Complex entropy and resultant information measures

Classical and nonclassical contributions to Author’s resultant Shannon- and Fisher-type measures of the information content in general electronic state φ ( r ) = R ( r ) exp [ i ϕ ( r ) ] , due to the state probability density p ( r ) = R ( r ) 2 and its phase ϕ ( r ) or current j ( r ) = ħ / m p ( r ) ∇ ϕ r distributions, respectively, are reexamined. The components of the overall entropy, S [ φ ] ≡ - ∫ p ( r ) [ ln p ( r ) + 2 ϕ ( r ) ] d r ≡ S [ p ] + S [ ϕ ] , are shown to determine the real and imaginary parts of the state complex Shannon entropy, H [ φ ] ≡ - 2 φ | ln φ | φ = S p + i S [ ϕ ] , a natural quantum-amplitude generalization of the classical Shannon entropy. Its contributions are related to the associated terms in the state resultant Fisher information, I [ φ ] ≡ - 4 ⟨ φ | ∇ 2 | φ ⟩ ≡ ∫ p ( r ) { [ ∇ ln p ( r ) ] 2 + [ 2 ∇ ϕ r ] 2 } d r ≡ I [ p ] + I [ ϕ ] = I [ p ] + ∫ p ( r ) [ ( 2 m / ħ ) j ( r ) / p ( r ) ] 2 d r ≡ I [ p ] + I [ j ] , and the gradient entropy: I ~ [ φ ] ≡ ⟨ φ | [ ( ∇ ln p ) 2 + ( i 2 ∇ ϕ ) 2 ] | φ ⟩ = I [ p ] - I [ ϕ ] = I ~ [ p ] + I ~ [ ϕ ] .