The researchers determine a unified topological speed limit for the evolution of physical states

The researchers determine a unified topological speed limit for the evolution of physical states

The topological speed limit indicates that the topological structure restricts the speed of physical systems such as chemical reactions and many-body interacting bosonic systems. Credit: Vu & Saito

Physical systems evolve at a certain rate, which depends on various factors, including the so-called topological structure of the system (ie, the spatial properties that are preserved over time despite any physical changes that occur). However, existing methods for determining the rate at which physical systems change over time do not account for these structural properties.

Two researchers at Keio University in Japan recently determined a rate limit for the evolution of physical states that also takes into account the topological structure of a system and its underlying dynamics. This speed limit, outlined in a paper published in Physical Review Letterscould have many valuable applications for the study and development of various physical systems, including quantum technologies.

“Finding how quickly the state of a system can change is a central topic in classical and quantum mechanics that has attracted great interest from scientists,” Tan Van Vu and Keiji Saito, the researchers who carried out the study, told Phys. org. “Understanding the timing control mechanism is relevant to the design of fast devices such as quantum computers.”

The notion that there is a limit to the operating time required for a system to transition from one physical state to another was first introduced decades ago by Leonid Isaakovich Mandelstam and Igor Tamm. Since then, other research teams have explored this idea further, finding similar limits that can be applied to different types of physical systems.

“These limits, called ‘rate limits,’ set the ultimate rates at which a system can evolve to a distinct state, and have found a variety of applications,” explained Vu and Saito. “However, conventional speed limits have the shortcoming of not providing meaningful limits as the system size increases. One explanation is that the topological nature of the dynamics, arising from the network structure of the underlying dynamics, has not been properly taken into account. account.”

The key objective of Vu and Saito’s recent work was to devise a new rate limit that takes into account both the topological structure of a physical system and the underlying dynamics. This could ultimately help establish tight quantitative limits, potentially revealing the physical mechanism underlying state-to-state transformations. Remarkably, this cannot be achieved using any of the rate limiting methodologies introduced so far.

“Our idea is to use a generalized version of the discrete Wasserstein distance to quantify the distance between states,” said Vu and Saito. “The Wasserstein distance comes from the idea of ​​quantifying how much and how far a bunch of goods must be transported to create another lump of goods out of one piece. This distance, widely used in optimal transport theory, encodes topological information and can grow proportionally to the size of the system”.

To derive the unified topological rate limit, Vu and Saito mapped the time evolution of physical states to the optimal transport problem by exploiting the properties of the optimal transport distance. As part of their study, they also demonstrated the validity of their approach by applying it to networks of chemical reactions and interacting with many-body quantum systems.

“In our view, the most notable finding of our study is the discovery of the topological speed limit that provides accurate predictions for operational times and can be applied to a wide range of dynamics,” said Vu and Saito.

The new topological speed limit introduced by this team of researchers could eventually be applied to research in various fields of physics, potentially improving the current understanding of various systems, in some cases facilitating their use for the development of new technologies. For example, it allows the creation of a rate formula for chemical reactions, as well as the establishment of universal constraints on the speed of bosonic transport and communication through spin systems.

“In the future, we plan to explore further applications of the derived topological speed limit from different directions,” added Vu and Saito. “Using the rate limit to better understand the underlying mechanisms of physical phenomena such as thermalization of closed and open systems is a promising approach.”

More information:
Tan Van Vu et al, Topological speed limit, Physical Review Letters (2023). DOI: 10.1103/PhysRevLett.130.010402

L. Mandelstam et al., The uncertainty relation between energy and time in non-relativistic quantum mechanics, Selected works (2011). DOI: 10.1007/978-3-642-74626-0_8

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