Theory of Quantum Information, Fall 2025
Debbie Leung
Email: wcleung(at)uwaterloo(dot)ca
Avantika Agarwal
Email: avantika.agarwal(at)uwaterloo(dot)ca
Amolak Ratan Kalra
Email: arkalra(at)uwaterloo(dot)ca
Tue/Thur 11:30-12:50, QNC 1201.
21 lectures Sept 04 - Nov 27, presentations Dec 2, 4, 9 (tentative)
no lectures Sept 30, Oct 2 (reading assignments), 14, 16 (reading week)
Instructor office hour: after class
TA office hour: dynamically decided
5 assignments (total 75%)
Term project (25%)
This page (main)
Crowdmark (assignement submission)
Piazza (2-way communication)
LEARN (mark registration)
Posted Nov 09, 2025, 18:21
Final file for A4 posted. Major changes from f23: Q1 changed, and parts of Q2, Q3 removed. Minor rewording and corrections of typos from the draft posted on piazza on Nov 07.
Posted Oct 14, 2025, 11:38
Final file for A2 posted. Changes: correcting two typos spotted and shared on piazza, putting 4 marks to Q1a instead of 3 marks, summarizing from the 2023 version how to relate parts (a)-(c) to the final structural theorem, and postponing the deadline to end of Tuesday Oct 21.
Posted Sept 21, 2025, 21:16
Part 1 Notes 3 p16 updated with normalization corrections. A1 updated with 4 normalization corrections (corresponding to piazza @17-19).
Posted Sept 16, 2025, 10:46
Part 1 Notes 3 updated with various changes.
Posted Sept 11, 2025, 00:06
Part 1 Notes 1 and 2 updated with minor corrections.
Posted Sept 03, 2025, 00:11
Website was set up with syllabus drafted.
Course description:
Students will learn(1) mathematical background for understanding quantum information,
(2) important aspects of quantum information including
(a) states, operations, their matrix representations,
(b) measures of distance for quantum states and operations;
(c) quantum Shannon theory on how how data can be encoded, transmitted, and decoded via noiseless and noisy quantum channels;
(d) theory of entanglement including measures of entanglement and transformation rules,
(3) the mathematical language and tools for proving results in quantum information, and
(4) how physics can be translated into mathematics, and vice versa.
Course materials
Prerequisite: QIC 710 Chapters 1-12, 28-34, 38-39 Prerequisite: CO481/CS467/PHYS467 Topics 1-5, 8 Textbook by Nielsen and Chuang Textbook by John Watrous Units 1,3
Lecture notes for F2011 offering Textbook
This will follow the fall 2023 offering very closely with files updated closer to scheduled coverage. Schedule and coverage are subject to minor tweaks.
Part 1 -- Mathematical preliminaries and representation of states and operations
Sept 4-23 (6 lectures) (corresponding to lectures 1-3,5-6 of F2011 offering)
Registers and states (Sec 3.1.1) Complex Euclidean space, direct sum and tensor product (Sec 1.1, 1.2.3-1.2.4) Linear Operators, tensor product (Sec 1.2, 2.2.1) The vec function Notes 1 (updated Sept 11)
Eigenvectors and eigenvalues (Sec 1.3.2) Important classes of operators (Sec 1.4) The spectral theorem (Sec 1.5.1) Functions of normal operators (Sec 1.5.2) The singular value theorem (Sec 2.1) Schatten norms of operators (Sec 2.3) Compact sets, convexity (Sec 2.5) Notes 2 (updated Sept 11)
Quantum states (Sec 3.1.2) Measurements (Sec 3.1.3) Information complete measurements (Sec 3.2) (reading exercise) Helstrom-Holevo theorem (Sec 3.4) Product measurements (Sec 3.1.3) Channels (Sec 3.1.4) Instruments, partial measurements (Sec 6.1, Sec 3.4) Mixed unitary channels, depolarizing channel, Weyl operators, teleportation (Sec 6.2.3, Sec 6.3) Notes 3 (updated Sept 21)
A1 (due Sept 26) (updated Sept 21) Representations and characterizations of Quantum channelsLinear representation Natural representation Choi representation Equivalence of natural and Choi representations with Kraus and Stinespring representations Characterizations of complete positivity Characterizations of trace-preservation Characterizations of Quantum channels
Part 2 -- Distance between states and operations, and semidefinite programming
Sept 25 - Oct 28 (6 lectures + 1 reading)Purifications and fidelity (lecture 4 of F2011 offering)Reductions, extensions, and purifications Equivalence of purifications Fidelity and Uhlmann's Theorem Alberti’s theorem The Fuchs–van de Graaf inequalities Notes 1 (by Kohdai Kuroiwa)
A2 (old, from 2023, for reference) A2 (updated Oct 14, 2025, due Oct 21) Semidefinite programming (lectures 7-8 in F2011 offering)We will follow lectures 1-2 from Jamie Sikora's S2019 course at PI. All videos and lecture notes can be found here.
