Presence Of Conservation Laws Research Articles

Measurement disturbance and conservation laws in quantum mechanics

Measurement error and disturbance, in the presence of conservation laws, are analysed in general operational terms. We provide novel quantitative bounds demonstrating necessary conditions under which accurate or non-disturbing measurements can be achieved, highlighting an interesting interplay between incompatibility, unsharpness, and coherence. From here we obtain a substantial generalisation of the Wigner-Araki-Yanase (WAY) theorem. Our findings are further refined through the analysis of the fixed-point set of the measurement channel, some extra structure of which is characterised here for the first time.

Variational learning of Euler–Lagrange dynamics from data

The principle of least action is one of the most fundamental physical principle. It says that among all possible motions connecting two points in a phase space, the system will exhibit those motions which extremise an action functional. Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equations, are related to the existence of an action functional. Incorporating variational structure into learning algorithms for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predictions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position data of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this, we introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position observations only, based on variational backward error analysis.

Restrictions on realizable unitary operations imposed by symmetry and locality

According to an elementary result in quantum computing, any unitary transformation on a composite system can be generated using 2-local unitaries, i.e., those that act only on two subsystems. Beside its fundamental importance in quantum computing, this result can also be regarded as a statement about the dynamics of systems with local Hamiltonians: although locality puts various constraints on the short-term dynamics, it does not restrict the possible unitary evolutions that a composite system with a general local Hamiltonian can experience after a sufficiently long time. We ask if such universality remains valid in the presence of conservation laws and global symmetries. In particular, can k-local symmetric unitaries on a composite system generate all symmetric unitaries on that system? Interestingly, it turns out that the answer is negative in the case of continuous symmetries, such as U(1) and SU(2): generic symmetric unitaries cannot be implemented, even approximately, using local symmetric unitaries. In fact, the difference between the dimensions of the manifold of all symmetric unitaries and the submanifold of unitaries generated by k-local symmetric unitaries, constantly increases with the system size. On the other hand, we find that this no-go theorem can be circumvented using ancillary qubits. For instance, any unitary invariant under rotations around z can be implemented using Hamiltonian XX+YY together with local Z Hamiltonian on the ancillary qubit. Moreover, any globally energy-conserving unitary on a composite system can be implemented using a sequence of 2-local energy-conserving unitaries, provided that one can use a single ancillary qubit (catalyst).

Symplectic integration of learned Hamiltonian systems.

Hamiltonian systems are differential equations that describe systems in classical mechanics, plasma physics, and sampling problems. They exhibit many structural properties, such as a lack of attractors and the presence of conservation laws. To predict Hamiltonian dynamics based on discrete trajectory observations, the incorporation of prior knowledge about Hamiltonian structure greatly improves predictions. This is typically done by learning the system's Hamiltonian and then integrating the Hamiltonian vector field with a symplectic integrator. For this, however, Hamiltonian data need to be approximated based on trajectory observations. Moreover, the numerical integrator introduces an additional discretization error. In this article, we show that an inverse modified Hamiltonian structure adapted to the geometric integrator can be learned directly from observations. A separate approximation step for the Hamiltonian data is avoided. The inverse modified data compensate for the discretization error such that the discretization error is eliminated. The technique is developed for Gaussian processes.

Reference Frames Which Separately Store Noncommuting Conserved Quantities.

Even in the presence of conservation laws, one can perform arbitrary transformations on a system if given access to a suitable reference frame, since conserved quantities may be exchanged between the system and the frame. Here we explore whether these quantities can be separated into different parts of the reference frame, with each part acting as a "battery" for a distinct quantity. For systems composed of spin-1/2 particles, we show that the components of angular momentum S_{x}, S_{y}, and S_{z} (noncommuting conserved quantities) may be separated in this way, and also provide several extensions of this result. These results also play a key role in the quantum thermodynamics of noncommuting conserved quantities.

Coherence and Asymmetry Cannot be Broadcast.

In the presence of conservation laws, superpositions of eigenstates of the corresponding conserved quantities cannot be generated by quantum dynamics. Thus, any such coherence represents a potentially valuable resource of asymmetry, which can be used, for example, to enhance the precision of quantum metrology or to enable state transitions in quantum thermodynamics. Here we ask if such superpositions, already present in a reference system, can be broadcast to other systems, thereby distributing asymmetry indefinitely at the expense of creating correlations. We prove a no-go theorem showing that this is forbidden by quantum mechanics in every finite-dimensional system. In doing so, we also answer some open questions in the quantum information literature concerning the sharing of timing information of a clock and the possibility of catalysis in quantum thermodynamics. We also prove that even weaker forms of broadcasting, of which Åberg's "catalytic coherence" is a particular example, can only occur in the presence of infinite-dimensional reference systems. Our results set fundamental limits on the creation and manipulation of quantum coherence and shed light on the possibilities and limitations of quantum reference frames to act catalytically without being degraded.

Traveling fronts bifurcating from stable layers in the presence of conservation laws

We study traveling waves bifurcating from stable standing layers in systems where a reaction-diffusion equation couples to a scalar conservation law. We prove the existence of weekly decaying traveling fronts that emerge in the presence of a weakly stable direction on a center manifold. Moreover, we show the existence of bifurcating traveling waves of constant mass. The main difficulty is to prove the smoothness of the ansatz in exponentially weighted spaces required to apply the Lyapunov-Schmidt methods.

