Quantum information measures of the Aharonov–Bohm ring in uniform magnetic fields

Abstract

Shannon quantum information entropies S ρ,γ , Fisher informations I ρ,γ , Onicescu energies O ρ,γ and complexities e S O are calculated both in the position (subscript ρ) and momentum (γ) spaces for the azimuthally symmetric twodimensional nanoring that is placed into the combination of the transverse uniform magnetic field B and the Aharonov-Bohm (AB) flux φ AB and whose potential profile is modeled by the superposition of the quadratic and inverse quadratic dependencies on the radius r. The increasing intensity B flattens momentum waveforms Φ nm (k) and in the limit of the infinitely large fields they turn to zero: Φ nm (k) → 0 at B → ∞, what means that the position wave functions Ψ nm (r), which are their Fourier counterparts, tend in this limit to the δ-functions. Position (momentum) Shannon entropy depends on the field B as a negative (positive) logarithm of ω ef f ≡ ω 2 0 + ω 2 c /4 1/2 , where ω 0 determines the quadratic steepness of the confining potential and ω c is a cyclotron frequency. This makes the sum S ρ nm + S γ nm a field-independent quantity that increases with the principal n and azimuthal m quantum numbers and does satisfy entropic uncertainty relation. Position Fisher information does not depend on m, linearly increases with n and varies as ω ef f whereas its n and m dependent Onicescu counterpart O ρ nm changes as ω −1 ef f . The products I ρ nm I γ nm and O ρ nm O γ nm are B-independent quantities. A dependence of the measures on the ring geometry is discussed. It is argued that a variation of the position Shannon entropy or Onicescu energy with the AB field uniquely determines an associated persistent current as a function of φ AB at B = 0. An inverse statement is correct too.

Rényi and Tsallis Entropies of the Aharonov–Bohm Ring in Uniform Magnetic Fields

“…Similar to the Shannon case [2], the whole dependence of the Rényi position and momentum entropies on the uniform magnetic field B is embedded in the terms ±2 ln r e f f . Concerning the Tsallis functionals, a dimensional incompatibility of the two items in Equations (24) precludes their direct application for the continuous probability distributions, suggesting instead the forms presented in the corresponding uncertainty relation, Equation 20, but below we will continue to write them, keeping in mind that it is just a formal representation only.…”

“…In an attempt to expand quantum-information theory to the study of the quantum rings (QRs) [1], a recent analysis [2] addressed an influence of the combination of the transverse uniform magnetic field B and the AB flux φ AB [3] on the position and momentum components of, among others, Shannon entropy [4] of the one-particle orbitals of the flat 2D annulus whose rotationally symmetric potential profile U(r) is modeled in the position of polar coordinates (r, ϕ r ) by the superposition of the quadratic and inverse quadratic dependencies on the radius r [5][6][7][8][9][10][11][12][13][14][15]:…”

“…where δ nn is a Kronecker delta. It was shown [2] that Shannon position (momentum) quantum information entropy decreases (increases) with the growing field B as ±2 ln r e f f , which physically means that in the corresponding space, there is more (less) information about the particle location (intensity of motion). As a result, the sum S ρ + S γ , describing a total amount of the simultaneous information about the charge carrier, cannot be altered by the uniform magnetic component, and it always satisfies the fundamental restriction [18,19] S ρ + S γ ≥ l(1 + ln π).…”

“…For many years, physicists and mathematicians have been looking for and discussing generalizations of the Shannon measures from Equations (2). Notably, two of the most famous and frequently used outcomes of these endeavors are one-parameter functionals of the Rényi R ρ,γ (α) [21,22] (To tell the Rényi entropies from the radial part of the position waveform (Equation (6a)), we will always write the former ones with the subscript ρ or γ denoting a corresponding space.…”

“…For the QR, the position (momentum) Onicescu energy increases (decreases) with the uniform field as r −2 e f f r 2 e f f , which makes the product O ρ O γ similar to the sum S ρ + S γ , a B-independent quantity [2]. Sobolev inequality of the conjugated Fourier transforms [27] α π…”