🔬 applied physics

Certifying Nonlocal Spin Quantum Entanglement from Local Tunneling Spectra

This paper demonstrates that local tunneling spectra can certify nonlocal spin entanglement without coherent control or absolute calibration by utilizing an exact spin sum rule and total-spin conservation to derive a lower bound on a collective entanglement witness from inelastic spin-flip weights.

Original authors: Payman Kazemikhah, Ruolun Zhang, Xiangyu Luo, Byunghoon Kim, Yu-Peng Zhu, Md. Shafayat Hossain, M. Zahid Hasan

Published 2026-10-20

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CC BY 4.0

Original authors: Payman Kazemikhah, Ruolun Zhang, Xiangyu Luo, Byunghoon Kim, Yu-Peng Zhu, Md. Shafayat Hossain, M. Zahid Hasan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Quantum entanglement is one of the most profound features of the microscopic world, a phenomenon where two particles become so deeply linked that the state of one instantly influences the other, regardless of the distance between them. This connection is not just a theoretical curiosity; it is the fundamental resource that powers future technologies like quantum computers and ultra-secure communication. However, proving that two specific particles in a solid material are actually entangled is notoriously difficult. In a complex material, atoms are surrounded by a sea of other electrons and vibrations, making it hard to tell if two chosen spins are truly talking to each other or just reacting to the same noisy environment. Traditionally, scientists have relied on techniques that require precise, coherent control over the particles, often involving complex pulses of energy to force them into specific states. Without such control, the local measurements usually taken in a lab seemed insufficient to certify a nonlocal connection, as the data appeared to contain too many unknown variables and missing pieces of information.

A team of researchers has now demonstrated that this apparent limitation can be overcome using only standard, equilibrium measurements. They showed that it is possible to certify spin entanglement between two atoms by analyzing the local tunneling spectra of each atom individually, without needing to manipulate the spins with external fields or calibrate the absolute intensity of the signals. The key insight is that the laws of physics impose strict constraints on how these spins can behave. Specifically, a spin has a fixed total amount of "fluctuation weight," a concept that describes how much the spin can wiggle or change direction. When the researchers measured the electrical conductance through a pair of coupled atoms, they found that the high-energy part of the signal provided a built-in reference point. This reference allowed them to normalize the data internally, effectively canceling out the unknown factors that usually plague such measurements. By comparing the strength of the inelastic spin-flip signal against this fixed total weight, they could calculate a lower bound for a quantity known as the quantum Fisher information, which serves as a direct measure of entanglement.

The researchers then applied a conservation law to connect the two separate measurements. Because the total spin of the pair is conserved, the fluctuations of one atom are mathematically tied to the fluctuations of the other. By combining the local bounds derived from each atom's individual spectrum, they could construct a rigorous lower bound for the collective entanglement of the pair. This method also accounts for the fact that the atoms are coupled to a metallic substrate, which can sometimes mimic entanglement by acting as a shared environment. The team developed a way to measure how much the total spin "leaks" into this substrate by looking at the width of the energy steps in the spectrum. This leakage parameter allowed them to distinguish between three distinct scenarios: a simple product state where the atoms are independent, a state where each atom is entangled with its own private environment, and the true state where the two atoms are entangled with each other.

The study establishes a clear hierarchy of thresholds to identify these states. If the calculated collective value exceeds a specific limit, it proves the atoms are not just independent. If it exceeds a higher limit, it rules out the possibility that they are merely correlated through their environment. Finally, crossing the highest threshold certifies that the two spins themselves are genuinely entangled, a state that is essential for using them as qubits in quantum computing. The researchers tested their method extensively with simulations that included realistic experimental noise and instrumental broadening. They found that even with a small amount of noise, the certificate held true with a very high confidence level, successfully identifying entangled states in a wide range of conditions. This work provides a practical route for extracting nonlocal quantum information from local equilibrium spectroscopy, meaning that scientists can now verify the presence of entanglement in engineered spin clusters using standard scanning tunneling microscopes, without the need for complex coherent control protocols.

Technical Summary: Certifying Nonlocal Spin Quantum Entanglement from Local Tunneling Spectra

Problem Statement
Quantum entanglement is a nonlocal resource essential for quantum technologies and a defining feature of many-body states. However, experimentally establishing entanglement in materials is challenging because the relevant degrees of freedom are embedded in larger systems, and strong correlations alone do not prove that two specific spins are entangled. While Quantum Fisher Information (QFI) provides a rigorous link between measurable fluctuations and entanglement, standard methods like neutron scattering probe collective fluctuations averaged over many sites. Conversely, Scanning Tunneling Microscopy (STM) and Inelastic Electron Tunneling Spectroscopy (IETS) offer atomic resolution, measuring fluctuations of single sites. A fundamental mismatch exists: entanglement witnesses typically require correlations between different spins, whereas IETS measures single-site spectra. Furthermore, local spectra lack absolute intensity calibration, contain unknown tunneling amplitudes, and are complicated by spin leakage into the substrate environment, which can mimic entanglement. Previous proposals to detect entanglement at the atomic scale often relied on coherent control (e.g., ESR-STM), leaving it unclear if ordinary equilibrium tunneling spectra contain sufficient information.