General linear response formula for non integrable systems obeying the Vlasov equation

Long-range interacting N-particle systems get trapped into long-living out-of-equilibrium stationary states called quasi-stationary states (QSS). We study here the response to a small external perturbation when such systems are settled into a QSS. In the N → ∞ limit the system is described by the Vlasov equation and QSS are mapped into stable stationary solutions of such equation. We consider this problem in the context of a model that has recently attracted considerable attention, the Hamiltonian mean field (HMF) model. For such a model, stationary inhomogeneous and homogeneous states determine an integrable dynamics in the mean-field effective potential and an action-angle transformation allows one to derive an exact linear response formula. However, such a result would be of limited interest if restricted to the integrable case. In this paper, we show how to derive a general linear response formula which does not use integrability as a requirement. The presence of conservation laws (mass, energy, momentum, etc.) and of further Casimir invariants can be imposed a posteriori. We perform an analysis of the infinite time asymptotics of the response formula for a specific observable, the magnetization in the HMF model, as a result of the application of an external magnetic field, for two stationary stable distributions: the Boltzmann-Gibbs equilibrium distribution and the Fermi-Dirac one. When compared with numerical simulations the predictions of the theory are very good away from the transition energy from inhomogeneous to homogeneous states.

Layers in the Presence of Conservation Laws

We study standing layers in systems where a reaction-diffusion equation couples to a scalar conservation law. Our results give spectral stability and instability results depending only on relative monotonicity of the two components of the system. We also prove the robustness of layers and their stability properties. Our results classify stability properties of layers in most such systems. Our method is based on tracking the point spectrum during a homotopy to a simple, decoupled system. Main difficulty is the possibility of eigenvalues disappearing in a branch point of the essential spectrum. This phenomenon is investigated using a Lyapunov–Schmidt reduction method on exponentially weighted spaces combined with a matching procedure for the far-field.

Instability of spikes in the presence of conservation laws

We show that spikes are unstable in a class of scalar reaction–diffusion equations coupled to a general conservation law. Our class includes the Keller–Segel model for chemotaxis, phase-field models and models for chemical reactions in closed chemical reactors.

Discussion on: “Passivity and Structure Preserving Order Reduction of Linear Port-Hamiltonian Systems Using Krylov Subspaces”

The port-Hamiltonian approach to modeling and control of complex physical systems has arisen as a systematic and unifying framework during the last 20 years. The port-Hamiltonian modeling captures the physical properties of the considered system including the energy dissipation, stability and passivity properties as well as the presence of conservation laws. Another important issue the port-Hamiltonian approach deals with is the interconnection of the physical system with other physical systems creating the so-called physical network. In real applications the dimensions of such interconnected port-Hamiltonian state-space systems rapidly grow both for lumped- and (spatially discretized) distributed-parameter models. Therefore an important issue concerns (structure preserving) model reduction of these high-dimensional models for further analysis and control.

QUANTUM COSMOLOGY FROM INDUCED SCALAR–TENSOR THEORY OF GRAVITY AND THE ROLE OF CONSERVATION LAWS

In this paper, it is shown that the so-called Hartle criterion to select correlated regions in configuration space of dynamical variables, is directly connected to the presence of conservation laws, that is the existence of conservation laws in minisuperspace quantum cosmology constitutes a selection rule to recover observable universes. In this context, we develop the quantum cosmology for a scalar–tensor theory of gravity, induced from the reduction procedure of a 5D-unification theory. A class of exact solutions for Wheeler–DeWitt equation is derived and their connection to classical observable universes is discussed.

Self-similarity in random collision processes.

Kinetics of collision processes with linear mixing rules are investigated analytically. The velocity distribution becomes self-similar in the long-time limit and the similarity functions have algebraic or stretched exponential tails. The characteristic exponents are roots of transcendental equations and vary continuously with the mixing parameters. In the presence of conservation laws, the velocity distributions become universal.

Solutions for vectorial beam coupling under ac field in cubic photorefractive crystals

We develop a theory of vectorial wave coupling in cubic photorefractive crystals placed in an alternating ac field to enhance the nonlinear response. It is proven in the general case that despite essential differences between the diffusion and the ac nonlocal responses, the latter keeps the light interference fringes straight during the interaction. This fundamental feature allows, under certain restrictions, to reduce the nonlinear problem of vectorial coupling to the known linear problem of vectorial Bragg diffraction from a spatially uniform grating, which admits an exact solution. As a result, the nonlinear vectorial problem can be effectively solved for a number of practically important cases. These cases include nontrivial polarization effects and also the high-contrast effects. The presence of conservation laws involving the polarization degrees of freedom is shown. A number of particular cases relevant to experiments with BTO crystals are considered.

Growth of the $H^s$-norm for the modified KdV equation

We study the growth of the $H^s$-norm for solutions of the modified Korteweg-de Vries equation, corresponding to data in $H^s$ for noninteger values of $s$ in the case where global solutions exist. The presence of conservation laws and the local existence theory permit us to obtain upper "polynomial" bounds, for the $H^s$-norm of these solutions, with power depending on the distance to the closest integer to $s$.

Singular Lyapunov spectra and conservation laws.

We give analytic arguments and numerical evidence to show that the presence of conservation laws can produce a singularity in the spectrum of Lyapunov exponents for extended dynamical systems of low spatial dimensionality. This phenomenon can be used, e.g., for finding hidden conservation laws. (c) 1995 American Institute of Physics.

Disturbance, conservation laws and the uncertainty principle

The interpretation of the uncertainty principle in terms of a measurement of a single observable disturbing other observables, originating in Heisenberg's 1927 paper, is shown to be derivable from an uncertainty principle for joint measurements of incompatible observables. The latter also limits measurements of a single observable in the presence of conservation laws.