The so-called Darboux III oscillator is an exactly solvable N-dimensional nonlinear oscillator defined on a radially symmetric space with non-constant negative curvature. This oscillator can be interpreted as a smooth (super)integrable deformation of the usual N-dimensional harmonic oscillator in terms of a non-negative parameter λ which is directly related to the curvature of the underlying space. In this paper, a detailed study of the Shannon information entropy for the quantum version of the Darboux III oscillator is presented, and the interplay between entropy and curvature is analysed. In particular, analytical results for the Shannon entropy in the position space can be found in the N-dimensional case, and the known results for the quantum states of the N-dimensional harmonic oscillator are recovered in the limit of vanishing curvature λ→0. However, the Fourier transform of the Darboux III wave functions cannot be computed in exact form, thus preventing the analytical study of the information entropy in momentum space. Nevertheless, we have computed the latter numerically both in the one and three-dimensional cases and we have found that by increasing the absolute value of the negative curvature (through a larger λ parameter) the information entropy in position space increases, while in momentum space it becomes smaller. This result is indeed consistent with the spreading properties of the wave functions of this quantum nonlinear oscillator, which are explicitly shown. The sum of the entropies in position and momentum spaces has been also analysed in terms of the curvature: for all excited states such total entropy decreases with λ, but for the ground state the total entropy is minimized when λ vanishes, and the corresponding uncertainty relation is always fulfilled.
The so-called Darboux III oscillator is an exactly solvable N-dimensional nonlinear oscillator defined on a radially symmetric space with non-constant negative curvature. This oscillator can be interpreted as a smooth (super)integrable deformation of the usual N-dimensional harmonic oscillator in terms of a non-negative parameter λ which is directly related to the curvature of the underlying space. In this paper, a detailed study of the Shannon information entropy for the quantum version of the Darboux III oscillator is presented, and the interplay between entropy and curvature is analysed. In particular, analytical results for the Shannon entropy in the position space can be found in the N-dimensional case, and the known results for the quantum states of the N-dimensional harmonic oscillator are recovered in the limit of vanishing curvature λ→0. However, the Fourier transform of the Darboux III wave functions cannot be computed in exact form, thus preventing the analytical study of the information entropy in momentum space. Nevertheless, we have computed the latter numerically both in the one and three-dimensional cases and we have found that by increasing the absolute value of the negative curvature (through a larger λ parameter) the information entropy in position space increases, while in momentum space it becomes smaller. This result is indeed consistent with the spreading properties of the wave functions of this quantum nonlinear oscillator, which are explicitly shown. The sum of the entropies in position and momentum spaces has been also analysed in terms of the curvature: for all excited states such total entropy decreases with λ, but for the ground state the total entropy is minimized when λ vanishes, and the corresponding uncertainty relation is always fulfilled.
The Fock-Darwin (FD) quantum system describes the motion on the plane of a charged particle under the action of an isotropic oscillator potential together with a perpendicular constant magnetic field. When the isotropic oscillator is suppressed, the FD system leads to the Landau Hamiltonian with infinitely degenerate Landau levels. The Fock-Darwin-Darboux (FDD) system is the generalisation of the FD system to a particle moving on the Darboux III space, which is a conformally flat surface with non-constant negative curvature. We present a systematic study of some information-theoretic entropy and dispersion-like measures for these quantum systems. Since both systems are exactly solvable, analytical expressions for Shannon, Rényi and Tsallis entropies, among others, can be obtained. We show that for the FD system, its information-theoretic measures are formally the same as the ones for the harmonic oscillator, provided a modified effective frequency depending on the magnetic field is introduced. In the FDD case, the nonlinear nature of the underlying manifold precludes the existence of a simple closed form for the wave-function on momentum space, which is numerically analysed. We compare the numerical behaviour of the different entropy measures and we analyse the interplay arising in the FDD system between the curvature parameter and the magnetic field. In particular, it is shown that the Landau system on the Darboux III space has no infinitely degenerate Landau levels.
The time-dependent behaviour of harmonic oscillator squeezed states is examined from the viewpoint of position and momentum space Shannon entropies and their sum. Analytical expressions for these entropies are obtained from a symplectic group approach. The entropy sum returns to the entropic uncertainty bound at different time intervals, and is characterized by the presence of either a maximum (minimum) in the position space Shannon entropy and a minimum (maximum) in the momentum space one. The maximal value of the entropy sum is characterized by equality of position and momentum Shannon entropies. The transition of the entropy sum from the bound to its maximal value is studied, and relations between position and momentum space Shannon entropies are probed. The relation between the position and momentum space entropies is deformed as the value of the squeezing parameter increases. An analysis of the dependence of the Shannon entropies on the squeezing parameter shows different behaviours at different times. Both position and momentum space Shannon entropies exhibit a linear dependence on the parameter when their sum lies on the uncertainty bound. Here, the entropy sum is independent of the parameter. At other times, the relation between the entropies and squeezing parameter is non-linear, and extremal points are present. The entropy sum here is seen to be an increasing function of the parameter. Kullback–Leibler distances between squeezed and unsqueezed densities illustrate that the position or momentum space densities of the squeezed system can revert back to the unsqueezed ones during the time evolution. The phase-space Wigner function, entropy and mutual information are also examined and discussed. The statistical correlation as measured by mutual information is the entropy sum minus its value at the uncertainty bound.
