Properties of the Shannon entropy of arrays of elastic scatterers

Abstract

Recently, we used an effective spin concept to expound the analogy between spin-based quantum information processing and phase coherent charge transport through an array of elastic scatterers. Here, we extend that analogy by calculating an effective Shannon entropy for such an array and examining its various properties. For single-moded transport, the Shannon entropy is given by H bin ( | t | 2 ) = − | t | 2 log 2 | t | 2 − | r | 2 log 2 | r | 2 , where | t | 2 and | r | 2 are the transmission and reflection probabilities through the array of scatterers. A lower bound for H bin ( | t | 2 ) is found starting with the entropic quantum uncertainty principle. An important result is that although evanescent channels (modes) have | t | 2 = 1 − | r | 2 → 0 , so that their own contribution to H bin ( | t | 2 ) → 0 , they nevertheless have a profound influence on the total H bin ( | t | 2 ) of the array and its associated signal-to-noise ratio (SNR) since they renormalize the transmission probabilities of the propagating modes. This is reminiscent of the well-known fact that evanescent modes influence the conductance of a structure by renormalizing the transmission probabilities of the propagating modes. The numerical values of H bin ( | t | 2 ) and its SNR are strongly sensitive to the nature of the elastic scatterers, i.e., whether they are attractive (negative potential), repulsive (positive potential), or a combination of both. In samples with repulsive scatterers, the SNR can be tuned over a wide range by applying a potential through a gate to change the scattering potentials from repulsive to attractive by moving their energy levels with respect to the quasi-Fermi level in the sample. We also found that the mean free path of an electron traversing a random array of elastic scatterers is the length scale at which the sum of the cross-correlation coefficients of the effective spin components reaches a minimum. At that point, the sum of the effective Heisenberg and Zeeman Hamiltonians associated with effective spins describing the propagating channels, reaches a minimum. Hence the mean free path can be viewed as an order parameter for a phase transition.