Search Results (34)

In his recent paper, Vlatko Vedral claims that the Schrödinger picture can describe quantum systems as locally as the Heisenberg picture, relying on a product notation for the density matrix. Here, I refute that claim. I show that the so-called ‘local factors’ in the product notation do not correspond to individual systems and therefore fail to satisfy Einsteinian locality. Furthermore, the product notation does not track where local gates are applied. Finally, I expose internal inconsistencies in the argument: if, as is also stated, the Schrödinger-picture locality ultimately depends on the explicit bookkeeping of all operations, then the explanatory power of the product notation is de facto undermined. Full article

C+L+S multiband (MB) optical transmission has the potential to increase the capacity of optical transport networks, and thus, it is a possible solution to cope with the traffic increase expected in the years to come. However, the introduction of MB optical technology needs to come together with the needed tools that support network planning and operation. In particular, quality of transmission (QoT) estimation is needed for provisioning optical MB connections. In this paper, we concentrate on modelling MB optical transmission for provide fast and accurate QoT estimation and propose machine learning (ML) approaches based on neural networks, which can be easily integrated into an optical layer digital twin (DT) solution. We start by considering approaches that can be used for accurate signal propagation modelling. Even though solutions such as the split-step Fourier method (SSFM) for solving the nonlinear Schrödinger equation (NLSE) have limited application for QoT estimation during provisioning because of their very high complexity and time consumption, they could be used to generate datasets for ML model creation. However, even that can be hard to carry out on a fully loaded MB system with hundreds of channels. In addition, in MB optical transmission, interchannel stimulated Raman scattering (ISRS) becomes a major effect, which adds more complexity. In view of that, the fourth-order Runge–Kutta in the interaction picture (RK4IP) method, complemented with an adaptive step size algorithm to further reduce the computation time, is evaluated as an alternative to reduce time complexity. We show that RK4IP provided an accuracy comparable to that of the SSFM with reduced computation time, which enables its application for MB optical transmission simulation. Once datasets were generated using the adaptive step size RK4IP method, two ML modelling approaches were considered to be integrated in the OCATA DT, where models predict optical signal propagation in the time domain. Being able to predict the optical signal in the time domain, as it will be received after propagation, opens opportunities for automating network operation, including connection provisioning and failure management. In this paper, we focus on comparing the proposed ML modelling approaches in terms of the models’ general and QoT estimation accuracy. Full article

The choice between the Schrödinger and Heisenberg pictures can significantly impact the computational resources needed to solve a problem, even though they are equivalent formulations of quantum mechanics. Here, we present a method for analysing Bosonic quantum circuits based on the Heisenberg picture which allows, under certain conditions, a useful factoring of the evolution into signal and noise contributions, similar way to what can be achieved with classical communication systems. We provide examples which suggest that this approach may be particularly useful in analysing quantum computing systems based on the Gottesman–Kitaev–Preskill (GKP) qubits. Full article

We present an efficient numerical method to solve the time-dependent Schrödinger equation in the single-active electron picture for atoms interacting with intense optical laser fields. Our approach is based on a non-uniform radial grid with smoothly increasing steps for the electron distance from the residual ion. We study the accuracy and efficiency of the method, as well as its applicability to investigate strong-field ionization phenomena, the process of high-order harmonic generation, and the dynamics of highly excited Rydberg states. Full article

Quantum mechanics of unitary systems is considered in quasi-Hermitian representation and in the dynamical regime in which one has to take into account the ubiquitous presence of perturbations, random or specific. In this paper, it is shown that multiple technical obstacles encountered in such a context can be circumvented via just a mild amendment of the so-called Rayleigh–Schrödinger perturbation–expansion approach. In particular, the quasi-Hermitian formalism characterized by an enhancement of flexibility is shown to remain mathematically tractable while, on the phenomenological side, opening several new model-building horizons. It is emphasized that they include, i.a., the study of generic random perturbations and/or of multiple specific non-Hermitian toy models. In parallel, several paradoxes and open questions are shown to survive. Full article

This paper explains how the so-called Einstein locality is to be understood in the Schrödinger picture of quantum mechanics. This notion is fully compatible with the Bell non-locality exhibited by entangled states. Contrary to the belief that quantum mechanics is incomplete, it is, As a matter of fact, its overcompleteness, as exemplified by the different pictures of quantum physics, that points to the same underlying reality. Full article

