# The authors derive a unified topological speed limit for the evolution of physical states

Physical systems evolve at a certain speed, which depends on various factors including the so-called topological structure of the system (i.e. spatial properties that are maintained over time despite any physical changes that occur). Existing methods for determining the speed at which physical systems change over time, however, do not take into account these structural properties.

Two researchers at Keio University in Japan recently derived a speed limit for the evolution of physical states that also explains the system’s topological structure and underlying dynamics. This speed limit was specified in research published in *Physical review letters*could have many valuable applications for the study and development of various physical systems, including quantum technologies.

“Knowing how quickly a system’s state can change is a central topic in quantum mechanics and classical, and has attracted great interest among scientists,” Tan van Fu and Keiji Saito, the two researchers who conducted the study, told Phys.org. “Understanding the mechanism of time control is relevant to engineering fast devices such as quantum computers.”

The idea that there is a limit to the operational time required for a system to transition from one physical state to another was first introduced several decades ago by Leonid Isakovich Mandelstam and Igor Tam. Since then, other research teams have explored this idea further, finding similar limitations that can be applied to different kinds of physics systems.

Vu and Saito explain that “these limits, which are called ‘speed limits,’ specify the final rates at which the system can evolve into a recognizable state and have found a wide variety of applications.” However, traditional speed limits have the disadvantage of not Introducing meaningful limits as system size grows. One explanation is that the topological nature of the dynamics, which arises from the network structure of the underlying dynamics, has not been properly taken into account.”

A major goal of recent work by Vu and Saito has been to establish a new velocity limit that also takes into account the topological structure of a physical system and its underlying dynamics. This could eventually help establish strict quantum limits, which could reveal the physical mechanism underlying transitions from one state to another. Notably, this cannot be achieved using any of the velocity methodologies presented so far.

“Our idea is to use a generalized version of the discrete Wasserstein distance to determine the distance between states,” Fu and Saito said. “Wasserstein distance arises from the idea of determining the number and quantity of a stack of goods that must be moved to create another mass of goods from one mass. This distance, used extensively in optimal transport theory, encodes topological information and can grow proportionally to the size of the system.”

To derive the uniform topological speed limit, Vu and Saito map the time evolution of the physical states of the optimal transport problem, 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 chemical interaction networks and the interaction of many-body quantum systems.

“In our opinion, the most remarkable finding of our study is the discovery of a topological velocity limit that yields accurate predictions of run 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 different areas of physics, potentially improving current understanding of different systems, and in some cases facilitating their use to develop new technologies. For example, it allows creating a speed equation for chemical reactions, as well as setting global constraints on the speed of bosonic transport and communication through spin systems.

“In the future, we plan to explore more applications of topological velocity limits derived from different directions,” Fu and Saito added. “Using the speed limit to better understand the mechanisms underlying physical phenomena, such as the heat treatment of closed and open systems, is a promising approach.”

**more information:**

Tan Van Fu et al., Topological Speed Limit, *Physical review letters* (2023). DOI: 10.1103/PhysRevLett.130.010402

Mandelstam et al., Energy-time uncertainty relation in non-relativistic quantum mechanics, *Selected papers* (2011). DOI: 10.1007/978-3-642-74626-0_8

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**the quote**: Researchers derived a uniform topological speed limit for the evolution of physical states (2023, January 24) Retrieved January 25, 2023 from https://phys.org/news/2023-01-derive-topological-limit-evolution-physical.html

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