Inequalities for Tsallis relative entropy and generalized skew information

Abstract

Quantum entropy and skew information play important roles in quantum information science. They are defined by the trace of the positive operators so that the trace inequalities often have important roles to develop the mathematical theory in quantum information science. In this paper, we study some properties for information quantities in quantum system through trace inequalities. Especially, we give upper bounds and lower bounds of Tsallis relative entropy, which is a one-parameter extension of the relative entropy in quantum system. In addition, we compare the known bounds and the new bounds, for both upper and lower bounds, respectively. We also give an inequality for generalized skew information by introducing a generalized correlation measure.

Relative entropy and Tsallis entropy of two accretive operators

“…see [2,3,4] for instance. The Tsallis relative operator entropy is a parametric extension in the sense that lim λ→0 T λ (A|B) = S(A|B).…”

Uncertainty relation associated with a monotone pair skew information

“…Recently in [2], for a monotone pair ( f , g) of operator monotone functions, Furuichi introduced the ( f , g)-skew information by…”

“…For f (x) = x α and g(x) = x 1−α (0 < α < 1), I ρ, ( f ,g) reduces to I ρ,α in (1.2). For this information, Furuichi has shown the following trace inequality [2]:…”

“…β ( f ,g) Tr f (ρ)g(ρ)[A, B] 2 4β ( f ,g) l<m f (λ l )g(λ l ) − f (λ m )g(λ m ) |a ml b lm | 2 l<m f (λ l ) 2 − f (λ m ) 2 g(λ l ) 2 − g(λ m ) 2 |a ml b lm | Substituting ( f (λ l ) 2 − f (λ m ) 2 )(g(λ l ) 2 − g(λ m ) 2 ) = ( f (λ l )g(λ l ) + f (λ m )g(λ m )) 2 − ( f (λ l )g(λ m ) + f (λ m )g(λ l ))2 into the above, and by the Schwarz inequality, we haveβ ( f ,g) Tr f (ρ)g(ρ)[A, B] 2 l<m f (λ l )g(λ l ) + f (λ m )g(λ m ) − f (λ l )g(λ m ) + f (λ m )g(λ l ) |a ml | 2 × l<m f (λ l )g(λ l ) + f (λ m )g(λ m ) + f (λ l )g(λ m ) + f (λ m )g(λ l ) |b lm | 2 .…”

Some inequalities on generalized entropies

“…See [4,5,6,7,8,9,10,11,12,13,14] and references therein for recent advances and applications on the Tsallis entropy. We easily find that the Tsallis relative entropy is a special case of Csiszár f -divergence [15,16,17] defined for a convex function f on (0, ∞) with f (1) = 0 by…”