In the context of quantum information, both Shannon entropy and the different Jaynes entropy for a two-electron ``entangled artificial'' atom proposed by Moshinsky are studied here. The Jaynes entropy in this model is shown to be intimately related to the correlation kinetic energy. A generalization of the customary Shannon entropy based on the electron density is then proposed in terms of the electron pair correlation function. Both definitions behave in a similar way to the Jaynes entropy when considered as a function of the strength of the model electron-electron interaction.
In the context of quantum information, both Shannon entropy and the different Jaynes entropy for a two-electron ``entangled artificial'' atom proposed by Moshinsky are studied here. The Jaynes entropy in this model is shown to be intimately related to the correlation kinetic energy. A generalization of the customary Shannon entropy based on the electron density is then proposed in terms of the electron pair correlation function. Both definitions behave in a similar way to the Jaynes entropy when considered as a function of the strength of the model electron-electron interaction.
Different entropies which can characterize the inseparable quantum correlation in a two-electron entangled artificial atom proposed by Moshinsky are analyzed. The analysis is based on an exact representation for the trace of the q-order one-body reduced density matrix in terms of the interparticle coupling. The entanglement spectrum of the model system and the possible role of the entropy-maximum concept to practical applications of matrices are discussed as well. The ground-state energy, readily obtained from the Schr\"odinger Hamiltonian, is recalculated within a density-matrix-functional representation for its components.
We consider a nearly (or quasi) uniform gas of interacting electrons for which spin statistics play a crucial role. A previously developed procedure, based on the extension of the Levy-Lieb constrained search principle and Monte Carlo sampling of electron configurations in space, allows us to approximate the form of the kinetic-energy functional. For a spinless electron gas, this procedure led to a correlation term, which had the form of the Shannon entropy, but the resulting kinetic-energy functional does not satisfy the Lieb-Thirring inequality, which is rigorous and one of the most general relations regarding the kinetic energy. In this paper, we show that when the fermionic character of the electrons is included via a statistical spin approach, our procedure leads to correlation terms, which also have the form of the Shannon entropy and the resulting kinetic-energy functional does satisfy the Lieb-Thirring inequality. In this way we further strengthen the connection between Shannon entropy and electron correlation and, more generally, between information theory and quantum mechanics.
In this paper, we have presented an analytical model of two electron systems consisting of a many particle correlated wave function with some variational parameters α, λ and µ and used it to quantify the electron-electron correlation described by the wave function containing explicitly r12 (inter atomic distance between two electrons) dependent term. The single particle wave functions and the charge densities have been extracted from the said correlated wave function both for the uncorrelated and correlated systems in coordinate space and its momentum analogs have been obtained by taking the Fourier transform of the coordinate analogs. We have computed and presented the results of the numerical values of the theoretic information entropies of the Shannon entropy, Fisher information entropy, Shannon power and the FisherShannon product. The numerical values are consistently found to satisfy the Beckner, Bialynicki-Birula and Mycielski (BBM ) inequality relation; Stam-Cramer-Rao inequalities or Fisher based uncertainty relation and Fisher-Shannon product relation for the uncorrelated and correlated systems in both the coordinate and momentum spaces.
Position–momentum correlation in the Moshinsky atom is studied by examining the mutual information between Wigner functions and position/momentum space densities, at the one- and two-particle levels. We show that this correlation is independent of the strength of the interparticle potential at the two-particle level. At the one-particle level, the correlation decreases as the potential is increased. Shannon entropies of one- and two-particle Wigner functions are also discussed as measures of the localization in phase-space distributions.
In this chapter, we discuss several information measures that are important for quantifying the amount of information and correlations that are present in quantum systems. The first fundamental measure that we introduce is the von Neumann entropy (or simply quantum entropy ). It is the quantum generalization of the Shannon entropy, but it captures both classical and quantum uncertainty in a quantum state.1 The quantum entropy gives meaning to the notion of an information qubit . This notion is different from that of the physical qubit, which is the description of a quantum state of an electron or a photon. The information qubit is the fundamental quantum informational unit of measure, determining how much quantum information is present in a quantum system. The initial definitions here are analogous to the classical definitions of entropy, but we soon discover a radical departure from the intuitive classical notions from the previous chapter: the conditional quantum entropy can be negative for certain quantum states. In the classical world, this negativity simply does not occur, but it takes on a special meaning in quantum information theory. Pure quantum states that are entangled have stronger-than-classical correlations and are examples of states that have negative conditional entropy. The negative of the conditional quantum entropy is so important in quantum information theory that we even have a special name for it: the coherent information. We discover that the coherent information obeys a quantum data-processing inequality, placing it on a firm footing as a particular informational measure of quantum correlations. We then define several other quantum information measures, such as quantum mutual information, that bear similar definitions as in the classical world, but with Shannon entropies replaced with quantum entropies. This replacement may seem to make quantum entropy somewhat trivial on the surface, but a simple calculation reveals that a maximally entangled state on two qubits registers two bits of quantum mutual information (recall that the largest the mutual information can be in the classical world is one bit for the case of two maximally correlated bits). We then discuss several entropy inequalities that play an important role in quantum information processing: the monotonicity of quantum relative entropy, strong subadditivity, the quantum data-processing inequalities, and continuity of quantum entropy.
We use the Shannon informational entropies as a tool to study the artificial helium atom, namely, two electrons confined in a quantum dot. We adopt configurations with spherical and cylindrical symmetries for the physical system of interest. Using the informational quantities, we analyze the effects of electronic confinement, we validate the entropic uncertainty relation, we identify that the Coulomb interaction potential between the electrons is no longer important for strong confinements, and we indicate/predict the avoided crossing phenomena. Finally, we carried out a density function analysis. When available, the results are compared with those in the literature.