A strengthened data processing inequality for the Belavkin-Staszewski relative entropy

Abstract

In this work, we provide a strengthening of the data processing inequality for the relative entropy introduced by Belavkin and Staszewski (BS-entropy). This extends previous results by Carlen and Vershynina for the relative entropy and other standard $f$-divergences. To this end, we provide two new equivalent conditions for the equality case of the data processing inequality for the BS-entropy. Subsequently, we extend our result to a larger class of maximal $f$-divergences. Here, we first focus on quantum channels which are conditional expectations onto subalgebras and use the Stinespring dilation to lift our results to arbitrary quantum channels.

arXiv comment25 pages, no figures. We managed to extend the main result from conditional expectations to arbitrary quantum channels (see Section 4 and Section 7). Added one reference

References in corpus (11)

• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Quantum conditional mutual information and approximate Markov chains
• Quantum f-divergences and error correction
• Monotonicity of quantum relative entropy revisited
• Jensen's Operator Inequality
• Universal recovery maps and approximate sufficiency of quantum relative entropy
• Different quantum f-divergences and the reversibility of quantum operations
• Multivariate Trace Inequalities
• Quantum conditional relative entropy and quasi-factorization of the relative entropy
• Recovery and the Data Processing Inequality for quasi-entropies
• Recoverability for Holevo's just-as-good fidelity

Cited by in corpus (11)

• Exponential decay of mutual information for Gibbs states of local Hamiltonians
• Continuity of quantum entropic quantities via almost convexity
• Recoverability for optimized quantum $f$-divergences
• Equality conditions of Data Processing Inequality for $\alpha$-$z$ Rényi relative entropies
• Geometric Rényi Divergence and its Applications in Quantum Channel Capacities
• General Continuity Bounds for Quantum Relative Entropies
• Weak quasi-factorization for the Belavkin-Staszewski relative entropy
• Observational entropy with general quantum priors
• Revisiting the equality conditions of the data processing inequality for the sandwiched Rényi divergence
• Conditional Independence of 1D Gibbs States with Applications to Efficient Learning
• Belavkin-Staszewski Quantum Markov Chains