The Fock-Darwin (FD) quantum system describes the motion on the plane of a charged particle under the action of an isotropic oscillator potential together with a perpendicular constant magnetic field. When the isotropic oscillator is suppressed, the FD system leads to the Landau Hamiltonian with infinitely degenerate Landau levels. The Fock-Darwin-Darboux (FDD) system is the generalisation of the FD system to a particle moving on the Darboux III space, which is a conformally flat surface with non-constant negative curvature. We present a systematic study of some information-theoretic entropy and dispersion-like measures for these quantum systems. Since both systems are exactly solvable, analytical expressions for Shannon, Rényi and Tsallis entropies, among others, can be obtained. We show that for the FD system, its information-theoretic measures are formally the same as the ones for the harmonic oscillator, provided a modified effective frequency depending on the magnetic field is introduced. In the FDD case, the nonlinear nature of the underlying manifold precludes the existence of a simple closed form for the wave-function on momentum space, which is numerically analysed. We compare the numerical behaviour of the different entropy measures and we analyse the interplay arising in the FDD system between the curvature parameter and the magnetic field. In particular, it is shown that the Landau system on the Darboux III space has no infinitely degenerate Landau levels.
Abstract The Shannon information entropy sum is examined as systems comprised of two-level harmonic oscillator states are passed through dephasing and depolarization quantum channels. We show how the entropy sum increases as the dephasing parameter is increased. The interpretation here is a loss of information in the system as the quantum coherences are removed. This result and interpretation also holds in the depolarization channel as the system transits to the maximally mixed state. The increase in the entropy sum is associated with a delocalization of both position and momentum densities. The behaviour of phase-space Wigner functions and their Shannon entropies is also addressed. Wigner negativity is seen to decrease and disappear with an increase of the dephasing parameter. Here, Wigner negativity is associated with the presence of quantum coherences. The imaginary component of the Wigner function entropy is shown to capture the changes in Wigner negativity. On the other hand, Wigner functions with diagonal input density matrices are non-negative in the depolarization channel. Their real-valued Shannon entropies increase with the depolarization parameter as the system approaches the maximally mixed state. The mutual information between the Wigner function and the product of position and momentum space densities is used to probe the statistical correlation between position and momentum variables. This measure decreases in both channels as the parameter increases. Thus, there is a loss of correlation as quantum coherences are eliminated in the dephasing case, and as the system transits to the maximally mixed case in the depolarizing channel.
With the development of quantum technology and information science, significant attention is focused on studying the quantum information properties in confined systems. In this work, we present a systematic investigation of quantum information measures – including position-momentum uncertainty (Δx,Δp), Shannon entropy (Sx, Sp), and Onicescu energy (Ox, Op) in a confined Robin harmonic oscillator (CRHO) with symmetric boundary conditions λL = λR = λ. It is interesting to find that the Heisenberg uncertainty relation ΔxΔp≥ℏ/2, which holds under the Dirichlet boundary condition (λ = 0), is violated for the ground state of the CRHO. In contrast, the entropic uncertainty relation (Sx + Sp ≥ 1 + lnπ) remains satisfied at any Robin length λ and quantum state, demonstrating its fundamental advantage and broader applicability in quantum systems. For a fixed Robin length, we observe that as the quantum well width increases within a finite range, the position-space Shannon entropy Sₓ increases, while the Onicescu energy Oₓ decreases. Despite their opposing tendencies, both Shannon entropy and Onicescu energy reflect the same physical insight regarding the localisation-delocalisation property for the CRHO system. In addition, the dependence of these information measures on the Robin length is discussed, and their physical implications are highlighted. This study demonstrates that boundary conditions play a critical role in quantum information metrics, thereby informing the design and optimisation of quantum devices such as quantum dots, nanoscale resonators and atom-based sensors.
The well-known Heisenberg–Robertson uncertainty relation for a pair of noncommuting observables, is expressed in terms of the product of variances and the commutator among the operators, computed for the quantum state of a system. Different modified commutation relations have been considered in the last years with the purpose of taking into account the effect of quantum gravity. Indeed it can be seen that letting [ X , P ] = i ℏ ( 1 + β P 2 ) implies the existence of a minimal length proportional to β . The Bialynicki-Birula–Mycielski entropic uncertainty relation in terms of Shannon entropies is also seen to be deformed in the presence of a minimal length, corresponding to a strictly positive deformation parameter β . Generalized entropies can be implemented. Indeed, results for the sum of position and (auxiliary) momentum Rényi entropies with conjugated indices have been provided recently for the ground and first excited state. We present numerical findings for conjugated pairs of entropic indices, for the lowest lying levels of the deformed harmonic oscillator system in 1D, taking into account the position distribution for the wavefunction and the actual momentum.