MIT 8.372 / 18.S996 - Quantum Information Science 3

Fall 2022

Lecture: TR2:30-4, Room 3-370

Instructor: Aram W Harrow
Office hours: Tu 4-5 and Th 11-12, Room 6-416A

TA: Annie Wei
Office hours: Fri 10-11, Room 8-320

This is a third class in the MIT QIS sequence, following 8.370 and 8.371. It ran before in Fall 2020 as 8.S372/18.S996.

Psets and lecture notes will be made public. Registered students (including listeners) can access the canvas site and Piazza discussion boards. Any content only for registered students (such as pset solutions) will be labeled by the MIT icon.


This is a third course in quantum information and computing theory, focused on special topics that may change from year to year. This year the focus is on quantum information theory, both understanding the core theory of the field, as well as application to physics.

The first part of the course will introduce the main questions and tools of quantum information theory, such as entropies, capacities, hypothesis testing, decoupling, random states and unitaries, symmetry, and entanglement. The second part will apply these tools, along with those from quantum complexity theory and error correction, to questions in many-body physics.


Click on the due date to upload your completed homework.

Assignment Due Date Solutions Topic
Problem set 1 Fri, Sep 16 MIT trace distance and bit commitment
Problem set 2 Fri, Sep 23 MIT channels and types
Problem set 3 Fri, Sep 30 MIT gentle measurement, channel fidelity, entropy inequalities
Problem set 4 Fri, Oct 7 MIT Gibbs distributions and compression with side information
Problem set 5 Fri, Oct 14 MIT data compression converse
Problem set 6 Fri, Oct 21 MIT classical and entanglement-assisted channel capacities
Problem set 7 Fri, Oct 28 unital channels and additivity
Problem set 8 Fri, Nov 4 data hiding with Werner states
Project proposal Fri, Nov 18
Problem set 9 Fri, Dec 2distillation, monogamy and symmetry
Project Presentation Tues, Dec 13, 9am 3 min for individuals, 5 min for groups
Project paper Wed, Dec 14


  1. Sep 8, 2022: Introduction, bit commitment
    Review material: 8.371 lectures on density matrices, quantum operations
    Related reading: Chap 5 of [Wilde].

  2. Sep 13, 2022: purifications and the "no bit commitment" theorem
    Related reading: early review of the bit-commitment no-go paper.

  3. Sep 15, 2022: trace distance and fidelity
    Review materials: Chap 9 of [Wilde], Section 3.2 of [Wat].

  4. Sep 20, 2022: classical information theory: entropy and compression
    Review material: Chap 10 of [Wilde]
    Related reading: C. Shannon, A Mathematical Theory of Communication, Bell System Technical Journal, 1948. Shannon, The Bandwagon, IRE Transactions on Information Theory, 1956.

  5. Sep 22, 2022: quantum entropy and compression.
    Review material: Chap 11 of [Wilde]
    Related reading: algorithmic cooling on wikipedia.

  6. Sep 27, 2022: relative entropy and entropy inequalities
    Review material: Chaps 10 and 11 of [Wilde]

  7. Sep 29, 2022: quantum relative entropy
    Related reading: Igor Bjelakovic, Rainer Siegmund-Schultze, Quantum Stein's lemma revisited, inequalities for quantum entropies, and a concavity theorem of Lieb, quant-ph/0307170, 2012.

  8. Oct 4, 2022: hypothesis testing

  9. Oct 6, 2022: noisy channel coding
    Review material: Chap 2 of [Wilde]

  10. Oct 13, 2022: examples and achievability proof of the channel capacity
    Related reading: Tomohiro Ogawa, Hiroshi Nagaoka; A New Proof of the Channel Coding Theorem via Hypothesis Testing in Quantum Information Theory. arXiv:quant-ph/0208139, IEEE Trans. Inf. Th. 2002. Pranab Sen, Achieving the Han-Kobayashi inner bound for the quantum interference channel by sequential decoding. arXiv:1109.0802, IEEE Symp. on Inf. Th. 2012.

  11. Oct 18, 2022: Fannes's inequality. Converse to capacity theorem for classical and quantum channels.
    Review material: Chapters 16 and 20 of [Wilde]
    Related reading: Andreas Winter. Coding Theorem and Strong Converse for Quantum Channels. 1409.2536. IEEE Trans. Inf. Th. 1999. Andeas Winter. Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints. 1507.07775. Commun. Math. Phys 2016.

  12. Oct 20, 2022: Applications of HSW to tomography and random access codes.
    Review material: Jeongwan Haah, Aram W. Harrow, Zhengfeng Ji, Xiaodi Wu, Nengkun Yu. Sample-optimal tomography of quantum states. 1508.01797. STOC 2016 and IEEE Trans. Inf. Th. 2017. Scott Aaronson. The Learnability of Quantum States. quant-ph/0608142. Proc. Roc. Soc. A. 2007. Ashwin Nayak. Optimal lower bounds for quantum automata and random access codes. quant-ph/9904093. FOCS 1999.

  13. Oct 25, 2022

  14. Oct 27, 2022

  15. Nov 1, 2022

  16. Nov 3, 2022

  17. Nov 8, 2022

  18. Nov 10, 2022

  19. Nov 15, 2022

  20. Nov 17, 2022

  21. Nov 22, 2022

  22. Nov 29, 2022

  23. Dec 1, 2022

  24. Dec 6, 2022

  25. Dec 8, 2022

  26. Dec 13, 2022