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

Mixed-state extension of the holographic signal inequality

Joydeep Naskar

hep-thquant-phSubmitted May 26, 2026• Announced May 27, 2026• 0 citations• Revised Oct 27, 2026 (v3)
v1 · May 26, 2026v2 · Jun 3, 2026v3 · Oct 27, 2026
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Abstract

A novel inequality involving the residual information 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 a vanishing Markov gap. We can symmetrize this prescription, where instead of computing the residual information on the given mixed state $ρ_{ABC}$, we compute it on its canonical purification. The inequality is restored on the canonically purified state. 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; v3 PRD accepted version

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A novel inequality involving the residual information 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 a vanishing Markov gap. We can symmetrize this prescription, where instead of computing the residual information on the given mixed state ρABCρ_{ABC}ρABC​, we compute it on its canonical purification. The inequality is restored on the canonically purified state. Finally, we conjecture a new inequality for tripartite holographic states and give supporting evidence.
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.
Holographic states satisfy several entropic inequalities owing to the Ryu-Takayangi formula. A drawback of these inequalities is that they only use bipartite entanglement in their formulation. We investigate a recently proposed "GHZ-forbidding" inequality, built out of the reflected entropy and the tripartite multi-entropy, that holds for holographic states. We show that the inequality is either violated or saturated, but never strictly satisfied, by stabilizer states, thereby showing that stabilizer states are not holographic. As a consequence, we show that tripartite entanglement in the HaPPY code is not holographic.
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 zzz. 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 zzz 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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We show that in pure time-reflection-symmetric holographic states several known four-party entanglement signals vanish unless the triple information I3I_3I3​ is non-zero. In this sense, our results show that I3I_3I3​ is the strongest known signal of the presence of quadripartite entanglement. Additionally, I3I_3I3​ quantitatively bounds all four-party entanglement signals built from the multi-entropy. However, the residual entropy Q4Q_4Q4​, also a measure of four-party entanglement, is not bounded by I3I_3I3​, although I3=0I_3=0I3​=0 does imply Q4=0Q_4=0Q4​=0 for holographic states (except on a set of measure zero for which Q4Q_4Q4​ is ill-defined).
A bstract We propose a signal Δp(3) {\Delta }_p^{(3)} Δp(3)​ ∆ p 3 for tripartite correlations in finite-dimensional quantum systems and Δw(3) {\Delta }_w^{(3)} Δw(3)​ ∆ w 3 for holographic systems. We prove that Δp(3) {\Delta }_p^{(3)} Δp(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 Δp(n)(A1:⋯:An) {\Delta}_p^{(n)}\left({A}_1:\cdots :{A}_n\right) Δp(n)​(A1​:⋯:An​) Δ p n A 1 : ⋯ : A n .