Quantum Random Walks and Quantum Oscillator in an Infinite-Dimensional Phase Space
- URL: http://arxiv.org/abs/2406.10699v1
- Date: Sat, 15 Jun 2024 17:39:32 GMT
- Title: Quantum Random Walks and Quantum Oscillator in an Infinite-Dimensional Phase Space
- Authors: Vladimir Busovikov, Alexander Pechen, Vsevolod Sakbaev,
- Abstract summary: We consider quantum random walks in an infinite-dimensional phase space constructed using Weyl representation of the coordinate and momentum operators.
We find conditions for their strong continuity and establish properties of their generators.
- Score: 45.9982965995401
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: We consider quantum random walks in an infinite-dimensional phase space constructed using Weyl representation of the coordinate and momentum operators in the space of functions on a Hilbert space which are square integrable with respect to a shift-invariant measure. We study unitary groups of shift operators in the phase space and averaging of such shifts by Gaussian vectors, which form semigroups of self-adjoint contractions: we find conditions for their strong continuity and establish properties of their generators. Significant differences in their properties allow us to show the absence of the Fourier transform as a unitary transformation that implements the unitary equivalence of these compressive semigroups. Next, we prove the Taylor formula for a certain special subset of smooth functions for shifting to a non-finite vector. It allows us to prove convergence of quantum random walks in the coordinate representation to the evolution of a diffusion process, as well as convergence of quantum random walks in both coordinate and momentum representations to the evolution semigroup of a quantum oscillator in an infinite-dimensional phase space. We find the special essential common domain of generators of semigroups arising in averaging of random shift operators both in position and momentum representations. The invariance of this common domain with respect to both semigroups allows to establish properties of a convex combination of both generators. That convex combination are Hamiltonians of infinite-dimentional quantum oscillators. Thus, we obtain that a Weyl representation of a random walk in an infinite dimensional phase space describes the semigroup of self-adjoint contractions whose generator is the Hamiltonian of an infinite dimensional harmonic oscillator.
Related papers
- Exceptional points and quantum phase transition in a fermionic extension of the Swanson oscillator [8.84834042985207]
We propose a fermionic extension of a non-Hermitian quantum system consisting of a general representation of a quadratic Hamiltonian.
The model admits a quantum phase transition - we discuss the two phases and also demonstrate that the ground-state entanglement entropy exhibits a discontinuous jump.
arXiv Detail & Related papers (2024-01-30T17:20:34Z) - Noncommutativity in Configuration Space Induced by A Conjugate Magnetic
Field in Phase Space [0.0]
An external magnetic field in configuration space coupled to a quantum dynamics induces noncommutativity in its velocity momentum space.
A rationale for noncommutativity is explored herein for an arbitrary configuration space of Euclidean geometry.
arXiv Detail & Related papers (2024-01-08T14:02:32Z) - On reconstruction of states from evolution induced by quantum dynamical
semigroups perturbed by covariant measures [50.24983453990065]
We show the ability to restore states of quantum systems from evolution induced by quantum dynamical semigroups perturbed by covariant measures.
Our procedure describes reconstruction of quantum states transmitted via quantum channels and as a particular example can be applied to reconstruction of photonic states transmitted via optical fibers.
arXiv Detail & Related papers (2023-12-02T09:56:00Z) - Energy preserving evolutions over Bosonic systems [0.4604003661048266]
We investigate perturbations of quantum dynamical semigroups that operate on continuous variable (CV) systems.
We show that the level sets of operators with bounded first moments are admissible subspaces of the evolution.
We provide a new scheme for deriving continuity bounds on the energy-constrained capacities of Markovian perturbations of Quantum dynamical semigroups.
arXiv Detail & Related papers (2023-07-25T20:13:30Z) - Exotic quantum liquids in Bose-Hubbard models with spatially-modulated
symmetries [0.0]
We investigate the effect that spatially modulated continuous conserved quantities can have on quantum ground states.
We show that such systems feature a non-trivial Hilbert space fragmentation for momenta incommensurate with the lattice.
We conjecture that a Berezinskii-Kosterlitz-Thouless-type transition is driven by the unbinding of vortices along the temporal direction.
arXiv Detail & Related papers (2023-07-17T18:14:54Z) - Measurement phase transitions in the no-click limit as quantum phase
transitions of a non-hermitean vacuum [77.34726150561087]
We study phase transitions occurring in the stationary state of the dynamics of integrable many-body non-Hermitian Hamiltonians.
We observe that the entanglement phase transitions occurring in the stationary state have the same nature as that occurring in the vacuum of the non-hermitian Hamiltonian.
arXiv Detail & Related papers (2023-01-18T09:26:02Z) - Fermion production at the boundary of an expanding universe: a cold-atom
gravitational analogue [68.8204255655161]
We study the phenomenon of cosmological particle production of Dirac fermions in a Friedman-Robertson-Walker spacetime.
We present a scheme for the quantum simulation of this gravitational analogue by means of ultra-cold atoms in Raman optical lattices.
arXiv Detail & Related papers (2022-12-02T18:28:23Z) - Unified Fourier-based Kernel and Nonlinearity Design for Equivariant
Networks on Homogeneous Spaces [52.424621227687894]
We introduce a unified framework for group equivariant networks on homogeneous spaces.
We take advantage of the sparsity of Fourier coefficients of the lifted feature fields.
We show that other methods treating features as the Fourier coefficients in the stabilizer subgroup are special cases of our activation.
arXiv Detail & Related papers (2022-06-16T17:59:01Z) - Entanglement dynamics of spins using a few complex trajectories [77.34726150561087]
We consider two spins initially prepared in a product of coherent states and study their entanglement dynamics.
We adopt an approach that allowed the derivation of a semiclassical formula for the linear entropy of the reduced density operator.
arXiv Detail & Related papers (2021-08-13T01:44:24Z) - Covariant Quantum Mechanics and Quantum Spacetime [0.0]
The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation.
Explicit wavefunction description is given without any restriction of the variable domains, yet with a finite integral inner product.
arXiv Detail & Related papers (2020-02-04T08:55:56Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.