Theory of Quantum Information, Fall 2023
Debbie Leung
Email: wcleung(at)uwaterloo(dot)ca
Kohdai Kuroiwa
Email: kkuroiwa(at)uwaterloo(dot)ca
Tue/Thur 11:30-12:50, QNC 1201.
22 lectures in Sept 07 - Nov 30 (excluding Oct 26)
Instructor office hour: after class or by appointment
TA office hour: dynamically decided
Piazza (email Debbie for invitation)
5 assignments (total 75%)
Term project (25%)
Posted Nov 25, 2023, 18:50
Assignment 5 uploaded, due end of Dec 15. Getting the conceptual ideas right will greatly reduce the technical work. You can already start Q1 (depends lightly on Part 4 Lec 3, but you already know state discrimination, and informal definition of LOCC, so, you can start). Q2 will require the definition of PPT in the Part 4 Lec 4, or LN 2011 Chap 18, but otherwise does not rely too much on Part 4 Lec 3-4 in a big way. Similar results on SEP in Part 4 Lec 2 may be more relevant and that's already covered.
Posted Nov 20, 2023, 21:50
Notes for Part 4 lectures 1-4 uploaded.
Posted Nov 15, 2023, 18:50
Notes for Part 3 lecture 4 revised, and that for Part 3 lecture 5 uploaded.
Posted Nov 14, 2023, 01:25
Notes for Part 3 lecture 4, A4, Term project submission link posted.
Posted Nov 08, 2023, 23:55
Notes for Part 3 lecture 2 slightly revised and merged with that for lecture 3.
Posted Nov 07, 2023, 01:30
Note revised schedule for A4, A5, and term project presentations.
Posted Oct 31, 2023, 00:30
Notes for Oct 24 lecture replaced, notes for Oct 31 uploaded.
Posted Oct 20, 2023, 17:10
NO LECTURE ON THUR OCT 26, replaced by reading ex of Chapters 20-21 of LN2011.
Posted Oct 20, 2023, 17:05
Notes for Oct 24 lecture and A3 uploaded here. Crowdmark links will be sent shortly.
Posted Oct 17, 2023, 01:50
Supplementary notes for Oct 05 lecture, notes for Oct 17 and 19 lectures are uploaded.
Posted Oct 01, 2023, 22:30
Links are provided to the notes for lectures 8, 9. A2 due date changed.
Posted Sep 24, 2023, 22:20
Notes for lecture 7 posted. A2 draft posted.
Posted Sep 17, 2023, 14:20
Notes for lecture 6 posted. A1 due date postponed to Sept 29. Minor edits in syllabus and scheduling.
Posted Sep 12, 2023, 00:45
Notes for lecture 1.5 and A1 uploaded. Invitations for piazza and A1 submissions sent.
Posted Sep 06, 2023, 17:25
Notes for lectures 1-3 uploaded.
Posted Aug 10, 2023, 11:11
Website was set up.
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 Prerequisite: CO481/CS467/PHYS467 Textbook by Nielsen and Chuang
Lecture notes for F2011 offering Textbook
Part 1 -- Mathematical preliminaries and representation of states and operations
Lectures 1-3,5-6 of F2011 offering, Sept 7-26 (6 lectures).
Lecture 1:Lecture 1.5: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) Lecture 2: (actual 3)The vec function Lecture 3-4: (actual 4-5)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) A1 (due Sept 29) Lecture 6: Representations and characterizations of Quantum channelsQuantum 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) Linear 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
Lectures 4, 7-8, 20-21 of F2011 offering, Sept 28 - Oct 19 (5 lectures).
Lectures 1-2: Purifications and fidelity (by Kohdai Kuroiwa and DL)A2 (due Oct 16) Lecture 3-5: Semidefinite programming (Lectures 7-8 in F2011 offering)Reductions, extensions, and purifications Equivalence of purifications Fidelity and Uhlmann's Theorem Alberti’s theorem The Fuchs–van de Graaf inequalities 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 for part 2 lecture 3-4 (from Jamie Sikora)
Supplementary Notes for part 2 lecture 3
Notes for part 2 lecture 4 (from DL)
Notes for part 2 lecture 5 (from Jamie Sikora)
Notes for part 2 lecture 5 (from DL)
A3 (due Oct 31) Lecture 6: Channel distinguishability and the completely bounded normMaterial close to lecture 20 in F2011 offering Lecture 7: SDP for the completely bounded norm
Material close to lecture 21 in F2011 offering
Part 3 -- Encoding and retrieving information from quantum systems
Lecture 1: Shannon entropy and Shannon's noiseless coding theoremLecture 2: von Neumann entropy and quantum data compressionIID source Asymptotic Equipartition Theorem (AEP) Classical 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.
Lecture 3:Entanglement dilution and concentration Entropy of entanglement Asymptotic pure bipartite state transformation under LOCC Notes for both lectures 2 and 3
Lecture 4:A4 (out Nov 14, due Nov 24) Lecture 5:Highlights from Lectures 10-12 in LN 2011 (details: reading assignment) 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
Highlights of lectures 13-19 of F2011 offering, plus more examples and discussions from several arXiv papers. Lecture 1: Separable operatorsLecture 2: Separable class of operations and LOCC operationsSeparable vs entangled Horodecki criterion for separability Entanglement witness Separable ball around the maximally mixed state Lecture 3: the complexity of LOCCEntanglement 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 Lecture 4: Entanglement measures, PPT statesLOCC 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 A5 (out Nov 25, due Dec 15)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
Note: lecture plans are subject to minor changes, can take longer to cover materials than planned. The coverage for 3-4 lectures have not yet been allocated for this reason.
Assessment materials
Assignments will be posted with the syllabus. Please submit solutions to Crowdmark. Due dates are Fridays one week after the intended coverage. Tentatively, A1 covers up to Sept 21, due Sept 29, A2 covers up to Sept 28, due Oct 6 (or 13), A3 covers up to Oct 19, due Oct 27, A4 cover up to Nov 9, due Nov 17, A5 cover up to Nov 23, due Nov 31.
Each student chooses a body of research on a subject, 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 4-8, abstract due 2 days before presentation.