w and entanglement of purification E p , i.e., E w = E P in the semiclassical limit, we apply the tripartite correlation signal to study the structures of tripartite entanglement in AdS 3 /CFT 2 , especially for pure AdS 3 . We comment on a generalization to n -partite correlation signals

{\Delta}_p^{(n)}\left({A}_1:\cdots :{A}_n\right)

Δ p n A 1 : ⋯ : A n .