In this paper, we present the theoretical framework of quantum speed limits (QSLs) in terms of trajectory ensembles in phase space. This indicates that the QSL can be thought of as the summation of the connecting harmonic oscillators: the connections between the points of the system in phase space and the trajectory ensemble. Two typical models, the time-dependent harmonic oscillator and the undriven harmonic oscillator coupled to a thermal bath, are investigated by employing the theoretical framework. Our results from this perspective are in agreement with previous treatments.
In this paper, we present the theoretical framework of quantum speed limits (QSLs) in terms of trajectory ensembles in phase space. This indicates that the QSL can be thought of as the summation of the connecting harmonic oscillators: the connections between the points of the system in phase space and the trajectory ensemble. Two typical models, the time-dependent harmonic oscillator and the undriven harmonic oscillator coupled to a thermal bath, are investigated by employing the theoretical framework. Our results from this perspective are in agreement with previous treatments.
As one of the fundamental questions, quantum speed limit (QSL) is of vital role in the community of quantum physics, which sets the ultimate, maximal speed with which a quantum system can evolve. QSL originated from the understanding of the Heisenberg uncertainty relation. In 1945, Mandelstam and Tamm found the time Δt in the celebrated relation ΔEΔt≥ℏ could be thought as the minimum time that a quantum system needs to evolve from an initial state to the orthogonal state for unitary evolution. QSL sets the lower bound of the evolution. They established the relationship between the minimum evolution time ( τQSL ) and the energy uncertainty of the system (ΔE), namely τQSL=πℏ/(2ΔE) , the Mandelstam-Tamm (MT) bound. Later, Margolus and Levitin derived another different bound for a closed system which connected the τQSL with the average energy of the system ( 〈E〉 ), i.e., τQSL=πℏ/(2〈E〉) , the Margolus-Levitin (ML) bound. Combining the MT and ML bounds together, we can have a unified bound for a unitary system and orthogonal states. In the case of open system dynamics, because there is energy and/or coherence exchange between the system and the environment, the orthogonal final state of the initial state can hardly be reached any longer. An investigation of the QSL on non-unitary quantum evolution needs to be considered. The generalized QSLs are established by employing the distance between the distinguishable states. Different metrics define different distances, leading to different QSL bounds. We show a few typical generalized QSL bounds based on different metrics. And the unified QSL bound for non-unitary evolution is obtained by combining the generalized MT and ML bounds under the same metric. The generations and developments of QSL have successfully generalized the MT bounds. Even the ML bounds have been attempted to generalize, the current studies of ML bound are thought as the significantly harder task, and keep as the open questions. The influences of external environment and quantum system dynamics which include the role of entanglement in QSL for open dynamics and many-body systems, the relation between the maximum interaction speed and QSL in quantum spin systems, the non-Markovianity effect of the environment on accelerating the speed of evolution have been investigated. Furthermore, by employing the gauge invariant distance in the Riemannian metric, we establish a distinct QSL bound, which relates the quantum speed limit time to the geometry phase, and presents the geometric and gauge invariant properties of it. The bound shows the impacts of the transmission modes of the state vectors on the evolution path in the manifold. Different from the established MT and ML bounds, our bound establishes the relation between the QSL and the Pancharatnam connection 1-form, and our results show that the QSL can be controlled via manipulating the generalized Pahcharatnam connection. Also, our results of QSL are holonomy via the generalized Pahcharatnam connection, which show us a possible technique to speed up the holonomic quantum computation. The bound is also interesting to the metrology and non-equilibrium thermodynamics, etc. QSL has also been studied in the quantum representation of Wigner function. In the future research of quantum speed limit, there are many issues that need to be considered, such as the generalized ML bound, the unified QSL bound under different metrics, the application of maximum rate of quantum information derived from the QSL, and the relation between the quantum speed limit and the quantum heat engine, etc.
Quantum speed limits (QSLs) provide an upper bound for the speed of evolution of quantum states in any physical process. Based on the Stratonovich-Weyl correspondence, we derive a universal QSL bound in arbitrary phase spaces that is applicable for both continuous variable systems and finite-dimensional discrete quantum systems. This QSL bound allows the determination of speed limit bounds in specific phase spaces that are tighter than those in Wigner phase space or Hilbert space under the same metric, as illustrated by several typical examples, e.g., a single-mode free field and N-level quantum systems in phase spaces. This QSL bound also provides an experimentally realizable way to examine the speed limit in phase spaces relevant to applications in quantum information and quantum optics.
Limitations to the speed of evolution of quantum systems, typically referred to as quantum speed limits (QSLs), have important consequences for quantum control problems. However, in its standard formulation, is not straightforward to obtain meaningful QSL bounds for time-dependent Hamiltonians with unknown control parameters. In this paper we present a short introductory overview of quantum speed limit for unitary dynamics and its connection to quantum control. We then analyze potential methods for obtaining new bounds on control times inspired by the QSL. We finally extend the work in [Poggi, Lombardo and WisniackiEPL104 40005 (2013)] by studying the properties and limitations of these new bounds in the context of a driven two-level quantum system.
When is the quantum speed limit (QSL) really quantum? Typically, a vanishing QSL time is an indicator of an emergent classicality. However, it is still not entirely understood what precise aspects of classicality lead to diverging quantum speeds. Here, we show that vanishing QSL times (or, equivalently, diverging quantum speeds) can be traced back to reduced uncertainty in quantum observables. We illustrate this mechanism by developing a QSL formalism for continuous variable quantum systems undergoing general Gaussian dynamics. For these systems, we show that three typical scenarios leading to vanishing QSL times, namely large squeezing, small effective Planck's constant, and large particle number, can be fundamentally connected to each other. Finally, by studying the dynamics of open quantum systems and mixed states, we show that the addition of classical noise typically increases the QSL time.
We introduce action quantum speed limits (QSLs) as a family of bounds on the minimal time to connect two states that, unlike the usual geometric approach, crucially depend on how the path is traversed, i.e., on the instantaneous speed. The two approaches provide consistent bounds when the instantaneous speed is optimized along a fixed path and we demonstrate this explicitly for the case of a thermalizing qubit employing techniques from optimal control theory. In addition, we critically analyze the interpretation of QSLs based on different choices of metric establishing that, in general, these open-system QSL times provide an indication of optimality with respect to the geodesic path, rather than necessarily being indicative of an achievable minimal time.
When is the quantum speed limit (QSL) really quantum?While vanishing QSL times often indicate emergent classical behavior, it is still not entirely understood what precise aspects of classicality are at the origin of this dynamical feature.Here, we show that vanishing QSL times (or, equivalently, diverging quantum speeds) can be traced back to reduced uncertainty in quantum observables and can thus be understood as a consequence of emerging classicality for these particular observables.We illustrate this mechanism by developing a QSL formalism for continuous-variable quantum systems undergoing general Gaussian dynamics.For these systems, we show that three typical scenarios leading to vanishing QSL times, namely large squeezing, small effective Planck's constant, and large particle number, can be fundamentally connected to each other.In contrast, by studying the dynamics of open quantum systems and mixed states, we show that the classicality that emerges due to incoherent mixing of states from the addition of classical noise typically increases the QSL time.