Abstract
In this paper, some new inequalities for convex functions of self-adjoint operators are obtained. As applications, we present some inequalities for quantum $f$-divergence of trace class operators in Hilbert Spaces.
Highlights
(i) If P = Q, the equality holds in the first part of (1.4)
If f is strictly convex at 1, the equality holds in the first part of (1.4) if and only if P = Q; (ii) If Q ⊥ P, the equality holds in the second part of (1.4)
The following result is a refinement of the second inequality in Theorem 1.4
Summary
The following two theorems contain the most basic properties of f -divergences. Let f : [0, ∞) → R be a continuous convex function on [0, ∞). The following result is a refinement of the second inequality in Theorem 1.4 (see [3, Theorem 3]). Let f be a continuous convex function on [0, ∞) with f (1) = 0 (f is normalised) and f (0) + f ∗ (0) < ∞. In what follows we recall some facts we need concerning the trace of operators and quantum f -divergence for trace class operators in infinite dimensional complex Hilbert spaces
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