No GHZ in holography — what happens when the regions leave gaps?

Entanglement between two parties is well understood. Give it a third party and the subject becomes surprisingly hard.

For two systems in a pure state, one number says it all: the entanglement entropy, which measures how mixed either half looks on its own. With three systems no single number is enough, because there are genuinely different ways for three parties to be entangled.

The standard examples use three qubits. In the GHZ state, (|000⟩ + |111⟩)/√2, the entanglement is all or nothing. Throw away any one qubit and the remaining two are left only classically correlated: they always agree, but they are not entangled. In the W state, (|001⟩ + |010⟩ + |100⟩)/√3, the entanglement is spread out. Throw away one qubit and the other two are still entangled with each other.

No sequence of local operations, even with luck on your side, turns one of these into the other. They are different kinds of three-party entanglement, and a natural question is which kinds a given physical system prefers.

For one class of systems we now have a sharp answer, at least when the three parties make up the whole system. This post is about what happens when they don't.

The systems in question are holographic. In the AdS/CFT correspondence, a quantum field theory living on the boundary of a space is equivalent to a theory of gravity inside it, in the "bulk". A remarkable entry in the dictionary between the two is the Ryu–Takayanagi formula. The entanglement entropy of a region A of the boundary equals the area of the smallest surface in the bulk that ends on the edges of A:

Here G_N is Newton's constant. In the simplest setting, a time slice of three-dimensional anti-de Sitter space, the bulk is a disk, the boundary is its circle, and the "smallest surface" is a curve, a geodesic of the hyperbolic disk. The region enclosed between A and its curve is called the entanglement wedge of A. It is the part of the bulk that region A knows about.

Because minimal surfaces have to fit together consistently, holographic entropies obey inequalities that general quantum states can violate. The best known is the monogamy of mutual information. Much of my earlier work, on the holographic entropy cone, is about cataloguing and proving such inequalities. But they are all built from ordinary entropies, which only measure how one group of parties is entangled with the rest. To see three-party entanglement directly, you need other tools.

Two quantities do the job here. Both have a simple geometric meaning in holography.

The first is the multi-entropy, a three-party generalization of the entanglement entropy. In the bulk it is the length of the shortest Y-shaped network of geodesics that connects the three junctions between the regions to a single bulk point. Subtracting the ordinary entropies of the three regions, with the right weights, leaves the genuine multi-entropy, GM(A:B:C). It is designed to be blind to entanglement that is only between pairs. For holographic states it is positive and, remarkably, finite: the infinities from the edges of the regions cancel.

The second quantity is the Markov gap, written R(A:B). It compares two ways of measuring the correlation between A and B: the reflected entropy, and the familiar mutual information, I(A:B) = S(A) + S(B) − S(AB). The difference is never negative. Geometrically, the reflected entropy is twice the narrowest cut across the entanglement wedge of AB, called its cross-section.

What makes the Markov gap useful here is a simple fact about GHZ states: their Markov gap is zero. So is that of the generalized GHZ states. A Markov gap of zero is a fingerprint of GHZ-like correlations.

In 2025, Balasubramanian, Kang, Cummings, Murdia and Ross proposed an inequality between these two quantities for holographic states. When A, B and C together make up the whole boundary, so that the state on ABC is pure, they showed that

I will call it the holographic signal inequality. Its consequence is striking. If a holographic state had purely GHZ-like three-party entanglement, its Markov gap would vanish while its genuine multi-entropy stayed positive, and the inequality would fail. So purely GHZ-like tripartite entanglement is forbidden in holography. The three-party entanglement of a spacetime cannot all be of the all-or-nothing kind.

The rule was stated for three regions that cover the whole boundary. Real questions rarely come that tidy.

Now leave gaps between the regions. The leftover part of the boundary acts like an environment, so the state of A, B and C on their own is mixed. Mixed states are the generic case: any three regions you pick in a larger system are in one. My recent paper asks whether the signal inequality survives this.

The first step is to extend both sides to mixed states. The left side needs nothing new, because the Markov gap only ever involves the two-party state of A and B. For the right side there is a standard trick, the canonical purification. Any mixed state can be written as a pure state of a doubled system, the original plus a mirror copy. In holography the doubling has a picture: take the entanglement wedge and glue it to its own reflection. Measuring the multi-entropy of the doubled state gives the reflected multi-entropy of Yuan, Li and Zhou, and from it a reflected genuine multi-entropy GM_R. When the gaps close, it smoothly turns back into the ordinary GM. The most direct extension of the rule is then

It fails. To see why, arrange the regions so that A and B are small and far apart, while C is large and fills most of the space between them on the other side, with only narrow gaps.

On the left, the wedge of AB has no connected cut to measure. Its cross-section and the mutual information are both zero, so the Markov gap is zero. On the right, the wedge of ABC is connected, and in that phase the reflected genuine multi-entropy is bounded below by a fixed positive amount. The left side of the inequality is zero while the right side is positive.

The reason is worth spelling out. The Markov gap only ever looks at A and B. Here, the three-way correlations among the regions run through C, and a test that throws C away cannot see them. Choosing a different purification does not help either: as long as the purified state is itself holographic, its genuine multi-entropy is positive and the inequality still fails.

The fix is to treat both sides the same way. In what I call the symmetric extension, you first purify the whole mixed state of ABC, doubling every region: A and its mirror A*, and so on. Then you forget about C and its mirror, and compute the Markov gap between AA* and BB* in what remains.

That two-party state is not the original state of A and B. It remembers that C was part of the picture when the doubling happened. Geometrically, the doubled wedge of AA*BB* is connected even though the original wedge of AB is not. Its cross-section, which I call the C-excised reflected entropy, is no longer zero. For geometries like the one above, the three-way correlations that the minimal extension missed are now visible, and the inequality holds again.

The paper ends with a conjecture. In earlier work with Ning Bao and Keiichiro Furuya, we defined a tripartite correlation signal, Δ_w, built from the three-party entanglement wedge cross-section by subtracting its two-party versions. I conjecture that for holographic states

For pure states Δ_w vanishes, and the conjecture reduces to the statement that the genuine multi-entropy is non-negative, which is already known to hold in holography. In the simplest geometries, a time slice of AdS₃, it becomes a comparison between two kinds of three-way minimal networks, and I propose a "sandwich" in which one is caught between the other and twice the other. For the geometry studied in the paper, the triangle inequality and properties of hyperbolic triangles show that the sandwich holds. That is evidence, not a proof.

Plenty is left open. The same construction can be pushed to four parties, where the minimal extension fails in the same way. Three qutrits can be entangled in twelve inequivalent ways and four qubits in nine, so one can ask which of these classes survive a Rényi version of the signal inequality. And replacing the entanglement wedge cross-section by the entanglement of purification gives a version of the conjecture that makes sense for any quantum system, holographic or not. Which states satisfy it is a good question.

Holography rules out the all-or-nothing kind of three-party entanglement. Finding out exactly what it allows, once the regions stop fitting together neatly, is the next step.