The common geometrical (symplectic) structures of classical mechanics, quantum mechanics, and classical thermodynamics are unveiled with three pictures. These cardinal theories, mainly at the non-relativistic approximation, are the cornerstones for studying chemical dynamics and chemical kinetics. Working in extended phase spaces, we show that the physical states of integrable dynamical systems are depicted by Lagrangian submanifolds embedded in phase space. Observable quantities are calculated by properly transforming the extended phase space onto a reduced space, and trajectories are integrated by solving Hamilton’s equations of motion. After defining a Riemannian metric, we can also estimate the length between two states. Local constants of motion are investigated by integrating Jacobi fields and solving the variational linear equations. Diagonalizing the symplectic fundamental matrix, eigenvalues equal to one reveal the number of constants of motion. For conservative systems, geometrical quantum mechanics has proved that solving the Schrödinger equation in extended Hilbert space, which incorporates the quantum phase, is equivalent to solving Hamilton’s equations in the projective Hilbert space. In classical thermodynamics, we take entropy and energy as canonical variables to construct the extended phase space and to represent the Lagrangian submanifold. Hamilton’s and variational equations are written and solved in the same fashion as in classical mechanics. Solvers based on high-order finite differences for numerically solving Hamilton’s, variational, and Schrödinger equations are described. Employing the Hénon–Heiles two-dimensional nonlinear model, representative results for time-dependent, quantum, and dissipative macroscopic systems are shown to illustrate concepts and methods. High-order finite-difference algorithms, despite their accuracy in low-dimensional systems, require substantial computer resources when they are applied to systems with many degrees of freedom, such as polyatomic molecules. We discuss recent research progress in employing Hamiltonian neural networks for solving Hamilton’s equations. It turns out that Hamiltonian geometry, shared with all physical theories, yields the necessary and sufficient conditions for the mutual assistance of humans and machines in deep-learning processes. Full article

In this work, we lay the foundations for computing the behavior of a quantum heat engine whose working medium consists of an ensemble of non-harmonic quantum oscillators. In order to enable this analysis, we develop a method based on the Schrödinger picture. We investigate different possible choices on the basis of expanding the density operator, as it is crucial to select a basis that will expedite the numerical integration of the time-evolution equation without compromising the accuracy of the computed results. For this purpose, we developed an estimation technique that allows us to quantify the error that is unavoidably introduced when time-evolving the density matrix expansion over a finite-dimensional basis. Using this and other ways of evaluating a specific choice of basis, we arrive at the conclusion that the basis of eigenstates of a harmonic Hamiltonian leads to the best computational performance. Additionally, we present a method to quantify and reduce the error that is introduced when extracting relevant physical information about the ensemble of oscillators. The techniques presented here are specific to quantum heat cycles; the coexistence within a cycle of time-dependent Hamiltonian and coupling with a thermal reservoir are particularly complex to handle for the non-harmonic case. The present investigation is paving the way for numerical analysis of non-harmonic quantum heat machines. Full article

A unitary-evolution process leading to an ultimate collapse and to a complete loss of observability alias quantum phase transition is studied. A specific solvable $N\mathrm{-}$state model is considered, characterized by a non-stationary non-Hermitian Hamiltonian. Our analysis uses quantum mechanics formulated in Schrödinger picture in which, in principle, only the knowledge of a complete set of observables (i.e., operators ${\mathrm{\Lambda }}_j$) enables one to guarantee the uniqueness of the related physical Hilbert space (i.e., of its inner-product metric $\mathrm{\Theta }$). Nevertheless, for the sake of simplicity, we only assume the knowledge of just a single input observable (viz., of the energy-representing Hamiltonian $H\equiv {\mathrm{\Lambda }}_1$). Then, out of all of the eligible and Hamiltonian-dependent “Hermitizing” inner-product metrics $\mathrm{\Theta }=\mathrm{\Theta }\left(H\right)$, we pick up just the simplest possible candidate. Naturally, this slightly restricts the scope of the theory, but in our present model, such a restriction is more than compensated for by the possibility of an alternative, phenomenologically better motivated constraint by which the time-dependence of the metric is required to be smooth. This opens a new model-building freedom which, in fact, enables us to force the system to reach the collapse, i.e., a genuine quantum catastrophe as a result of the mere conventional, strictly unitary evolution. Full article

Grassmann variables are used to formally transform a system with constraints into an unconstrained system. As a result, the Schrödinger equation arises instead of the Wheeler–DeWitt one. The Schrödinger equation describes a system’s evolution, but a definition of the scalar product is needed to calculate the mean values of the operators. We suggest an explicit formula for the scalar product related to the Klein–Gordon scalar product. The calculation of the mean values is compared with an etalon method in which a redundant degree of freedom is excluded. Nevertheless, we note that a complete correspondence with the etalon picture is not found. Apparently, the picture with Grassmann variables requires a further understanding of the underlying Hilbert space. Full article

