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.

Achieving effectively adiabatic dynamics is a ubiquitous goal in almost all areas of quantum physics.Here, we study the speed with which a quantum system can be driven when employing transitionless quantum driving. As a main result, we establish a rigorous link between this speed, the quantum speed limit, and the (energetic) cost of implementing such a shortcut to adiabaticity. Interestingly, this link elucidates a trade-off between speed and cost, namely that instantaneous manipulation is impossible as it requires an infinite cost. These findings are illustrated for two experimentally relevant systems -the parametric oscillator and the Landau-Zener model -which reveal that the spectral gap governs the quantum speed limit as well as the cost for realizing the shortcut.

We study transitionless quantum driving in an infinite-range many-body system described by the LipkinMeshkov-Glick model. Despite the correlation length being always infinite the closing of the gap at the critical point makes the driving Hamiltonian of increasing complexity also in this case. To this aim we develop a hybrid strategy combining a shortcut to adiabaticity and optimal control that allows us to achieve remarkably good performance in suppressing the defect production across the phase transition. DOI: 10.1103/PhysRevLett.114.177206 PACS numbers: 75.10.Jm, 05.30.Rt, 64.60.Ht, 73.43.Nq The dynamical evolution of a quantum system often has to be tailored so that a given initial state is transformed into a suitably chosen target one. In cases such as this, the use of techniques for quantum optimal control can be key in engineering an efficient protocol. Over the years, formal control methods have been devised, both in the classical and quantum scenario [1]. To date, optimal control has been proven beneficial in a multitude of fields, ranging from molecular physics to quantum information processing or high precision measurements [2]. Only very recently, however, this framework has been extended so as to cope with the rich phenomenology and complexity of quantum many-body systems [3,4]. In this context, quantum optimal control has been shown to be crucial for the design of schemes for the preparation of many-body quantum states [3,5,6], the exploration of the experimentally achievable limits in quantum interferometry [7], and the cooling of quantum systems [8].Needless to say, quantum optimal control is not the only way to design the dynamical evolution of a quantum system, and one could consider simpler (suboptimal) ways to drive the desired dynamics. For instance, using the adiabatic theorem we are able to constrain a quantum system to remain in an eigenstate during any evolution. However, in order for such a technique to be accurate, it should operate on a rather long time scale. Unwanted transitions between the state we would like to confine the system into and other ones in its spectrum, which are induced by the unavoidably finite-speed nature of an evolution and ultimately limit the precision of the adiabatic dynamics, can be suppressed by adding suitable corrections to the Hamiltonian guiding the evolution [9,10]. This form of quantum control, named the shortcut to adiabaticity (STA), has been considered in a variety of different situations, and recently reviewed in Ref. [11]. An experimental implementation using cold atomic gases has been reported in Ref. [12].Recently, the idea of a STA has been extended to quantum many-body systems, a context where it can be potentially very beneficial. The STA was first employed in the suppression of defects produced when crossing a quantum phase transition in the paradigm model embodied by the one-dimensional Ising model [13]. Despite such potential, a crucial feature that emerges from the use of a STA in many-body scenarios is the inherent complexity of the ...

Unitary processes allow for the transfer of work to and from Hamiltonian systems. However, to achieve non-zero power for the practical extraction of work, these processes must be performed within a finite-time, which inevitably induces excitations in the system. We show that depending on the time-scale of the process and the physical realization of the external driving employed, the use of counterdiabatic quantum driving to extract more work is not always effective. We also show that by virtue of the two-time energy measurement definition of quantum work, the cost of counterdiabatic driving can be significantly reduced by selecting a restricted form of the driving Hamiltonian that depends on the outcome of the first energy measurement. Lastly, we introduce a measure, the exigency, that quantifies the need for an external driving to preserve quantum adiabaticity which does not require knowledge of the explicit form of the counterdiabatic drivings, and can thus always be computed. We apply our analysis to systems ranging from a two-level Landau-Zener problem to many-body problems, namely the quantum Ising and Lipkin-Meshkov-Glick models.

