Quantum speed limits: from Heisenberg's uncertainty principle to optimal quantum control (1705.08023v2)

Abstract: One of the most widely known building blocks of modern physics is Heisenberg's indeterminacy principle. Among the different statements of this fundamental property of the full quantum mechanical nature of physical reality, the uncertainty relation for energy and time has a special place. Its interpretation and its consequences have inspired continued research efforts for almost a century. In its modern formulation, the uncertainty relation is understood as setting a fundamental bound on how fast any quantum system can evolve. In this Topical Review we describe important milestones, such as the Mandelstam-Tamm and the Margolus-Levitin bounds on the quantum speed limit, and summarise recent applications in a variety of current research fields -- including quantum information theory, quantum computing, and quantum thermodynamics amongst several others. To bring order and to provide an access point into the many different notions and concepts, we have grouped the various approaches into the minimal time approach and the geometric approach, where the former relies on quantum control theory, and the latter arises from measuring the distinguishability of quantum states. Due to the volume of the literature, this Topical Review can only present a snapshot of the current state-of-the-art and can never be fully comprehensive. Therefore, we highlight but a few works hoping that our selection can serve as a representative starting point for the interested reader.

• The paper reviews how quantum speed limits, derived from Heisenberg’s uncertainty principle, set fundamental bounds on the minimum evolution time of quantum systems.
• The paper compares the Mandelstam-Tamm and Margolus-Levitin bounds to analyze how energy and Hamiltonian variance dictate quantum evolution speeds.
• The paper explores how open system dynamics and non-Markovian effects can potentially accelerate quantum processes, impacting quantum computing and thermodynamics.

An Academic Perspective on Quantum Speed Limits and Their Implications

The paper authored by Sebastian Deffner and Steve Campbell, titled "Quantum speed limits: from Heisenberg's uncertainty principle to optimal quantum control," presents an in-depth analysis of quantum speed limits (QSLs), a fundamental concept rooted in quantum mechanics and the Heisenberg uncertainty principle. This comprehensive review serves as a critical resource for researchers seeking to understand the theoretical foundations and practical implications of QSLs in various quantum systems.

Quantum speed limits define the minimum time required for a quantum system to evolve between two states, providing a fundamental bound on the speed of quantum evolution. These limits stem from the energy-time uncertainty relation, one of the formulations of Heisenberg's uncertainty principle. The paper systematically reviews the historical context, starting from the Mandelstam-Tamm bound and the Margolus-Levitin bound, each offering different interpretations of QSLs based on energy and Hamiltonian variance considerations.

The significance of QSLs extends to several fields, including quantum information theory, quantum computing, and quantum thermodynamics. In quantum computing, QSLs establish limits on the rate of computation, determined by the available energy, thereby influencing the design and optimization of quantum algorithms. In quantum thermodynamics, the speed of quantum processes dictates the maximal efficiency and power of quantum engines, connecting QSLs to foundational thermodynamic principles.

The paper also explores the implications of QSLs in open quantum systems, considering both Markovian and non-Markovian dynamics. The inclusion of environmental interactions adds complexity to the dynamics and necessitates the development of geometric approaches to evaluate QSLs for mixed states and open systems. These approaches reveal the potential for non-Markovian environments to enhance the evolution speed, a counterintuitive result that suggests that memory effects could be leveraged for faster quantum processes.

In addition to theoretical insights, the paper speculates on the future applicability of QSLs across emerging quantum technologies. For instance, QSLs could play a pivotal role in optimizing quantum communication channels and designing quantum control protocols that reach the limits of possible speeds without exceeding energy constraints.

The review makes explicit the interconnectedness of QSLs with other fundamental limits such as the Cramer-Rao bound in quantum metrology and Landauer's bound in quantum information theory. These connections emphasize the integral nature of QSLs within the broader framework of quantum mechanics, extending beyond mere theoretical curiosity to become essential considerations in quantum technology development.

In summary, the paper by Deffner and Campbell provides a thorough examination of quantum speed limits from their theoretical underpinnings to practical implications. It highlights the relevance of QSLs in defining the ultimate capabilities and limitations of quantum systems, suggesting that future research will likely deepen our understanding of these quantum bounds and exploit them in the development of advanced quantum technologies. As quantum mechanics continues to evolve, QSLs will remain a pivotal topic of inquiry, driving innovations that harness the full potential of quantum theory.