Quantum Kinematics Research Articles

Universality of quantum time dilation

Time dilation is a difference in measured time between two clocks that either move with different velocities or experience different gravitational potentials. Both of these effects stem from the theory of relativity and are usually associated with classically defined trajectories, characterized by position, momentum, and acceleration. However, when spatial degrees of freedom are treated in a quantum way and a clock is allowed to be in a coherent superposition of either two momenta or two heights, additional quantum corrections to classical time dilation appear, called kinematic and gravitational quantum time dilations, respectively. We show that similarly to its classical counterpart, kinematic quantum time dilation is universal for any clock mechanism, while gravitational quantum time dilation is not. We also show that although both of these effects reduce to incoherent averaging of different classical time dilation contributions, there exists an additional quantum time dilation effect that has no classical analog and can be extracted from higher-order corrections to the system’s Hamiltonian.

Schrödinger dynamics in length-scale hierarchy: from spatial rescaling to Huygens-like proliferation of Gaussian wavepackets

Studying possible laws, rules, and mechanisms of time-evolution of quantum wavefunctions leads to deeper understanding about the essential nature of the Schrödinger dynamics and interpretation on what the quantum wavefunctions are. As such, we attempt to clarify the mechanical and geometrical processes of deformation and bifurcation of a Gaussian wavepacket of the Maslov type from the viewpoint of length-scale hierarchy in the wavepacket size relative to the range of relevant potential functions. Following the well-known semiclassical view that (1) Newtonian mechanics gives a phase space geometry, which is to be projected onto configuration space to determine the basic amplitude of a wavefunction (the primitive semiclassical mechanics), our study proceeds as follows. (2) The quantum diffusion arising from the quantum kinematics makes the Gaussian exponent complex-valued, which consequently broadens the Gaussian amplitude and brings about a specific quantum phase. (3) The wavepacket is naturally led to bifurcation (branching), when the packet size gets comparable with or larger than the potential range. (4) Coupling between the bifurcation and quantum diffusion induces the Huygens-principle like wave dynamics. (5) All these four processes are collectively put into a path integral form. We discuss some theoretical consequences from the above analyses, such as (i) a contrast between the δ-function-like divergence of a wavefunctions at focal points and the mesoscopic finite-speed shrink of a Gaussian packet without instantaneous collapse, (ii) the mechanism of release of the zero-point energy to external dynamics and that of tunneling, (iii) relation between the resultant stochastic quantum paths and wave dynamics, and so on.

Control Landscape of Measurement-Assisted Transition Probability for a Three-Level Quantum System with Dynamical Symmetry

Quantum systems with dynamical symmetries have conserved quantities that are preserved under coherent control. Therefore, such systems cannot be completely controlled by means of only coherent control. In particular, for such systems, the maximum transition probability between some pairs of states over all coherent controls can be less than one. However, incoherent control can break this dynamical symmetry and increase the maximum attainable transition probability. The simplest example of such a situation occurs in a three-level quantum system with dynamical symmetry, for which the maximum probability of transition between the ground and intermediate states using only coherent control is 1/2, whereas it is about 0.687 using coherent control assisted by incoherent control implemented through the non-selective measurement of the ground state, as was previously analytically computed. In this work, we study and completely characterize all critical points of the kinematic quantum control landscape for this measurement-assisted transition probability, which is considered as a function of the kinematic control parameters (Euler angles). The measurement-driven control used in this work is different from both quantum feedback and Zeno-type control. We show that all critical points are global maxima, global minima, saddle points or second-order traps. For comparison, we study the transition probability between the ground and highest excited states, as well as the case when both these transition probabilities are assisted by incoherent control implemented through the measurement of the intermediate state.

