Complex Stochastic Optimal Control Foundation of Quantum Mechanics

• URL: http://arxiv.org/abs/2404.15964v6
• Date: Fri, 11 Oct 2024 09:20:44 GMT
• Title: Complex Stochastic Optimal Control Foundation of Quantum Mechanics
• Authors: Vasil Yordanov,
• Abstract summary: This paper explores the application of the HJB equation in the context of complex variables. It provides an in-depth investigation of the movement of quantum particles within the framework of optimal control theory.
• Score: 0.0
• License:
• Abstract: Recent studies have extended the use of the stochastic Hamilton-Jacobi-Bellman (HJB) equation to include complex variables for deriving quantum mechanical equations. However, these studies often assume that it is valid to apply the HJB equation directly to complex numbers, an approach that overlooks the fundamental problem of comparing complex numbers when finding optimal controls. This paper explores the application of the HJB equation in the context of complex variables. It provides an in-depth investigation of the stochastic movement of quantum particles within the framework of stochastic optimal control theory. We obtain the complex diffusion coefficient in the stochastic equation of motion using the Cauchy-Riemann theorem, considering that the particle's stochastic movement is described by two perfectly correlated real and imaginary stochastic processes. During the development of the covariant form of the HJB equation, we demonstrate that if the temporal stochastic increments of the two processes are perfectly correlated, then the spatial stochastic increments must be perfectly anti-correlated, and vice versa. The diffusion coefficient we derive has a form that enables the linearization of the HJB equation. The method for linearizing the HJB equation, along with the subsequent derivation of the Dirac equation, was developed in our previous work [V. Yordanov, Scientific Reports 14, 6507 (2024)]. These insights deepen our understanding of quantum dynamics and enhance the application of stochastic optimal control theory to quantum mechanics.

Related papers

• Efficiency of Dynamical Decoupling for (Almost) Any Spin-Boson Model [44.99833362998488]
  We analytically study the dynamical decoupling of a two-level system coupled with a structured bosonic environment. We find sufficient conditions under which dynamical decoupling works for such systems. Our bounds reproduce the correct scaling in various relevant system parameters.
  arXiv  Detail & Related papers  (2024-09-24T04:58:28Z)
• Closed-form solutions for the Salpeter equation [41.94295877935867]
  We study the propagator of the $1+1$ dimensional Salpeter Hamiltonian, describing a relativistic quantum particle with no spin. The analytical extension of the Hamiltonian in the complex plane allows us to formulate the equivalent problem, namely the B"aumer equation. This B"aumera corresponds to the Green function of a relativistic diffusion process that interpolates between Cauchy for small times and Gaussian diffusion for large times.
  arXiv  Detail & Related papers  (2024-06-26T15:52:39Z)
• Cavity QED materials: Comparison and validation of two linear response theories at arbitrary light-matter coupling strengths [41.94295877935867]
  We develop a linear response theory for materials collectively coupled to a cavity that is valid in all regimes of light-matter coupling. We compare two different approaches to obtain thermal Green functions. We provide a detailed application of the theory to the Quantum Hall effect and to a collection of magnetic models.
  arXiv  Detail & Related papers  (2024-06-17T18:00:07Z)
• Stochastic modeling of superfluorescence in compact systems [0.0]
  We propose an approach to describe superfluorescence in compact ensembles of multi-level emitters in the presence of various incoherent processes. We present a series of numerical examples, comparing our solution to exact calculations and discussing the limits of applicability.
  arXiv  Detail & Related papers  (2023-12-11T17:18:08Z)
• Spectral chaos bounds from scaling theory of maximally efficient quantum-dynamical scrambling [49.1574468325115]
  A key conjecture about the evolution of complex quantum systems towards an ergodic steady state, known as scrambling, is that this process acquires universal features when it is most efficient. We develop a single- parameter scaling theory for the spectral statistics in this scenario, which embodies exact self-similarity of the spectral correlations along the complete scrambling dynamics. We establish that scaling predictions are matched by a privileged process, and serve as bounds for other dynamical scrambling scenarios, allowing one to quantify inefficient or incomplete scrambling on all timescales.
  arXiv  Detail & Related papers  (2023-10-17T15:41:50Z)
• Quasiclassical approaches to the generalized quantum master equation [0.0]
  The generalized quantum master equation (GQME) expresses correlation functions in terms of a non-Markovian equation of motion. We study the approximate solution of the GQME, obtained by calculating the kernels with two methods, namely Ehrenfest mean-field theory and spin mapping. We find that the accuracy of the predictions of the GQME depends strongly on the specific technique used to calculate the kernels.
  arXiv  Detail & Related papers  (2022-09-02T14:14:05Z)
• Decimation technique for open quantum systems: a case study with driven-dissipative bosonic chains [62.997667081978825]
  Unavoidable coupling of quantum systems to external degrees of freedom leads to dissipative (non-unitary) dynamics. We introduce a method to deal with these systems based on the calculation of (dissipative) lattice Green's function. We illustrate the power of this method with several examples of driven-dissipative bosonic chains of increasing complexity.
  arXiv  Detail & Related papers  (2022-02-15T19:00:09Z)
• Sampling, rates, and reaction currents through reverse stochastic quantization on quantum computers [0.0]
  We show how to tackle the problem using a suitably quantum computer. We propose a hybrid quantum-classical sampling scheme to escape local minima.
  arXiv  Detail & Related papers  (2021-08-25T18:04:52Z)
• Bernstein-Greene-Kruskal approach for the quantum Vlasov equation [91.3755431537592]
  The one-dimensional stationary quantum Vlasov equation is analyzed using the energy as one of the dynamical variables. In the semiclassical case where quantum tunneling effects are small, an infinite series solution is developed.
  arXiv  Detail & Related papers  (2021-02-18T20:55:04Z)
• Phase space theory for open quantum systems with local and collective dissipative processes [0.0]
  We investigate driven dissipative quantum dynamics of an ensemble of two-level systems given by a Markovian master equation with collective and noncollective dissipators. Our results expose, utilize and promote pioneered techniques in the context of laser theory.
  arXiv  Detail & Related papers  (2020-06-05T07:22:02Z)
• Computing quantum dynamics in the semiclassical regime [0.0]
  The Schr"odinger equation describes, inter alia, quantum dynamics of nuclei in a molecule. It poses the combined computational challenges of high oscillations and high dimensions. This paper reviews and studies numerical approaches that are robust to the small semiclassical parameter.
  arXiv  Detail & Related papers  (2020-02-03T09:43:01Z)

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.