Methodology
The authors propose a framework to certify spin entanglement using only local, equilibrium tunneling spectra without coherent driving or absolute intensity calibration. The method relies on three physical pillars:

Exact Spin Sum Rules: For a spin of fixed length $S$ , the total fluctuation weight is fixed at $S(S+1)$ . In tunneling spectroscopy, this weight is shared between elastic scattering and inelastic spin-flip excitations. Once all spin-flip channels are open, the high-bias conductance plateau ( $G_\infty$ ) serves as an internal normalization reference. This allows the inelastic step weight to be converted into a calibration-free lower bound on the local QFI ( $D_i$ ) via a scale-free ratio, eliminating dependence on unknown tunneling amplitudes ( $\tau_0, \tau_1$ ).
Total-Spin Conservation: For a spin dimer, total-spin conservation fixes the covariance between the two spins. This constraint converts the two local QFI bounds (derived from separate local spectra) into a lower bound on the collective QFI ( $W$ ).
Spectroscopic Leakage Bounds: Real systems couple to a substrate, violating strict spin conservation. The authors model this as a leakage parameter $\epsilon$ , representing the loss of total spin to the environment. They derive a spectroscopic bound for $\epsilon$ based on the excitation linewidth ( $\Gamma_E$ ) and the Kondo coupling strength, utilizing second-order perturbation theory and Numerical Renormalization Group (NRG) calculations to establish a rigorous upper bound ( $\epsilon_{max}$ ).

The framework incorporates instrumental broadening (lock-in modulation, voltage noise) and measurement noise explicitly. By formulating the problem as a linear programming optimization over non-negative spectral weights consistent with the noisy data, the authors derive a certificate with a stated confidence level.

Key Contributions and Results

Calibration-Free Local QFI: The paper derives a lower bound for the local QFI ( $D_i^{low}$ ) that depends only on the ratio of the inelastic step weight to the high-bias plateau. This bound is independent of the unknown tunneling strength and absolute intensity calibration.
Hierarchy of Entanglement Thresholds: The authors establish a hierarchy of thresholds for the collective QFI ( $W$ $W$ ) to distinguish between different physical states:
- $W \leq 4$ : Product states (separable spins).
- $4 < W \leq 6$ : Spins correlated only through their environments (e.g., Kondo screening by independent baths) but not entangled with each other.
- $W > 6$ : Indicates entanglement beyond simple environmental correlations.
- $W > 8$ : Certifies entanglement between the two spins themselves (the resource relevant for spin qubits), providing a lower bound on the concurrence.
Robustness to Noise and Leakage: The method explicitly accounts for spin leakage into the substrate. By bounding $\epsilon$ via the linewidth, the collective QFI bound remains rigorous even for open systems. Statistical analysis shows that with realistic noise levels (0.5% of conductance) and bias points, the certification holds with high confidence (e.g., >99% coverage in simulations).
Distinction from Mimics: The framework successfully distinguishes true two-spin entanglement from "false positives" such as:
- Néel product states: Which appear as $W=8$ in standard analyses but correct to $W=4$ when leakage is accounted for.
- Privately screened spins: Where each spin forms a singlet with a private bath partner, yielding $W=6$ but no inter-spin entanglement.
- Misidentified ions: A spin-1 ion misidentified as a dimer would yield $W=12$ , but the method requires independent identification of the spin cluster.

Significance
The paper establishes a route for extracting nonlocal quantum information from equilibrium local spectroscopy. It demonstrates that when exact sum rules fix the available fluctuation weight and conservation laws constrain how that weight is shared, local measurements contain sufficient information to certify nonlocal quantum properties. This approach offers a quantitative, statistically controlled measure of quantum correlations carried by individual spins without the need for coherent driving or absolute intensity calibration. The authors suggest that this method is within the reach of present atomic-scale spectroscopy (e.g., on transition-metal atoms on MgO/Ag(001) or magnetic atoms on Cu2N) and provides a necessary tool for testing entanglement in engineered spin networks before applying coherent operations.

Technical Summary: Certifying Nonlocal Spin Quantum Entanglement from Local Tunneling Spectra

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