Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userQuantum dynamics and Grams matrix
293
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We propose to analyse the statistical properties of a sequence of vectors using the spectrum of the associated Gram matrix. Such sequences arise e.g. by the repeated action of a deterministic kicked quantum dynamics on an initial condition or by a random process. We argue that, when the number of time-steps, suitably scaled with respect to $hbar$, increases, the limiting eigenvalue distribution of the Gram matrix reflects the possible quantum chaoticity of the original system as it tends to its classical limit. This idea is subsequently applied to study the long-time properties of sequences of random vectors at the time scale of the dimension of the Hilbert space of available states.
rate research
Read More
In this paper we review a proposed geometrical formulation of quantum mechanics. We argue that this geometrization makes available mathematical methods from classical mechanics to the quantum frame work. We apply this formulation to the study of separability and entanglement for states of composite quantum systems.
We focus on symmetries related to matrices and vectors appearing in the simulation of quantum many-body systems. Spin Hamiltonians have special matrix-symmetry properties such as persymmetry. Furthermore, the systems may exhibit physical symmetries translating into symmetry properties of the eigenvectors of interest. Both types of symmetry can be exploited in sparse representation formats such as Matrix Product States (MPS) for the desired eigenvectors. This paper summarizes symmetries of Hamiltonians for typical physical systems such as the Ising model and lists resulting properties of the related eigenvectors. Based on an overview of Matrix Product States (Tensor Trains or Tensor Chains) and their canonical normal forms we show how symmetry properties of the vector translate into relations between the MPS matrices and, in turn, which symmetry properties result from relations within the MPS matrices. In this context we analyze different kinds of symmetries and derive appropriate normal forms for MPS representing these symmetries. Exploiting such symmetries by using these normal forms will lead to a reduction in the number of degrees of freedom in the MPS matrices. This paper provides a uniform platform for both well-known and new results which are presented from the (multi-)linear algebra point of view.
In this paper we treat the time evolution of unitary elements in the N level system and consider the reduced dynamics from the unitary group U(N) to flag manifolds of the second type (in our terminology). Then we derive a set of differential equations of matrix Riccati types interacting with one another and present an important problem on a nonlinear superposition formula that the Riccati equation satisfies. Our result is a natural generalization of the paper {bf Chaturvedi et al} (arXiv : 0706.0964 [quant-ph]).
The iterative method of Sinkhorn allows, starting from an arbitrary real matrix with non-negative entries, to find a so-called scaled matrix which is doubly stochastic, i.e. a matrix with all entries in the interval (0, 1) and with all line sums equal to 1. We conjecture that a similar procedure exists, which allows, starting from an arbitrary unitary matrix, to find a scaled matrix which is unitary and has all line sums equal to 1. The existence of such algorithm guarantees a powerful decomposition of an arbitrary quantum circuit.
Upon revisiting the Hamiltonian structure of classical wavefunctions in Koopman-von Neumann theory, this paper addresses the long-standing problem of formulating a dynamical theory of classical-quantum coupling. The proposed model not only describes the influence of a classical system onto a quantum one, but also the reverse effect -- the quantum backreaction. These interactions are described by a new Hamiltonian wave equation overcoming shortcomings of currently employed models. For example, the density matrix of the quantum subsystem is always positive-definite. While the Liouville density of the classical subsystem is generally allowed to be unsigned, its sign is shown to be preserved in time for a specific infinite family of hybrid classical-quantum systems. The proposed description is illustrated and compared with previous theories using the exactly solvable model of a degenerate two-level quantum system coupled to a classical harmonic oscillator.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
