No Access Submitted: 04 September 2019 Accepted: 14 October 2019 Published Online: 04 December 2019
J. Math. Phys. 60, 122201 (2019);
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.
  1. 1. Beigi, S., “Sandwiched Rényi divergence satisfies data processing inequality,” J. Math. Phys. 54(12), 122202 (2013)., Google ScholarScitation, ISI
  2. 2. Datta, N., “Min- and max-relative entropies and a new entanglement monotone,” IEEE Trans. Inf. Theory 55(6), 2816–2826 (2009)., Google ScholarCrossref
  3. 3. Drescher, L. and Fawzi, O., “On simultaneous min-entropy smoothing,” in 2013 IEEE International Symposium on Information Theory (IEEE, 2013), pp. 161–165. Google ScholarCrossref
  4. 4. Dupuis, F., Kraemer, L., Faist, P., Renes, J. M., and Renner, R., “Generalized entropies,” in Proceedings of the XVIIth International Congress on Mathematical Physics (Aalborg, Denmark, 2012), pp. 134–153. Google Scholar
  5. 5. Frank, R. L. and Lieb, E. H., “Monotonicity of a relative Rényi entropy,” J. Math. Phys. 54(12), 122201 (2013)., Google ScholarScitation, ISI
  6. 6. Jain, R. and Nayak, A., “Short proofs of the quantum substate theorem,” IEEE Trans. Inf. Theory 58(6), 3664–3669 (2012)., Google ScholarCrossref
  7. 7. Jain, R., Radhakrishnan, J., and Sen, P., “Privacy and interaction in quantum communication complexity and a theorem about the relative entropy of quantum states,” in Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002 (IEEE, 2002), pp. 429–438. Google ScholarCrossref
  8. 8. Jain, R., Radhakrishnan, J., and Sen, P., “A property of quantum relative entropy with an application to privacy in quantum communication,” J. ACM 56(6), 33:1–33:32 (2009)., Google ScholarCrossref
  9. 9. Müller-Lennert, M., Dupuis, F., Szehr, O., Fehr, S., and Tomamichel, M., “On quantum Rényi entropies: A new generalization and some properties,” J. Math. Phys. 54(12), 122203 (2013)., Google ScholarScitation, ISI
  10. 10. Renner, R., “Security of quantum key distribution,” Ph.D. thesis, ETH Zurich, 2005; e-print arXiv:quant-ph/0512258. Google Scholar
  11. 11. Sen, P., “A one-shot quantum joint typicality lemma,” e-print arXiv:1806.07278 (2018). Google Scholar
  12. 12. Sion, M., “On general minimax theorems,” Pac. J. Math. 8, 171–176 (1958)., Google ScholarCrossref
  13. 13. Tomamichel, M., “A framework for non-asymptotic quantum information theory,” Ph.D. thesis, ETH Zurich, 2012; e-print arXiv:1203.2142. Google Scholar
  14. 14. Tomamichel, M., Colbeck, R., and Renner, R., “Duality between smooth min- and max-entropies,” IEEE Trans. Inf. Theory 56(9), 4674–4681 (2010)., Google ScholarCrossref
  15. 15. Tomamichel, M. and Hayashi, M., “A hierarchy of information quantities for finite block length analysis of quantum tasks,” IEEE Trans. Inf. Theory 59(11), 7693–7710 (2013)., Google ScholarCrossref
  16. 16. Tomamichel, M. and Leverrier, A., “A largely self-contained and complete security proof for quantum key distribution,” Quantum 1, 14 (2017)., Google ScholarCrossref
  17. 17. Wilde, M. M., Winter, A., and Yang, D., “Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Rényi relative entropy,” Commun. Math. Phys. 331(2), 593–622 (2014)., Google ScholarCrossref
  1. © 2019 Author(s). Published under license by AIP Publishing.