Convexity of quasi-entropy type functions: Lieb's and Ando's convexity theorems revisited

Abstract

Given a positive function $f$ on $(0,\infty )$ and a non-zero real parameter $\theta$, we consider a function $I_f^{\theta }(A,B,X)=Tr{X}^{\ast }(f(L_A{R}_B^{-1})R_B{)}^{\theta }(X)$ in three matrices $A,B>0$ and $X$. In the literature $\theta =\pm 1$ has been typical. The concept unifies various quantum information quantities such as quasi-entropy, monotone metrics, etc. We characterize joint convexity/concavity and monotonicity properties of the function $I_f^{\theta }$, thus unifying some known results for various quantum quantities.

27 pages

References in corpus (5)

• Metric adjusted skew information
• A Matrix Convexity Approach to Some Celebrated Quantum Inequalities
• A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity
• Quantum state discrimination bounds for finite sample size
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Cited by in corpus (5)

• Different quantum f-divergences and the reversibility of quantum operations
• Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems
• Contraction coefficients for noisy quantum channels
• Families of completely positive maps associated with monotone metrics
• Lower Bounds on Error Exponents via a New Quantum Decoder