κ-Galilean and κ-Carrollian noncommutative spaces of worldlines

The noncommutative spacetimes associated to the κ-Poincaré relativistic symmetries and their “non-relativistic” (Galilei) and “ultra-relativistic” (Carroll) limits are indistinguishable, since their coordinates satisfy the same algebra. In this work, we show that the three quantum kinematical models can be differentiated when looking at the associated spaces of time-like worldlines. Specifically, we construct the noncommutative spaces of time-like geodesics with κ-Galilei and κ-Carroll symmetries as contractions of the corresponding κ-Poincaré space and we show that these three spaces are defined by different algebras. In particular, the κ-Galilei space of worldlines resembles the so-called Euclidean Snyder model, while the κ-Carroll space turns out to be commutative. Furthermore, we identify the map between quantum spaces of geodesics and the corresponding noncommutative spacetimes, which requires to extend the space of geodesics by adding the noncommutative time coordinate.

Quantum kinematics in terms of observable quantities and the chirality of entangled two-qubit states.

We consider the kinematics of bipartite quantum states as determined by observable quantities, in particular the Bloch vectors of the subsystems. In examining the simplest case of a pair of two-level systems, there is a remarkable connection between the presence of non-classical correlations and the chirality of the two bases generated by the singular value decomposition of the correlation matrix of the Bloch vectors. We investigate the limits imposed by quantum mechanics of this effect and its relationship with other methods on quantifying the system's non-classical behavior.

A novel quantum model of forward kinematics based on quaternion/Pauli gate equivalence: Application to a six-jointed industrial robotic arm

A novel quantum-based kinematic model for computing the position and the orientation of the six-jointed IRB 140 ABB robotic arm is presented. Position is determined through successive elementary translations and rotations from the end effector down to the base frame, whereas orientation is derived from quaternion representation. The model is based on the equivalence between quaternions and Pauli matrices or gates. Based on comparison to experimental results, the quantum kinematic model is valid. Its main advantage is that it uses only one qubit to model both position and orientation, which represents a major saving in terms of computing resources and time. This model could guide development of other quantum models and simulation software to be used in future quantum computers and could also inform classical computing theory for robotic arm kinematics modeling.

STABILITY OF FLUID/PLASMA IN THE PRESENCE OF QUANTUM PHYSICS SATURATING A POROUS MEDIU

The present investigation deals with the quantum effects on the Rayleigh –Taylor instability in an infinitely electrically conducting inhomogeneous stratified incompressible viscoelastic fluid/plasma through a porous medium. The linear growth rate is derived for the case where a plasma with exponential density, viscosity, viscoelasticity and quantum parameter distribution is confined between two rigid planes. The solution of the linearized equations of the system together with the appropriate boundary conditions leads to derive the dispersion relation (the relation between the normalized growth rate and square normalized wavenumber) using normal mode technique. The behavior of growth rate with respect to quantum effect and kinematic viscoelasticity are examined in the presence of porous medium, medium permeability and kinematic viscoelasticity. It is observed that the quantum effects bring more stability for a certain wave number band on the growth rate on the unstable configuration.

Canonical loop quantization of the lowest-order projectable Horava gravity

The Hamiltonian formulation of the lowest-order projectable Horava gravity, namely the so-called $\lambda$-$R$ gravity, is studied. Since a preferred foliation has been chosen in projectable Horava gravity, there is no local Hamiltonian constraint in the theory. In contrast to general relativity, the constraint algebra of $\lambda$-$R$ gravity forms a Lie algebra. By canonical transformations, we further obtain the connection-dynamical formalism of the $\lambda$-R gravity theories with real $su(2)$-connections as configuration variables. This formalism enables us to extend the scheme of non-perturbative loop quantum gravity to the $\lambda$-$R$ gravity. While the quantum kinematical framework is the same as that for general relativity, the Hamiltonian constraint operator of loop quantum $\lambda$-$R$ gravity can be well defined in the diffeomorphism-invariant Hilbert space. Moreover, by introducing a global dust degree of freedom to represent a dynamical time, a physical Hamiltonian operator with respect to the dust can be defined and the physical states satisfying all the constraints are obtained.

