Unravelling the Li-Haldane Conjecture with the Projected Ensemble

Deep within the quantum world, certain materials behave in ways that defy our everyday intuition. These are topological phases of matter, exotic states where the rules of physics are written not in the arrangement of individual atoms, but in the global, invisible connections between them. Because these connections are non-local, traditional tools that look at small, local pieces of the material fail to see them. Instead, scientists have turned to quantum entanglement, a phenomenon where particles remain linked across vast distances, to diagnose these hidden orders. A landmark idea known as the Li–Haldane conjecture proposed that if you look at the "entanglement spectrum"—a specific mathematical map of how a piece of a material is connected to the rest—you would see a pattern that perfectly matches the physics of the material's edge. This discovery linked the mysterious interior of the material to the behavior of its boundary, becoming a cornerstone for understanding fractional quantum Hall states, which are fluids of electrons that flow without resistance under extreme conditions.

For years, this connection has been a powerful tool, but it has treated the quantum information inside the material as a single, averaged whole. A new study by researchers at the University of Cambridge and the National University of Singapore suggests that this view is too coarse. By peering deeper into the data, they have uncovered a hidden, intricate hierarchy within these quantum states. They found that the entanglement spectrum is not just a static picture, but a structure that can be unraveled and reorganized by the act of measurement itself. When researchers measure parts of the system surrounding a specific region, the quantum states in that region do not just collapse randomly; they sort themselves into highly specific, smaller groups. The size and shape of these groups follow a precise, universal rule that was previously invisible. This discovery, which the authors call the measurement-resolved Li–Haldane conjecture, reveals that a single quantum state contains a nested library of topological information, waiting to be unlocked by how we choose to look at it.

To understand how this works, imagine a fluid of electrons trapped on a sphere, a common setup for studying these exotic states. The researchers divided this sphere into two parts: a central region they wanted to study, and a surrounding ring of orbitals they would measure. In the standard approach, scientists would simply ignore the details of the surrounding ring, effectively averaging over all possible outcomes to create a reduced picture of the center. This averaged picture produces the familiar entanglement spectrum that confirmed the original Li–Haldane conjecture. However, the new study took a different path. Instead of discarding the information from the surrounding ring, they kept the specific results of their measurements. They asked: if we know exactly how the electrons are arranged in the outer ring, what does the quantum state in the center look like?

The answer was a revelation. When the researchers conditioned their view of the center on specific measurement outcomes from the outside, the quantum states in the center did not spread out randomly. Instead, they collapsed into much smaller, lower-dimensional spaces. In one specific example involving a state known as the Moore-Read state, the researchers found that the quantum states, which normally spanned a three-dimensional space, would shrink down to a two-dimensional line or even a single point, depending entirely on the pattern of electrons measured in the outer ring. This phenomenon, which they term "rank collapse," showed that the quantum information was not a uniform blob, but a structured hierarchy. The size of the space the states occupied was not arbitrary; it was dictated by a precise counting rule from a branch of physics called conformal field theory, which describes how these systems behave at their edges.

What makes this finding particularly striking is how the structure rearranges itself. In the case of the Moore-Read state, which involves complex, non-Abelian physics, the researchers discovered a surprising twist. When they measured the outer ring and found a pattern with an even number of electrons, the inner region's states collapsed into a space whose size matched the counting rules for an odd-electron pattern, and vice versa. It was as if the act of measuring the outside forced the inside to reorganize according to the rules of a different, opposite version of itself. This hidden relationship between different "parity" sectors of the theory had never been seen before. It suggests that the entanglement spectrum contains a rich internal architecture, where different layers of topological order are nested inside one another, only revealed when the system is probed with the right level of detail.

The researchers also tested whether this delicate structure was just a mathematical curiosity of idealized models or if it existed in the messy reality of actual physical interactions. They simulated the system with realistic Coulomb interactions, the natural repulsive force between electrons, which often disrupts such clean quantum patterns. Remarkably, the structure survived. Even in these more realistic, interacting ground states, the quantum states still collapsed into the same specific subspaces predicted by the theory. While the states were not as perfectly sharp as in the ideal models, they clustered tightly around the predicted locations, showing that this hierarchical organization is a robust feature of the topological order itself. The study further showed that this structure is rigid; it does not depend on the specific way the measurements are performed, as long as the measurements respect the fundamental symmetries of the system. This "rank rigidity" means that the hidden hierarchy is an intrinsic property of the material, not an artifact of the experimental setup.

This work fundamentally changes how scientists can probe the quantum world. It demonstrates that the act of measurement is not merely a passive observation that destroys quantum information, but a tool that can actively resolve finer structures within that information. By moving beyond the averaged view of the reduced density matrix, researchers can now access a new layer of universal fingerprints that define topological phases. The study confirms that the Li–Haldane conjecture is not just a statement about the total number of states, but a gateway to a deeper, measurement-dependent organization of the quantum world. For the first time, scientists have a method to see the internal geometry of these topological states, revealing a universe of structure hidden within the entanglement of a few electrons.