Sharma-mittal Entropy Advances Quantum Speed Limits for Finite-Dimensional Systems

The fundamental limits of how quickly quantum systems change remain a central question in physics, and researchers now investigate these speed limits with a new approach based on a versatile measure of entropy. Dong-Ping Xuan, Zhi-Xi Wang, and Shao-Ming Fei, all from Capital Normal University, present a novel class of quantum speed limits formulated using the Sharma-Mittal entropy, a tool applicable to a wide range of quantum processes, even those not governed by traditional rules of quantum mechanics. This work establishes intrinsic bounds on the minimum time needed for a system to evolve, offering insights into the dynamics of both simple qubits and complex many-body systems like the XXZ spin chain, and potentially impacting fields from quantum metrology to the precise control of quantum states. The team’s entropy-based speed limits provide a powerful new way to characterise the ultimate constraints on quantum evolution, advancing our understanding of how quickly information can be processed and manipulated in the quantum realm.

This work presents a class of quantum speed limits formulated in terms of the two-parameter Sharma-Mittal entropy, applicable to finite-dimensional systems evolving under general nonunitary dynamics. The research investigates how this entropy, a measure of uncertainty, can define fundamental limits on how quickly a quantum state can change, offering a valuable tool for analysing and optimising quantum processes with potential implications for quantum technologies and information processing.

In the single-qubit case, the quantum speed limits for both quantum channels and non-Hermitian dynamics are analysed in detail. For many-body systems, the role of sum-of-mean-energies-based bounds in characterising the reduced dynamics is explored, and the results are applied to the XXZ spin chain model. These entropy-based quantum speed limits characterise fundamental limits on quantum evolution speeds and may be employed in contexts including entropic uncertainty relations, quantum metrology, coherent control and quantum sensing.

Quantum Speed Limits, Open System Derivations

This document provides supplementary mathematical details and derivations for research investigating quantum speed limits in specific quantum systems, rigorously supporting the claims made in the main paper. The intended audience is researchers and graduate students already familiar with quantum information theory, open quantum systems, and related mathematical tools.

The core concepts explored include quantum speed limits, which define the minimum time for a quantum state to evolve, and open quantum systems, which interact with their environment leading to non-unitary evolution. Density matrices are used to describe quantum states, particularly in mixed states, while the Sharma-Mittal entropy serves as a measure of quantum uncertainty and entanglement. The paper demonstrates how this entropy relates to quantum speed limit bounds.

Specific quantum systems investigated include a PT-symmetric two-level system, exhibiting non-Hermitian Hamiltonian dynamics, and the spin-1/2 XXZ model, a common model in condensed matter physics and quantum information. Detailed calculations are provided for the dynamics of the PT-symmetric system, deriving expressions for the density matrix and analysing behaviour under varying conditions. The Sharma-Mittal entropy is used to derive a specific quantum speed limit bound for the spin-1/2 XXZ model, demonstrating the application of the general formula to a complex system.

The document explores the relationship between the Sharma-Mittal entropy and other entropy measures, such as Rényi, Tsallis, and von Neumann entropies, showing how the Sharma-Mittal entropy reduces to these standard measures under specific limits. Key mathematical techniques employed include density matrix formalism, Kraus operators, trace operations, eigenvalue analysis, and entropy calculations, alongside calculus, limits, and matrix algebra.

This work rigorously supports claims regarding quantum speed limits, enhances understanding of non-unitary dynamics, and provides a more general framework for quantifying quantum uncertainty. The findings have applications in the design and optimisation of quantum algorithms and devices, and contribute to the development of efficient quantum technologies.

Sharma-Mittal Entropy Defines Quantum Speed Limits

This research establishes new quantum speed limits, fundamental bounds on how quickly quantum systems can evolve, using the Sharma-Mittal entropy. Scientists developed a general framework for calculating these limits applicable to systems undergoing both standard and non-standard quantum dynamics, extending beyond traditional approaches. The team demonstrated that these entropy-based speed limits accurately characterise the minimum time required for a system to change its state, offering a powerful tool for understanding the dynamics of quantum processes.

The findings have implications for several areas of quantum information science, including entropic uncertainty relations, the precision of quantum measurements, and the control of quantum systems. By applying their method to the well-studied XXZ spin chain model, researchers showed how these limits function in complex, many-body scenarios, validating the approach’s broad applicability. The authors suggest future work could explore the conditions under which these limits can be approached and potentially overcome, as well as investigating their role in open quantum systems and non-Markovian dynamics.