The uncertainty principle is a cornerstone of modern physics, and its implications have a fundamental impact on theoretical and applied quantum mechanics. The aim of this thesis is to study and apply the uncertainty relations between time and energy, which are interpreted operationally as the speed limit of quantum evolution. A geometric approach is used to derive improved bounds for the evolution of isolated and open systems and obtain a constructive approach for efficient and fast quantum driving. These results are applied to set the limits for the rate of information transfer and processing, and the power of nanomachines.
The uncertainty principle is a cornerstone of modern physics, and its implications have a fundamental impact on theoretical and applied quantum mechanics. The aim of this thesis is to study and apply the uncertainty relations between time and energy, which are interpreted operationally as the speed limit of quantum evolution. A geometric approach is used to derive improved bounds for the evolution of isolated and open systems and obtain a constructive approach for efficient and fast quantum driving. These results are applied to set the limits for the rate of information transfer and processing, and the power of nanomachines.
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.
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 presence of noise or the interaction with an environment can radically change the dynamics of observables of an otherwise isolated quantum system.We derive a bound on the speed with which observables of open quantum systems evolve.This speed limit is divided into Mandelstam and Tamm's original time-energy uncertainty relation and a time-information uncertainty relation recently derived for classical systems, and both are generalized to open quantum systems.By isolating the coherent and incoherent contributions to the system dynamics, we derive both lower and upper bounds on the speed of evolution.We prove that the latter provide tighter limits on the speed of observables than previously known quantum speed limits and that a preferred basis of speed operators serves to completely characterize the observables that saturate the speed limits.We use this construction to bound the effect of incoherent dynamics on the evolution of an observable and to find the Hamiltonian that gives the maximum coherent speedup to the evolution of an observable.
We establish a comprehensive theoretical framework for coherent quantum speed limits (QSLs), deriving fundamental bounds on the rate of quantum evolution that explicitly isolate the contribution of quantum coherence. By applying Hölder's inequality for matrix norms to the Liouville-von Neumann equation, we construct two infinite families of QSLs for general unitary dynamics. These bounds are characterized by coherence measures based on Schatten p-norms and Hellinger distance, respectively, defined with respect to the instantaneous energy eigenbasis. Unlike traditional Mandelstam-Tamm bounds, our approach disentangles the quantum state's coherence structure from the Hamiltonian's energy scale. Using the Landau-Zener model accelerated by shortcuts to adiabaticity, we demonstrate that coherence functions as a critical kinematic resource: achieving faster evolution entails maintaining a state with high coherence relative to the instantaneous basis. Our results provide a resource-theoretic perspective on time-energy uncertainty, offering insights into the fundamental limits of quantum control and information processing.
Quantum speed limits establish fundamental bounds on the minimum time required for a quantum system to evolve from an initial state to a target state. Traditional formulations treat the evolution speed as a single global quantity governed by the system energy uncertainty or average energy. In this paper, we propose the theory of Quantum Speed-Limit Splitting, which posits that a quantum system undergoing complex dynamics does not necessarily evolve along a single speed limit, but may instead exhibit multiple competing, distinct state-evolution speed limits simultaneously. By defining a multi-valued velocity spectrum belonging to the set {v_1, v_2, ..., v_n}, we demonstrate that different geometric pathways in the underlying Hilbert space correspond to distinct temporal boundaries. We formulate the mathematical framework, replacing the traditional scalar evolution time with a set of multi-branch bounds {tau_1, tau_2, ...}. This theoretical extension provides a novel lens for viewing quantum dynamics, with profound implications for quantum control, thermodynamic cost, and the fundamental limits of information processing.
Uncertainty relations play a crucial role in quantum mechanics. Well-defined methods exist for the derivation of such uncertainties for pairs of observables. Other approaches also allow the formulation of time-energy uncertainty relations, even though time is not an operator in standard quantum mechanics. However, in these cases, different approaches are associated with different meanings and interpretations for these relations. The one of interest here revolves around the idea of whether quantum mechanics inherently imposes a fundamental minimum duration for energy measurements with a certain precision. In our study, we investigate within the Page and Wootters timeless framework how energy measurements modify the relative "flow of time'' between internal and external clocks. This provides a unified framework for discussing the subject, allowing us to recover previous results and derive new ones. In particular, we show that the duration of an energy measurement carried out by an external system cannot be performed arbitrarily fast from the perspective of the internal clock. Moreover, we show that during any energy measurement the evolution given by the internal clock is non-unitary.