Abstract A common feature of collapse models and an expected signature of the quantization of gravity at energies well below the Planck scale is the deviation from ordinary quantum-mechanical behavior. Here, we analyze the general consequences of such modifications from the point of view of quantum information theory and we anticipate applications to different quantum systems. We show that quantum systems undergo corrections to the quantum speed limit which, in turn, imply the modification of the Heisenberg limit for parameter estimation. Our results hold for a wide class of scenarios beyond ordinary quantum mechanics. For some nonlocal models inspired by quantum gravity, the bounds are found to oscillate in time, an effect that could be tested in future high-precision quantum experiments.
Abstract A common feature of collapse models and an expected signature of the quantization of gravity at energies well below the Planck scale is the deviation from ordinary quantum-mechanical behavior. Here, we analyze the general consequences of such modifications from the point of view of quantum information theory and we anticipate applications to different quantum systems. We show that quantum systems undergo corrections to the quantum speed limit which, in turn, imply the modification of the Heisenberg limit for parameter estimation. Our results hold for a wide class of scenarios beyond ordinary quantum mechanics. For some nonlocal models inspired by quantum gravity, the bounds are found to oscillate in time, an effect that could be tested in future high-precision quantum experiments.
Abstract Phenomenological models aiming to join gravity and quantum mechanics often predict effects that are potentially measurable in refined low-energy experiments. For instance, modified commutation relations between position and momentum, that account for a minimal scale length, yield a dynamics that can be codified in additional Hamiltonian terms. When applied to the paradigmatic case of a mechanical oscillator, such terms, at the lowest order in the deformation parameter, introduce a weak intrinsic nonlinearity and, consequently, deviations from the classical trajectory. This point of view has stimulated several experimental proposals and realizations, leading to meaningful upper limits to the deformation parameter. All such experiments are based on classical mechanical oscillators, i.e., excited from a thermal state. We remark indeed that decoherence, that plays a major role in distinguishing the classical from the quantum behavior of (macroscopic) systems, is not usually included in phenomenological quantum gravity models. However, it would not be surprising if peculiar features that are predicted by considering the joined roles of gravity and quantum physics should manifest themselves just on purely quantum objects. On the basis of this consideration, we propose experiments aiming to observe possible quantum gravity effects on macroscopic mechanical oscillators that are preliminary prepared in a high purity state, and we report on the status of their realization. Graphical abstract
Abstract Several approaches to quantum gravity lead to nonlocal modifications of fields’ dynamics. This, in turn, can give rise to nonlocal modifications of quantum mechanics at non-relativistic energies. Here, we analyze the nonlocal Schrödinger evolution of a quantum harmonic oscillator in one such scenario, where the problem can be addressed without the use of perturbation theory. We demonstrate that although deviations from standard quantum predictions occur at low occupation numbers, where they could potentially be detected or constrained by high-precision experiments, the classical limits of quantum probability densities and free energy remain unaffected up to energies comparable with the nonlocality scale. These results provide an example of nonlocal quantum dynamics compatible with classical predictions, suggesting massive quantum objects as a promising avenue for testing some phenomenological aspects of quantum gravity.
Many quantum-gravity scenarios predict a minute modification of the canonical commutator, known as the generalized uncertainty principle (GUP), whose low-energy signatures are, in principle, accessible to state-of-the-art laboratory tests. We compute first-order minimal-length corrections to the quantum speed limit (QSL) for three cases: uniform superpositions in an infinite square well, coherent harmonic-oscillator states, and squeezed-oscillator states. We identify a universal amplification law: for any pure state, the fractional shift of either speed limit scales linearly with β and algebraically with the state's effective Hilbert-space size. As the effective Hilbert-space dimension can be exceedingly large, the associated minimal-length signatures are amplified by several orders of magnitude. Using high-precision matter-wave timing data, we set a direct bound on the GUP parameter β, which quantifies minimal-length quantum-gravity effects. Our analysis indicates that phase-locked, short-time overlap fits on kilogram-scale optical-spring modes can tighten this bound by orders of magnitude. We outline two implementable measurement pipelines -- continuous back-action-evading single-quadrature readout and stroboscopic, phase-locked pulsed tomography -- that exploit this leverage, making QSL-based timing a practical, near-term probe of minimal-length physics on quantum-optical and optomechanical platforms.
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.
Abstract The laws of quantum mechanics allow to perform measurements whose precision supersedes results predicted by classical parameter estimation theory. That is, the precision bound imposed by the central limit theorem in the estimation of a broad class of parameters, like atomic frequencies in spectroscopy or external magnetic field in magnetometry, can be overcomewhen using quantum probes. Environmental noise, however, generally alters the ultimate precision that can be achieved in the estimation of an unknown parameter. This tutorial reviews recent theoretical work aimed at obtaining general precision bounds in the presence of an environment.We adopt a complementary approach,wherewe first analyze the problem within the general framework of describing the quantum systems in terms of quantum dynamical maps and then relate this abstract formalism to a microscopic description of the system’s dissipative time evolution.We will show that although some forms of noise do render quantum systems standard quantum limited, precision beyond classical bounds is still possible in the presence of different forms of local environmental fluctuations.
The Heisenberg limit provides a fundamental bound on the achievable estimation precision with a limited number of N resources used (e.g., atoms, photons, etc.). Using entangled quantum states makes it possible to scale the precision with N better than when resources would be used independently. Consequently, the optimal use of all resources involves accumulating them in a single execution of the experiment. Unfortunately, that implies that the most common theoretical tool used to analyze metrological protocols - quantum Fisher information (QFI) - does not allow for a reliable description of this problem, as it becomes operationally meaningful only with multiple repetitions of the experiment. In this thesis, using the formalism of Bayesian estimation and the minimax estimator, I derive asymptotically saturable bounds on the precision of the estimation for the case of noiseless unitary evolution. For the case where the number of resources N is strictly constrained, I show that the final measurement uncertainty is π times larger than would be implied by a naive use of QFI. I also analyze the case where a constraint is imposed only on the average amount of resources, the exact value of which may fluctuate (in which case QFI does not provide any universal bound for precision). In both cases, I study the asymptotic saturability and the rate of convergence of these bounds. In the following part, I analyze the problem of the Heisenberg limit when multiple parameters are measured simultaneously on the same physical system. In particular, I investigate the existence of a gain from measuring all parameters simultaneously compared to distributing the same amount of resources to measure them independently. I focus on two examples - the measurement of multiple phase shifts in a multi-arm interferometer and the measurement of three magnetic field components.