Schedule

UNM Physics and Astronomy, Summer 2025 course

  1. [05/21] Overview of classical and quantum information theory.

    2. [Mohammad Alhejji]

    Summary. We discuss in brief the origins of classical and quantum information theory and introduce the fundamental communication problems.
    Reading. Ref. 1 and Section 1.2 of Ref. 2.
    Homework. Claude Shannon – Father of the Information Age. Slides. PDF

  2. [05/28] The asymptotic equipartition property and Shannon’s classical data compression.

    3. [Srinidhi Pawar and Basie Seibert]

    Summary. We introduce discrete memoryless classical sources, typical sequences, and prove Shannon's source coding theorem.
    Reading. Chapter 3 of Ref. 3.
    Homework. PDF Slides. PDF Excel sheet. CSV

  3. [06/04] Typical subspaces and Schumacher's quantum data compression.

    4. [Mohsin Raza and Jalan Ziyad]

    Summary. We introduce discrete memoryless quantum sources, typical subspaces, and the gentle measurement lemma. We prove Schumacher's source coding theorem for pure quantum sources.
    Reading. Ref. 4 and Chapter 18 of Ref. 2. For the history, see Ref. 15. For the state of the art of quantum data compression, see Ref. 5.
    Homework. PDF Slides. PDF

  4. [06/11] Joint typicality and distributed classical source coding.

    5. [Shravan]

    Summary. We introduce jointly typical sequences, Fano's inequality, and classical data compression in the presence of side information. We prove the Slepian-Wolf distributed source coding theorem.
    Reading. Sections 2.11, 8.6 and 14.4 of Ref. 3, and Ref. 6.
    Homework. PDF Slides. PDF

  5. [06/18] Shannon’s channel coding theorem.

    6. [Leroy Fagan and Ariq Haqq]

    Summary. We prove Shannon's channel coding theorem for discrete memoryless channels.
    Reading. Sections 8.5-8.9 of Ref. 3.
    Homework. PDF Slides. PDF

  6. [06/25] Pure bipartite entanglement concentration.

    7. [Alejandro Garcia and Alejandro Rascon]

    Summary. We show how local operations and classical communication (LOCC) can be used to concentrate pure bipartite entanglement.
    Reading. Ref. 7 and Chapter 19 of Ref. 2.
    Homework. PDF Slides. PDF

  7. [07/02] The quantum relative entropy

    8. [Morteza Darvishi and Omar Nazim]

    Summary. We introduce the Umegaki (quantum) relative entropy and discuss its basic properties. We show that Holevo's bound on the classical capacity of quantum channels is implied by the monotonicity of relative entropy under partial trace.
    Reading. Ref. 8 and Ref. 9.
    Homework. PDF Slides. PDF

  8. [07/09] Strong subadditivity of the quantum entropy.

    9. [Cole Kelson-Packer]

    Summary. We prove that the quantum relative entropy is monotonic under quantum channels and show how that implies strong subadditivity of the quantum entropy, i.e., the quantum conditional mutual information is non-negative.
    Reading. Ref. 9, Ref. 10 and Ref. 16.
    Slides. PDF

  9. [07/23] Quantum hypothesis testing and the quantum Stein's lemma

    10. [Hariprasad Madathil]

    Summary. We introduce quantum hypothesis testing (or binary quantum state discrimination) and prove the quantum Stein's lemma.
    Reading. Ref. 11-13 and section 12.7 of Ref. 3.
    Slides. PDF

  10. [07/30] Quantum channel coding with entanglement assistance.

    11. [Ariel Shlosberg]

    Summary. We prove that the entanglement-assisted capacity of communication through a quantum channel is given by the quantum mutual information.
    Reading. Ref. 14 and Chapter 21 of Ref. 2.
    Slides. PDF

Bibliography

  1. C. E. Shannon, "A mathematical theory of communication," The Bell System Technical Journal, vol. 27, no. 3, pp. 379-423, July 1948. (reprint)
  2. M. M Wilde, "Quantum Information Theory," Cambridge University Press, 2013.
  3. Thomas M. Cover and Joy A. Thomas, "Elements of Information Theory", Wiley, 1991
  4. Benjamin Schumacher, "Quantum coding", Phys. Rev. A 51, 2738, April 1995.
  5. Zahra Baghali Khanian and Andreas Winter, "General Mixed-State Quantum Data Compression With and Without Entanglement Assistance", IEEE Transactions on Information Theory, vol. 68, no. 5, pp. 3130-3138, May 2022.
  6. David Slepian and Jack Wolf, "Noiseless coding of correlated information sources", IEEE Transactions on Information Theory, vol. 19, no. 4, pp. 471-480, July 1973.
  7. Charles H. Bennett, Herbert J. Bernstein, Sandu Popescu and Benjamin Schumacher, "Concentrating partial entanglement by local operations", Phys. Rev. A 53, 2046, April 1996.
  8. Vlatko Vedral, "The role of relative entropy in quantum information theory", Rev. Mod. Phys. 74, 197, March 2002.
  9. Eric Carlen, "Trace inequalities and quantum entropy: an introductory course", Lecture notes, March 2009.
  10. Michael A Nielsen and Denes Petz, "A simple proof of the strong subadditivity inequality", ArXiv preprint, August 2004.
  11. K. M. R. Audenaert, M. Nussbaum, A. Szkoła and F. Verstraete, "Asymptotic Error Rates in Quantum Hypothesis Testing", Commun. Math. Phys. 279, 251–283, 2008.
  12. Tomohiro Ogawa and Hiroshi Nagaoka, "Strong Converse and Stein’s Lemma in the Quantum Hypothesis Testing", IEEE Transactions on Information Theory, vol. 46, no. 7, pp. 2428-2433, November 2000.
  13. Mario Berta, Fernando GSL Brandão, Gilad Gour, Ludovico Lami, Martin B. Plenio, Bartosz Regula, and Marco Tomamichel. "The tangled state of quantum hypothesis testing." Nature Physics 20, no. 2, 2024.
  14. Charles H. Bennett, Peter W. Shor, John A. Smolin, and Ashish V. Thapliyal. "Entanglement-Assisted Classical Capacity of Noisy Quantum Channels." Phys. Rev. Lett. 83, 3081, October 1999.
  15. Dénes Petz, "Entropy, von Neumann and the von Neumann Entropy", "John von Neumann and the Foundations of Quantum Physics", M. Rédei and M. Stöltzner (eds), 2001.
  16. M. Ohya and Dénes Petz, "Quantum Entropy and Its Use", Springer Berlin, 1993.