Search results for "Hilbert"
showing 10 items of 331 documents
On non-self-adjoint operators defined by Riesz bases in Hilbert and rigged Hilbert spaces
2018
In this paper we discuss some results on non self-adjoint Hamiltonians with real discrete simple spectrum under the assumption that their eigenvectors form Riesz bases of a certain Hilbert space. Also, we exhibit a generalization of those results to the case of rigged Hilbert spaces, and we also consider the problem of the factorization of the aforementioned Hamiltonians in terms of generalized lowering and raising operators.
Searching for exceptional points and inspecting non-contractivity of trace distance in (anti-) PT -symmetric systems
2022
Non-Hermitian systems with parity-time ($\mathcal{PT}$) symmetry and anti-$\mathcal{PT}$ symmetry give rise to exceptional points (EPs) with intriguing properties related to, e.g., chiral transport and enhanced sensitivity, due to the coalescence of eigenvectors. In this paper, we propose a powerful and easily computable tool, based on the Hilbert-Schmidt speed (HSS), which does not require the diagonalization of the evolved density matrix, to detect exactly the EPs and hence the critical behavior of the (anti-)$\mathcal{PT}\!-$symmetric systems, especially high-dimensional ones. Our theoretical predictions, made without the need for modification of the Hilbert space, which is performed by …
Memory Effects in High-Dimensional Systems Faithfully Identified by Hilbert–Schmidt Speed-Based Witness
2022
A witness of non-Markovianity based on the Hilbert–Schmidt speed (HSS), a special type of quantum statistical speed, has been recently introduced for low-dimensional quantum systems. Such a non-Markovianity witness is particularly useful, being easily computable since no diagonalization of the system density matrix is required. We investigate the sensitivity of this HSS-based witness to detect non-Markovianity in various high-dimensional and multipartite open quantum systems with finite Hilbert spaces. We find that the time behaviors of the HSS-based witness are always in agreement with those of quantum negativity or quantum correlation measure. These results show that the HSS-based witness…
Hilbert–Schmidt speed as an efficient figure of merit for quantum estimation of phase encoded into the initial state of open n-qubit systems
2021
AbstractHilbert–Schmidt speed (HSS) is a special type of quantum statistical speed which is easily computable, since it does not require diagonalization of the system state. We find that, when both HSS and quantum Fisher information (QFI) are calculated with respect to the phase parameter encoded into the initial state of an n-qubit register, the zeros of the HSS dynamics are actually equal to those of the QFI dynamics. Moreover, the signs of the time-derivatives of both HSS and QFI exactly coincide. These findings, obtained via a thorough investigation of several paradigmatic open quantum systems, show that HSS and QFI exhibit the same qualitative time evolution. Therefore, HSS reveals its…
Linear optical implementation of nonlocal product states and their indistinguishability
2001
In a recent paper Bennett et al.[Phys. Rev.A 59, 1070 (1999)] have shown the existence of a basis of product states of a bipartite system with manifest non-local properties. In particular these states cannot be completely discriminated by means of bilocal measurements. In this paper we propose an optical realization of these states and we will show that they cannot be completely discriminate by means of a global measurement using only optical linear elements, conditional transformation and auxiliary photons.
Dynamics of a quantum particle interacting with a thermal bath and subject to an oscillating asymmetric bistable potential
2012
Exploiting the approach of the Feynman-Vernon influence functional [1] within the framework of the discrete variable representation (DVR) [2], we consider a quantum particle described by the Caldeira-Leggett model [3]. The particle, “moving” in an asymmetric bistable potential and subject to a periodical driving, interacts with a thermal bath of harmonic oscillators. In this conditions we study the dynamics of the particle by analyzing the time evolution of the populations in the DVR. Specifically we focalize on the position eigenstate located in the shallower well, i.e. metastable state, finding a non-monotonic behaviour of the corresponding population as a function of the frequency. Moreo…
Dynamics of a Driven Dissipative Quantum System
2013
We investigate the dynamics of a driven multilevel system, consisting of a particle in an asymmetric bistable potential, strongly interacting with a thermal bath according to the Caldeira-Leggett model. The populations in the discrete (position) variable representation (DVR), are obtained as solution of a Markovian approximated master equation, which is derived from a discretized path integral approach based on the Feynman-Vernon influence functional. By varying the environmental parameters (temperature and coupling strength) as well as the driving frequency and amplitude, we study the transient dynamics and stationary configuration of the system. In particular, we analyze the population of…
Transient Dynamics of a Driven Quantum Bistable System
2013
We study the transient dynamics and the asymptotic behaviour of a multilevel system in the strong dissipation regime. The system is modeled as a periodically driven quantum particle in an asymmetric double well potential, interacting with the bosonic heat bath of the Caldeira-Leggett model. The analytical approach used is non- perturbative in the particle-bath coupling and is based on a space-discretized path integral expression for the particle’s reduced density matrix. By a suitable approximation on the Feynman-Vernon influence functional a Markov-approximated master equation is obtained for the populations in the Discrete Variable Representation (DVR).
L p-Spaces and the Radon–Nikodym Theorem
2020
In this chapter, we study the spaces of functions whose pth power is integrable. In Section 7.2, we first derive some of the important inequalities (Holder, Minkowski, Jensen) and then in Section 7.3 investigate the case p=2 in more detail.
Discrete Time Signal Processing Framework with Support Vector Machines
2007
Digital signal processing (DSP) of time series using SVM has been addressed in the literature with a straightforward application of the SVM kernel regression, but the assumption of independently distributed samples in regression models is not fulfilled by a time-series problem. Therefore, a new branch of SVM algorithms has to be developed for the advantageous application of SVM concepts when we process data with underlying time-series structure. In this chapter, we summarize our past, present, and future proposal for the SVM-DSP frame-work, which consists of several principles for creating linear and nonlinear SVM algorithms devoted to DSP problems. First, the statement of linear signal mod…