Reference systems and their origins

It is demonstrated how the choice of a reference system origin influences our interpretation of the natural laws. Instead of placing it arbitrarily as is customary in mechanics, we locate it at an energy minimum, or zero point of material systems. In an example from celestial mechanics, the sun defines the point of minimum gravitational potential and preferred origin for the planetary system. As a comparison, Heisenberg's derivation of quantum kinematics is shown to be deficient, because microscopes do not define the point of minimum energy of an atomic system. To improve upon it, new interpretations of quantum kinematics and mechanics are proposed by locating the reference system origin at the zero point of an atomic system, where the Hamiltonian, H = T + V, equals zero. Experiments with atomic clocks are cited in support of the new physical model. The Schwarzschild solution relative to a mass point is proposed as the preferred interpretation of a general relativity theory due to the location of the origin at a zero point. These interpretations of long-standing theories provide new points of view to familiar problems.

Cylindrically symmetric 2 + 1 gravity in terms of global variables: Quantum dynamics

We perform quantization of a model in which gravity is coupled to a circular dust shell in [Formula: see text] spacetime dimensions. Canonical analysis shows that momentum space of this model is [Formula: see text]-space, and the global chart for it is provided by the Euler angles. In quantum kinematics, this results in non-commutativity in coordinate space and discreteness of the shell radius in timelike region, which includes the collapse point. At the level of quantum dynamics, we find transition amplitudes between zero and non-zero eigenvalues of the shell radius, which describe the rate of gravitational collapse (bounce). Their values are everywhere finite, which could be interpreted as resolution of the central singularity.

Dapor-Liegener formalism of loop quantum cosmology for Bianchi I spacetimes

We discuss the quantization of vacuum Bianchi I spacetimes in the modified formalism of loop quantum cosmology recently proposed by Dapor and Liegener. This modification is based on a regularization procedure where both the Euclidean and Lorentzian terms of the Hamiltonian are treated independently. Whereas the Euclidean part has already been dealt with in the literature for Bianchi I spacetimes, the Lorentzian one has not yet been represented quantum mechanically. After a brief review of the quantum kinematics and the quantization of the Euclidean sector, we represent the Lorentzian part of the Hamiltonian constraint by an operator according to the factor ordering rules of the Martin-Benito--Mena Marugan--Olmedo prescription. We study the general properties of this quantum operator and the superselection rules derived therefrom, resulting in an action similar to that of the Euclidean operator except that the orientation of the densitized triad is not preserved, a fact which leads to a generic enlargement of the superselection sectors. We conclude with an explanation of the mechanism that prevents this enlargement in the isotropic case and a comment on the effect of alternative prescriptions for the implementation of the improved dynamics.

On Clifford groups in quantum computing

The term Clifford group was introduced in 1998 by D. Gottesmann in his investigation of quantum error-correcting codes. The simplest Clifford group in multiqubit quantum computation is generated by a restricted set of unitary Clifford gates - the Hadamard, π/4-phase and controlled-X gates. Because of this restriction the Clifford model of quantum computation can be efficiently simulated on a classical computer (the Gottesmann-Knill theorem). However, this fact does not diminish the importance of the Clifford model, since it may serve as a suitable starting point for a full-fledged quantum computation.In the general case of a single or composite quantum system with finite-dimensional Hilbert space the finite Weyl-Heisenberg group of unitary operators defines the quantum kinematics and the states of the quantum register. Then the corresponding Clifford group is defined as the group of unitary operators leaving the Weyl-Heisenberg group invariant. The aim of this contribution is to show that our comprehensive results on symmetries of the Pauli gradings of quantum operator algebras - covering any single as well as composite finite quantum systems - directly correspond to Clifford groups defined as quotients with respect to U(1).

