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An international team led by Keun Su Kim at Yonsei University has directly measured the quantum metric tensor, also called the quantum distance, in a real material for the first time. Using angle-resolved photoemission spectroscopy (ARPES) at the U.S. Advanced Light Source, researchers mapped how electrons in black phosphorus interact with light and reconstructed the full quantum metric tensor of its valence band. Their findings, published in Science on June 5, 2025, combine theoretical work by Bohm-Jung Yang’s group at Seoul National University with experimental measurements by Kim’s group at Yonsei.
The quantum metric quantifies how similar or different two quantum states are—a quantum distance of one means states are identical, while zero indicates they are opposite. The quantum metric plays an important role in flat-band superconductors by enhancing the transition temperature. Physicists introduced this concept decades ago, but measuring it in solids has been challenging; previous experiments were restricted to artificial systems such as nitrogen-vacancy center qubits.
Why quantum distance matters
The quantum metric appears in diverse phenomena across condensed-matter physics. It relates to quantum fluctuations, dissipative responses, and plays roles in quantum phase transitions, orbital magnetism, localization and superfluidity. In quantum information theory, it is equivalent to the quantum Fisher information, a measure of multipartite entanglement.
Recent theoretical work showed the metric contributes to superconductivity: quantum geometry contributes to the coherence length and could explain recent observations in flat band systems. The geometric superfluid weight can even survive when the bands become completely flat and is responsible for the enhanced transition temperature of flat-band superconductors. These connections make the metric a key parameter for understanding emergent behavior in materials and designing fault-tolerant quantum technologies.
Black phosphorus as a testbed
Black phosphorus is the most stable allotrope of phosphorus and forms a layered crystal with strong intrinsic in-plane anisotropy. Thin black-phosphorus films exhibit high Hall mobilities—about 1,000 cm²/V·s along the light-mass x direction and 600 cm²/V·s along the heavy-mass y direction at 120 K. Field-effect transistors fabricated from 5 nm black-phosphorus flakes achieve on–off current ratios above 10⁵ with reasonable mobility at room temperature. Yang’s theory group found that one of the elemental layered crystals, black phosphorus, is an ideal material to study the quantum distance of electrons owing to its structural simplicity.
In November 2024, researchers from MIT and Seoul National University reported a reconstruction method for extracting the quantum geometric tensor from ARPES measurements. Researchers outlined a way to measure the momentum-resolved QGT of solids using angle-resolved photoemission spectroscopy (ARPES). That approach combines two complementary analyses of polarization- and spin-resolved ARPES data to determine both the quantum distance and the Berry curvature. Riccardo Comin of MIT told researchers that the method allows extraction of information about the electron wavefunction, not just its energy bands. The Yonsei experiment builds on this methodology by applying it to black phosphorus’s simple band structure.
Results and implications
During measurements at the Advanced Light Source, the team used polarized light to probe the pseudospin texture of black phosphorus’s valence band. By analyzing the momentum-resolved pseudospin textures of valence electrons in black phosphorus, they were able to reconstruct the quantum metric tensor with unprecedented precision. The result provides direct experimental evidence for the quantum metric in a solid and serves as a benchmark for future studies.
According to Kim, “Measuring the quantum distance is fundamentally important not only to understand anomalous quantum phenomena in solids, including special ones such as superconductors, but also to advance our quantum science and technologies.” The team expects the technique to help develop next-generation semiconductors, high-temperature superconductors and improved quantum computers.
While the measurement represents an important demonstration, it is not a direct blueprint for a quantum computer. The quantum metric is one of several quantities governing electronic behavior, and harnessing it to build devices will require considerable engineering.
Recent quantum computing developments show steady but measured progress. IBM’s 2025 roadmap targets systems with thousands of qubits, while industry revenue exceeded $1 billion in 2025. However, classical simulation of IBM’s 127-qubit Eagle processor experiments with greater accuracy than the quantum device itself achieved using tensor network algorithms on a laptop computer demonstrates that quantum advantage remains elusive for many problems.
By making an abstract quantity experimentally tangible, the Yonsei study provides a new tool for probing quantum geometry and highlights the growing interplay between materials science and quantum information. The work demonstrates that theoretical constructs once considered mere abstractions can be translated into measurable quantities, enriching our empirical understanding of quantum systems.
The research paper “Direct measurement of the quantum metric tensor in solids” appears in Science (DOI: 10.1126/science.ado6049).