We start with a methodological analysis of the notion of scientific theory and its interrelation with reality. This analysis is based on the works of Helmholtz, Hertz, Boltzmann, and Schrödinger (and reviews of D’Agostino). Following Helmholtz, Hertz established the “Bild conception” for scientific theories. Here, “Bild” (“picture”) carries the meaning “model” (mathematical). The main aim of natural sciences is construction of the causal theoretical models (CTMs) of natural phenomena. Hertz claimed that a CTM cannot be designed solely on the basis of observational data; it typically contains hidden quantities. Experimental data can be described by an observational model (OM), often based on the price of acausality. CTM-OM interrelation can be tricky. Schrödinger used the Bild concept to create a CTM for quantum mechanics (QM), and QM was treated as OM. We follow him and suggest a special CTM for QM, so-called prequantum classical statistical field theory (PCSFT). QM can be considered as a PCSFT image, but not as straightforward as in Bell’s model with hidden variables. The common interpretation of the violation of the Bell inequality is criticized from the perspective of the two-level structuring of scientific theories. Such critical analysis of von Neumann and Bell no-go theorems for hidden variables was performed already by De Broglie (and Lochak) in the 1970s. The Bild approach is applied to the two-level CTM-OM modeling of Brownian motion: the overdamped regime corresponds to OM. In classical mechanics, CTM=OM; on the one hand, this is very convenient; on the other hand, this exceptional coincidence blurred the general CTM-OM structuring of scientific theories. We briefly discuss ontic–epistemic structuring of scientific theories (Primas–Atmanspacher) and its relation to the Bild concept. Interestingly, Atmanspacher as well as Hertz claim that even classical physical theories should be presented on the basic of two-level structuring. Full article

A rigorous analysis is undertaken based on the analysis of both Galilean and Lorentz (Poincaré) invariance in field theories in general in order to (i) identify a unique analytical scheme for canonical pairs of Lagrangians, one of them having Galilean, the other one Poincaré invariance; and (ii) to obtain transition conditions for the state function purely from Hamilton’s principle and extended Noether’s theorem applied to the aforementioned symmetries. The general analysis is applied on Schrödinger and Klein–Gordon theory, identifying them as a canonical pair in the sense of (i) and exemplified for the scattering problem for both theories for a particle beam against a potential step in order to show that the transition conditions that result according to (ii) in a ‘natural’ way reproduce the well-known ‘methodical’ continuity conditions commonly found in the literature, at least in relevant cases, closing a relevant argumentation gap in quantum mechanical scattering problems. Full article

A molecular formalism based on a decomposed energy space constructed by a modular basis of matter and radiation is proposed for relativistic quantum mechanics. In the proposed formalism, matter radiation interactions are incorporated via the dynamical transformation of the coupled particle/antiparticle pair in a multistate quantum mechanical framework. This picture generalizes relativistic quantum mechanics at minimal cost, unlike quantum field theories, and the relativistic energy–momentum relation is interpreted as energy transformations among different modules through a multistate Schrödinger equation. The application of two-state and four-state systems using a time-dependent Schrödinger equation with pair states as a basis leads to well-defined solutions equivalent to those obtained from the Klein–Gordon equation and the Dirac equation. In addition, the particle–antiparticle relationship is well manifested through a particle conjugation group. This work provides new insights into the underlying molecular mechanism of relativistic dynamics and the rational design of new pathways for energy transformation. Full article

In the conventional (so-called Schrödinger-picture) formulation of quantum theory the operators of observables are chosen self-adjoint and time-independent. In the recent innovation of the theory, the operators can be not only non-Hermitian but also time-dependent. The formalism (called non-Hermitian interaction-picture, NIP) requires a separate description of the evolution of the time-dependent states $\psi \left(t\right)$ (using Schrödinger-type equations) as well as of the time-dependent observables ${\Lambda }_j\left(t\right)$, $j=1,2,\dots ,K$ (using Heisenberg-type equations). In the unitary-evolution dynamical regime of our interest, both of the respective generators of the evolution (viz., in our notation, the Schrödingerian generator $G\left(t\right)$ and the Heisenbergian generator $\Sigma \left(t\right)$) have, in general, complex spectra. Only the spectrum of their superposition remains real. Thus, only the observable superposition $H\left(t\right)=G\left(t\right)+\Sigma \left(t\right)$ (representing the instantaneous energies) should be called Hamiltonian. In applications, nevertheless, the mathematically consistent models can be based not only on the initial knowledge of the energy operator $H\left(t\right)$ (forming a “dynamical” model-building strategy) but also, alternatively, on the knowledge of the Coriolis force $\Sigma \left(t\right)$ (forming a “kinematical” model-building strategy), or on the initial knowledge of the Schrödingerian generator $G\left(t\right)$ (forming, for some reason, one of the most popular strategies in the literature). In our present paper, every such choice (marked as “one”, “two” or “three”, respectively) is shown to lead to a construction recipe with a specific range of applicability. Full article

Quantum teleportation is the name of a problem: How can the real-valued parameters encoding the state at Alice’s location make their way to Bob’s location via shared entanglement and only two bits of classical communication? Without an explanation, teleportation appears to be a conjuring trick. Investigating the phenomenon with Schrödinger states and reduced density matrices shall always leave loose ends because they are not local and complete descriptions of quantum systems. Upon demonstrating that the Heisenberg picture admits a local and complete description, Deutsch and Hayden rendered its explanatory power manifest by revealing the trick behind teleportation, namely, by providing an entirely local account. Their analysis is re-exposed and further developed. Full article