arXiv:1408.4394

Symmetry examples in open quantum dynamics

Thomas F. Jordan, San Ha Seo

Published Aug 19, 2014• 0 citations

Abstract

Dependent symmetries, symmetries that depend on the situation of the subsystem in a larger closed system, are explored by looking at simple examples. This is a new kind of symmetry in the open quantum dynamics of a subsystem Each symmetry implies a particular form for the results of the open dynamics. The forms exhibit the symmetries very simply. It is shown directly, without assuming anything about the symmetry, that the dynamics produces the form, but knowing the symmetry and the form it implies can reduce what needs to be done to work out the dynamics; pieces can be deduced from the symmetry rather that calculated from the dynamics. Symmetries can be related to constants of the motion in new ways. A quantity might be a dependent constant of the motion, constant only for particular situations of the subsystem in the larger system. In particular, a generator of dependent symmetries could represent a quantity that is a dependent constant of the motion for the same situations as for the symmetries. The examples present a variety of possibilities. Sometimes a generator of dependent symmetries does represent a dependent constant of the motion. Sometimes it does not. Sometimes no quantity is a dependent constant of the motion. Sometimes every quantity is.

DOI: 10.48550/arxiv.1408.4394

Related papers

Symmetries in open quantum dynamics

Thomas F. Jordan

Simple examples are used to introduce and examine symmetries of open quantum dynamics that can be described by unitary operators. For the Hamiltonian dynamics of an entire closed system, the symmetry takes the expected form which, when the Hamiltonian has a lower bound, says that the unitary symmetry operator commutes with the Hamiltonian operator. There are many more symmetries that are only for the open dynamics of a subsystem. Examples show how these symmetries alone can reveal properties of the dynamics and reduce what needs to be done to work out the dynamics. A symmetry of the open dynamics of a subsystem can even imply properties of the dynamics for the entire system that are not implied by the symmetries of the dynamics of the entire system. The symmetries are generally not related to constants of the motion for the open dynamics of the subsystem. There are many symmetries that cannot be seen in the Schrodinger picture as symmetries of dynamical maps of density matrices for the subsystem. There are symmetries of the open dynamics of a subsystem that depend only on the dynamics. In the simplest examples, these are also symmetries of the dynamics of the entire system. There are many more symmetries, of a new kind, that also depend on correlations, or absence of correlations, between the subsystem and the rest of the entire system, or on the state of the rest of the entire system.

arXiv:1408.3758Aug 16, 20141 citation

Symmetries in quantum mechanics

Giampiero V. M. Esposito, Giuseppe Marmo, Gennaro Miele +1 more

The meaning of symmetries is studied in some detail, and a study is performed of transformations which preserve the description, of transformations of frames and the corresponding quantum symmetries. The second part of the chapter outlines Galilei transformations, time translations, spatial reflection and time reversal Meaning of symmetries Symmetries and constants of motion have already appeared here and there in this book, most notably in Appendix 3.A. Recall that, therein, we have defined symmetries as those transformations that generate new solutions out of known solutions; more generally, that permute solutions among themselves. Constants of motion are functions on the carrier space which, when restricted to solutions, acquire a constant value depending on the solution considered. This point of view may be viewed as a dynamical point of view, i.e. depending on the specific system under consideration. There is, however, another meaning of symmetry, which deals with the symmetry of the description that ensures the covariance of the equations of motion, e.g. transformations that map equations of motion in Hamiltonian form into other equations still in Hamiltonian form, or transformations that map second-order differential equations into second-order ones, i.e. equations admitting a Lagrangian description into new ones also admitting a Lagrangian description. Transformations of the dynamical type emerge out of the previous ones when we further require that the Hamiltonian is preserved or the Lagrangian is preserved up to a total time derivative.

Cambridge University Press eBooksOct 31, 20141 citation

Dynamical symmetries in physics

Hájek, Pavel

The aim of this thesis is to provide a definition of dynamical symme- try and to study its properties within simple quantum systems. In particular, I investigate Kepler's problem and the isotropic harmonic oscillator. Dynamical sy- mmetry is a kind of higher symmetry which is broken in a specific way. Definition of dynamical group and quantum mechanical system is presented. Subsequently, a definition of quantum degrees of freedom and quantum integrability is propo- sed. I mention briefly a possibility of finding the generators of dynamical group by considering time dependent constants of motion.

Jan 1, 20110 citations

One qubit almost completely reveals the dynamics of two

Thomas F. Jordan, Anil Shaji, E. C. G. Sudarshan

From the time dependence of states of one of them, the dynamics of two interacting qubits is determined to be one of two possibilities that differ only by a change of signs of parameters in the Hamiltonian. The only exception is a simple particular case where several parameters in the Hamiltonian are zero and one of the remaining nonzero parameters has no effect on the time dependence of the states of the one qubit. The mean values that describe the initial state of the other qubit and of the correlations between the two qubits also are generally determined to within a change of signs by the time dependence of the states of the one qubit, but with many more exceptions. An example demonstrates all the results. Feedback in the equations of motion that allows time dependence in a subsystem to determine the dynamics of the larger system can occur in both classical and quantum mechanics. The role of quantum mechanics here is just to identify qubits as the simplest objects to consider and specify the form that equations of motion for two interacting qubits can take.

arXiv:quant-ph/0611141Physical Review AJul 2, 20077 citations

Nonlinear quantum spin dynamics: Symmetries without conservation laws

Thomas F. Jordan

Spin examples demonstrate a nonlinear quantum mechanics. The Hamiltonian operator depends on the state. It contains mean values together with operators. Symmetries depend on these mean values. The requirement for a symmetry operator is not simply that it commutes with the Hamiltonian operator. Rotation invariance does not imply that for the angular-momentum operator. The Hamiltonian operator can change as the state changes. There may be no conserved energy. In examples for spin 1/2 and spin 1, the angle between precessing spin and magnetic field changes. This suggests possibilities for new precision tests of linearity.

Physical Review AJun 1, 19946 citations

Constructing Dynamical Symmetries

James Russell Hamilton, Raphael David Levine, F. Remacle

Dynamical symmetries, operators that do not quite commute with the Hamiltonian, extend the role of ordinary symmetries. They also provide an interesting insight on constants of the motion. Motivated by progress in quantum technologies we discuss a practical algebraic approach to computing such time-dependent operators. Explicitly we expand them as a linear combination with time dependent coefficients of time-dependent Schrödinger operators. We propose possible applications in determining quantum mechanical distributions of maximal entropy and to the dynamics of systems of coherently coupled coherent two state systems. We suggest that this generates an Ising-like Hamiltonian where each ’spin’ is a state and therefore of relevance to quantum computing based on qubit architecture.

Nov 14, 20231 citation