The Twofold Role of Observables in Classical and Quantum Kinematics

Observables have a dual nature in both classical and quantum kinematics: they are at the same time \emph{quantities}, allowing to separate states by means of their numerical values, and \emph{generators of transformations}, establishing relations between different states. In this work, we show how this two-fold role of observables constitutes a key feature in the conceptual analysis of classical and quantum kinematics, shedding a new light on the distinguishing feature of the quantum at the kinematical level. We first take a look at the algebraic description of both classical and quantum observables in terms of Jordan-Lie algebras and show how the two algebraic structures are the precise mathematical manifestation of the two-fold role of observables. Then, we turn to the geometric reformulation of quantum kinematics in terms of K\"ahler manifolds. A key achievement of this reformulation is to show that the two-fold role of observables is the constitutive ingredient defining what an observable is. Moreover, it points to the fact that, from the restricted point of view of the transformational role of observables, classical and quantum kinematics behave in exactly the same way. Finally, we present Landsman's general framework of Poisson spaces with transition probability, which highlights with unmatched clarity that the crucial difference between the two kinematics lies in the way the two roles of observables are related to each other.

Another look through Heisenberg’s microscope

Heisenberg introduced his famous uncertainty relations in a seminal 1927 paper entitled The Physical Content of Quantum Kinematics and Mechanics. He motivated his arguments with a gedanken experiment, a gamma ray microscope to measure the position of a particle. A primary result was that, due to the quantum nature of light, there is an inherent uncertainty in the determinations of the particle’s position and momentum dictated by an indeterminacy relation, . Heisenberg offered this demonstration as ‘a direct physical interpretation of the [quantum mechanical] equation ’ but considered the indeterminacy relation to be much more than this. He also argued that it implies limitations on the very meanings of position and momentum and emphasised that these limitations are the source of the statistical character of quantum mechanics. In addition, Heisenberg hoped but was unable to demonstrate that the laws of quantum mechanics could be derived directly from the uncertainty relation. In this paper, we revisit Heisenberg’s microscope and argue that the Schrödinger equation for a free particle does indeed follow from the indeterminacy relation together with reasonable statistical assumptions.

Probing mesoscopic crystals with electrons: One-step simultaneous inelastic and elastic scattering theory

A consistent theory of electron energy-loss spectroscopy (EELS) includes two indispensable elements: (i) electronic response of the target system and (ii) quantum kinematics of probing electrons. While for the bulk materials and their surfaces separating these two aspects and focusing on the former is the usual satisfactory practice (the energy-loss functions formalism), we show that, for quasi-2D crystals, the interplay of the system's electronic response and the details of the probe's motion affects EEL spectra dramatically, and it must be taken into account for the reliable interpretation of the experiment. To this end, we come up with a unified theory which, on the same footing, treats both the long- and short-range scattering, within both the transmission and reflection experimental setups. Our calculations performed for graphene reveal a phenomenon of a strong coupling between the $\pi+\sigma$ plasmon excitation and elastic {\it scattering resonances}. Freed from the conventions of the energy-loss functions formalism, our theory serves as a consistent and systematic means of the understanding of the EELS of quasi-2D materials.

Kinematic projective quantum states for loop quantum gravity coupled to tensor fields

We present a construction of kinematic quantum states for theories of tensor fields of an arbitrary sort. The construction is based on projective techniques by Kijowski. Applying projective quantum states for Loop Quantum Gravity (LQG) obtained by Lanéry and Thiemann we construct quantum states for LQG coupled to tensor fields.

