Untangling quantum entanglement with new calculation formulas

Osaka Metropolitan University physicists have developed new, simpler formulas to quantify quantum entanglement in strongly correlated electron systems and applied them to study several nanoscale materials. Their results offer fresh perspectives into quantum behaviors in materials with different physical characteristics, contributing to advances in quantum technologies.

Quantum entanglement is a unique phenomenon in which two particles, once connected, remain linked no matter how far apart they are in space. This fundamental feature plays a vital role in emerging technologies such as quantum computing and quantum cryptography.

Whilst significant progress has been made in understanding this so-called spooky phenomenon, scientists still find themselves tangled in its intricacies.

"Previous studies have largely focused on the universal properties of quantum entanglement in materials exhibiting magnetism or superconductivity," said Yunori Nishikawa, a lecturer at Osaka Metropolitan University's Graduate School of Science and lead author of the study.

The team, instead, went local: They zeroed in on quantum entanglement between one or two arbitrarily selected atoms within a strongly correlated electron system and their surrounding environment (the rest of the system).

Strongly correlated electron systems are materials in which electron-electron interactions dominate the system's behavior, leading to rich, complex and often highly entangled quantum states. These systems serve as fertile grounds for exploring quantum entanglement.

The researchers derived formulas to calculate key quantum informative quantities, including entanglement entropy (which quantifies how entangled a system is), mutual information (which measures shared information between two parts of the system), and relative entropy (which gauges differences between quantum states). These quantities are critical for understanding how different parts of a quantum system interact with and influence each other.

"It was a pleasant surprise when we found that the formula* for entanglement entropy could be rendered in a surprisingly simple expression," Nishikawa said.

To test their approach, the team applied their formulas to different material systems, including nanoscale artificial magnetic materials arranged in a linear chain and dilute magnetic alloys. Their analysis revealed counterintuitive patterns of quantum entanglement in the nanoscale artificial magnetic systems. In the dilute magnetic alloys, they successfully identified quantum relative entropy as a key quantity for capturing the Kondo effect, a phenomenon in which a magnetic impurity is screened by conduction electrons.

"The behavior of quantum entanglement in nanoscale artificial magnetic materials defied our initial expectations, opening new avenues for understanding quantum interactions," Nishikawa said.

The study paves the way for deeper explorations of quantum entanglement that could drive advancements in quantum technologies.

"Our formulas can also be applied to systems with various other physical properties," Nishikawa said. "We hope to inspire further research and provide new insights into quantum behaviors in different materials."

*The formula to calculate entanglement entropy is as follows:

S=-n↑n↓logn↑n↓-h↑h↓logh↑h↓-n↑h↑logn↑h↑-n↓h↓logn↓h↓

in which 𝑛_↑, 𝑛_↓ are the numbers of up- and down-spin electrons and h_↑, h_↓ are the numbers of up and down holes (operators) within the target atom.