Upper and Lower Bounds to Two and Three-Electron Systems

Nicolais L. Guevara, Robin P. Sagar, Rodolfo O. Esquivel

The one-electron Shannon information entropy sum is reformulated in terms of a single entropic quantity dependent on a one-electron phase space quasiprobability density. This entropy is shown to form an upper bound for the entropy of the one-electron Wigner distribution. Two-electron entropies in position and momentum space, and their sum, are introduced, discussed, calculated, and compared to their one-electron counterparts for neutral atoms. The effect of electron correlation on the two-electron entropies is examined for the helium isoelectronic series. A lower bound for the two-electron entropy sum is developed for systems with an even number of electrons. Calculations illustrate that this bound may also be used for systems with an odd number of electrons. This two-electron entropy sum is then recast in terms of a two-electron phase space quasiprobability density. We show that the original Bialynicki-Birula and Mycielski information inequality for the N-electron wave function may also be formulated in terms of an N-electron phase space density. Upper bounds for the two-electron entropies in terms of the one-electron entropies are reported and verified with numerical calculations.

G. A. Gallup

Tong Zhang, Gaojie Chen, Rui Wang

This paper investigates the sum-secure degrees-of-freedom (SDoF) of three-user multiple-input multiple-output (MIMO) broadcast channel with confidential messages (BCCM) and delayed channel state information at the transmitter (CSIT). Specifically, we obtain non-trivial sum-SDoF upper and lower bounds. Firstly, we derive the sum-SDoF upper bound by means of statistical equivalence property, security constraints, and permutations. Then, for the sum-SDoF lower bound, we leverage the artificial noise transmission and interference re-transmission to design two transmission schemes, which have holistic and sequential higher-order symbol generation, respectively. For these two schemes, we propose the redundancy reduction approach for security analysis, by which the minimal duration of artificial noise transmission phase of the scheme is obtained. To eliminate the redundant equations in security analysis, this approach first identifies the constituent equations, and then analyzes the rank of assemble of them. As a result, both the proposed sum-SDoF upper and lower bounds are tighter than the existing sum-SDoF upper and lower bounds, respectively. Furthermore, the proposed lower bound showcases a three-user coding gain.

S. Fraga, Gulzari L. Malli, L. H. Thomas

Fu-Gang Zhang

Abstract In this paper, we discuss quantum uncertainty relations of Tsallis relative α entropy coherence for a single qubit system based on three mutually unbiased bases. For α ∈ 1 2 , 1 ∪ ( 1 , 2 ] , the upper and lower bounds of sums of coherence are obtained. However, the above results cannot be verified directly for any α ∈ 0 , 1 2 . Hence, we only consider the special case of α = 1 n + 1 , where n is a positive integer, and we obtain the upper and lower bounds. By comparing the upper and lower bounds, we find that the upper bound is equal to the lower bound for the special α = 1 2 , and the differences between the upper and the lower bounds will increase as α increases. Furthermore, we discuss the tendency of the sum of coherence, and find that it has the same tendency with respect to the different θ or φ, which is opposite to the uncertainty relations based on the Rényi entropy and Tsallis entropy.

Marianna S. Safronova, A.S. Safronova