We study the dynamics of a quantum system whose interaction with an environment is described by a collision model, i.e. the open dynamics is modelled through sequences of unitary interactions between the system and the individual constituents of the environment, termed "ancillas", which are subsequently traced out. In this setting non-Markovianity is introduced by allowing for additional unitary interactions between the ancillas. For this model, we identify the relevant system-environment correlations that lead to a non-Markovian evolution. Through an equivalent picture of the open dynamics, we introduce the notion of "memory depth" where these correlations are established between the system and a suitably sized memory rendering the overall system+memory evolution Markovian. We extend our analysis to show that while most system-environment correlations are irrelevant for the dynamical characterization of the process, they generally play an important role in the thermodynamic description. Finally, we show that under an energy-preserving system-environment interaction, a non-monotonic time behaviour of the heat flux serves as an indicator of non-Markovian behaviour.

With the advent of quantum technologies comes the requirement of building quantum components able to store energy to be used whenever necessary, i.e. quantum batteries. In this paper we exploit an adiabatic protocol to ensure a stable charged state of a three-level quantum battery which allows to avoid the spontaneous discharging regime. We study the effects of the most relevant sources of noise on the charging process and, as an experimental proposal, we discuss superconducting transmon qubits. In addition we study the self-discharging of our quantum battery where it is shown that spectrum engineering can be used to delay such phenomena.

We perform an extensive study of the properties of global quantum correlations in finite-size one-dimensional quantum spin models at finite temperature. By adopting a recently proposed measure for global quantum correlations (Rulli and Sarandy 2011 Phys. Rev. A 84 042109), called global discord, we show that critical points can be neatly detected even for many-body systems that are not in their ground state. We consider the transverse Ising model, the cluster-Ising model where three-body couplings compete with an Ising-like interaction, and the nearest-neighbor XX Hamiltonian in transverse magnetic field. These models embody our canonical examples showing the sensitivity of global quantum discord close to criticality. For the Ising model, we find a universal scaling of global discord with the critical exponents pertaining to the Ising universality class.Entanglement and criticality in quantum many-body systems have been shown to be strongly and intimately connected concepts [1,2]. The body of work performed with the aim of grasping the implications that critical changes in the ground state of a given Hamiltonian model have for the sharing of entanglement by the parties of a quantum many-body systems is now quite substantial [3]. This has resulted in important progress in our understanding of the interplay between critical phenomena of interacting many-body systems and the setting up of genuinely quantum features. In turn, such success has proven the effectiveness of the cross-fertilization of quantum statistical mechanics by techniques and interpretations that are typical of quantum information theory.Yet, it has recently emerged that the way correlations of non-classical nature manifest themselves is not necessarily coincident with entanglement, and a much broader definition of quantum correlations should be given [4,5]. This is encompassed very effectively in the formulation of so-called quantum discord as a measure striving to capture the above-mentioned broadness of quantum correlations [6]. In analogy with the case of entanglement, the relation between quantum discord and the features of quantum many-body models is fundamentally interesting for the understanding of the role that the settlement of quantumness of correlations play in determining the critical properties of such models. A systematic analysis in this sense, which has only recently been considered [7][8][9][10][11][12][13], is thus highly desirable. This is even more important given that some of the investigations performed so far have indicated that quantum discord is more sensible than entanglement in revealing quantum critical points [9], even for systems that are not at zero temperature [14]. This is a particularly relevant result, whose validity should also be checked for models that are both finite sized and at finite temperature. The motivations for such an endeavor stem from the fact that, likely, the properties of quantum many-body systems will be addressed experimentally in systems consisting of, for instance, cold atoms loaded in optica...

We investigate a thermodynamic cycle using a Bose-Einstein condensate (BEC) with nonlinear interactions as the working medium. Exploiting Feshbach resonances to change the interaction strength of the BEC allows us to produce work by expanding and compressing the gas. To ensure a large power output from this engine these strokes must be performed on a short timescale, however such non-adiabatic strokes can create irreversible work which degrades the engine's efficiency. To combat this, we design a shortcut to adiabaticity which can achieve an adiabatic-like evolution within a finite time, therefore significantly reducing the out-of-equilibrium excitations in the BEC. We investigate the effect of the shortcut to adiabaticity on the efficiency and power output of the engine and show that the tunable nonlinearity strength, modulated by Feshbach resonances, serves as a useful tool to enhance the system's performance.