Basic definitions and examples Duality theory (weak and strong duality, complementary slackness) Quantum state discrimination / exclusion Trace distance Fidelity (Uhlmann's and Alberti's theorems) Notes 4 (p11 revised Oct 22, 2025)
Notes 6 (p8 revised Oct 22, 2025)
A3 (revised Oct 23, 2025, due Nov 03 9am) Channel distinguishability and the completely bounded norm (lecture 20 in F2011 offering)SDP for the completely bounded norm (lecture 21 in F2011 offering)
Part 3 -- Encoding and retrieving information from quantum systems
Oct 30 - Nov 13 (5 lectures) (lectures 9-11 of F2011 offering)
Shannon entropy and Shannon's noiseless coding theoremIID source Asymptotic Equipartition Theorem (AEP) Classical data compression von Neumann entropy and quantum data compression von Neumann entropy Typical space of a tensor power state The "Transmit the typical space" protocol Quantum iid source Quantum data compression Direct coding theorem: Schumacher compression Weak Converse Note that Part 3 lectures 1-2 correspond to the materials in Lecture 9 in LN2011, but various models and proofs differ substantially. Assessments should follow Part 3 lectures 1-2 wherever appropriate.
Entanglement dilution and concentration Entropy of entanglement Asymptotic pure bipartite state transformation under LOCC Highlights from Lectures 10-12 in LN 2011 (details: reading assignment) A4 (out Nov 09, due Nov 24) Shannon's noisy coding theorem and mutual information Accessible information Holevo information Holevo's theorem HSW theorem (informal statement) Nayak's bound (reading assignment)
Part 4 -- LOCC and entanglement theory
Nov 18-27 (4 lectures) (highlights of lectures 13-19 of F2011 offering, plus more examples and discussions)Separable operatorsSeparable vs entangled Horodecki criterion for separability Entanglement witness Separable ball around the maximally mixed state Separable class of operations and LOCC operations Entanglement rank of bipartite mixed states Definition of the separable operations Separable operations cannot increase entanglement rank Equivalent conditions for separable operations LOCC informal definition LOCC as a subset of SEP Examples for LOCC transformations of bipartite pure states teleportation entanglement dilution and concentration Lo-Popescu reduction Nielsen's majorization characterization The complexity of LOCC LOCC measurements impossibility to discriminate too many maximally entangled states any two orthogonal states can be perfectly discriminated nonlocality without entanglement (product states cannot be discriminated by LOCC) Irreversibility of LOCC LOCC, even with closure, is a proper subset of SEP Example of entanglement of assistance Random distillation and non-closure of LOCC LOCC formal definition conditions for finite intermediate measurements and classical communication Entanglement measures, PPT states Entanglement of formation, entanglement cost, distillable entanglement Partial transpose States with positive partial transpose (PPT states) Unextendible product bases and PPT bound entangled states Non-distillability of PPT states PPT channels and apps in entanglement theory and quantum channel capacities A5 (out Nov 25, due Dec 15)
Assignments will be posted with the syllabus. Please submit solutions to Crowdmark.
Each student chooses a body of research, submits a 3-page asbtract (as in QIP) and presents a 25-min talk with 5-min Q-n-A period. Please arrange topic with instructor before Nov 10. Presentations Dec 2-9, abstract due 2 days before presentation, and slides the night before.
Last updated Sept 3, 2025.