Noncommutative Relativistic Spacetimes and Worldlines from 2 + 1 Quantum (Anti-)de Sitter Groups

Theκ-deformation of the (2 + 1)D anti-de Sitter, Poincaré, and de Sitter groups is presented through a unified approach in which the curvature of the spacetime (or the cosmological constant) is considered as an explicit parameter. The Drinfel’d-double and the Poisson–Lie structure underlying theκ-deformation are explicitly given, and the three quantum kinematical groups are obtained as quantizations of such Poisson–Lie algebras. As a consequence, the noncommutative (2 + 1)D spacetimes that generalize theκ-Minkowski space to the (anti-)de Sitter ones are obtained. Moreover, noncommutative 4D spaces of (time-like) geodesics can be defined, and they can be interpreted as a novel possibility to introduce noncommutative worldlines. Furthermore, quantum (anti-)de Sitter algebras are presented both in the known basis related to 2 + 1 quantum gravity and in a new one which generalizes the bicrossproduct one. In this framework, the quantum deformation parameter is related to the Planck length, and the existence of a kind of “duality” between the cosmological constant and the Planck scale is also envisaged.

High-level support for simulations in astro- and elementary particle physics

To fully exploit the potential of modern supereomputing systems, when performing particle and astrophysics simulations, the special user support unite called simulation laboratory “Elementary- and Astro-Particle Physics” (Sim-Lab E&A Particle) is established at Steinbuch Centre for Computing of the Karlsruhe Institute for Technology. SimLab is providing advanced support to scientific groups in developing the simulation software and porting them into up-to-date super-computing infrastructures. Some details of how the SimLab is governed and cooperation with developers and users of codes are presented in this work in progress paper including examples about code THiSMPI for simulation of particle accelerations in supernovae shock fronts and the KB3D code for solving quantum kinematic equations that could be used in simulations of heavy ion collisions or laser excited semiconductor systems.

A quantum kinematics for asymptotically flat gravity

We construct a quantum kinematics for asymptotically flat gravity based on the Koslowski–Sahlmann (KS) representation. The KS representation is a generalization of the representation underlying loop quantum gravity (LQG) which supports, in addition to the usual LQG operators, the action of ‘background exponential operators’, which are connection dependent operators labelled by ‘background’ su(2) electric fields. KS states have, in addition to the LQG state label corresponding to one dimensional excitations of the triad, a label corresponding to a ‘background’ electric field that describes three dimensional excitations of the triad. Asymptotic behaviour in quantum theory is controlled through asymptotic conditions on the background electric fields that label the states and the background electric fields that label the operators. Asymptotic conditions on the triad are imposed as conditions on the background electric field state label while confining the LQG spin net graph labels to compact sets. We show that KS states can be realised as wave functions on a quantum configuration space of generalized connections and that the asymptotic behaviour of each such generalized connection is determined by that of the background electric fields which label the background exponential operators. Similar to the spatially compact case, the Gauss law and diffeomorphism constraints are then imposed through group averaging techniques to obtain a large sector of gauge invariant states. It is shown that this sector supports a unitary action of the group of asymptotic rotations and translations and that, as anticipated by Friedman and Sorkin, for appropriate spatial topology, this sector contains states that display fermionic behaviour under rotations.

Limits of optimal control yields achievable with quantum controllers

In quantum optimal control theory, kinematic bounds are the minimum and maximum values of the control objective achievable for any physically realizable system dynamics. For a given initial state of the system, these bounds depend on the nature and state of the controller. We consider a general situation where the controlled quantum system is coupled to both an external classical field (referred to as a classical controller) and an auxiliary quantum system (referred to as a quantum controller). In this general situation, the kinematic bound is between the classical kinematic bound (CKB), corresponding to the case when only the classical controller is available, and the quantum kinematic bound (QKB), corresponding to the ultimate physical limit of the objective's value. Specifically, when the control objective is the expectation value of a quantum observable (a Hermitian operator on the system's Hilbert space), the QKBs are the minimum and maximum eigenvalues of this operator. We present, both qualitatively and quantitatively, the necessary and sufficient conditions for surpassing the CKB and reaching the QKB, through the use of a quantum controller. The general conditions are illustrated by examples in which the system and controller are initially in thermal states. The obtained results provide a basis for the design of quantum controllers capable of maximizing the control yield and reaching the ultimate physical limit.