One-shot information theory entertains a plethora of entropic quantities, such as the smooth max-divergence, hypothesis testing divergence, and information spectrum divergence, that characterize various operational tasks in quantum information theory and are used to analyze their asymptotic behavior. Tight inequalities between these quantities are thus of immediate interest. In this note, we use a minimax approach (appearing previously, for example, in the proofs of the quantum substate theorem), to simplify the quantum problem to a commutative one, which allows us to derive such inequalities. Our derivations are conceptually different from previous arguments and in some cases lead to tighter relations. We hope that the approach discussed here can lead to progress in open problems in quantum Shannon theory and exemplify this by applying it to a simple case of the joint smoothing problem.
The work was done when A.A. was affiliated to the Centre for Quantum Technologies, National University of Singapore. We thank David Sutter for help with the Proof of Theorem 3 for normalized trace distance. A.A. and R.J. were supported by the Singapore Ministry of Education and the National Research Foundation through Grant No. “NRF2017-NRF-ANR004 VanQuTe”. R.J. was also supported by the VAJRA Faculty Scheme of the Science and Engineering Board (SERB), Department of Science and Technology (DST), Government of India.
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