Lists of references to topics in mathematics and quantum information science

On this page I am collecting selected references to topics in mathematics and quantum information theory that intersect with my own research interests. I compiled these lists having graduate students in mind who want to start working on questions related to these topics. Hopefully, they are also useful for more experienced researchers.

List of lists

• Introductions to quantum information
• Entanglement theory
• Haar measure and k-designs
• Quantum channel capacities
• Representation theory and its applications in quantum information theory

How to navigate these lists

I’ve tried to include both introductory reading material (textbooks, lecture notes, review articles, etc.) and more advanced material (papers). Every topic below starts with a subsection “Introductory reading” that is a good place to start. Except for the first section, which covers introductions to quantum information as a whole.

Please keep in mind that, while aimed at a general audience, these lists still mostly reflect my personal view of these topics and are heavily biased towards my own understanding of the material and personal preferences!

Please get in touch with me if you have any questions on how to start with these topics, or if I have forgotten any resources you think should be included!

Introductions to quantum information

This section lists some introductory texts and reading material to get started in the field of quantum information. Since this field is multidisciplinary by its very nature, spanning Mathematics, Physics, Computer Science, and Information Theory, one can approach it from different angles this list is trying to cover. The following legend is used: Math Mathematics; Phys Physics; CS Computer Science; IT Information Theory.

• [AS17] Aubrun, Szarek: Alice and Bob Meet Banach (textbook) Math
• [BF23] Bertlmann, Friis: Modern Quantum Theory (textbook) Phys
• [HVM23] Hajdušek, Van Meter: Quantum Communications (textbook) Phys IT
• [KW20] Khatri, Wilde: Principles of Quantum Communication Theory: A Modern Approach (textbook) IT
• [NC10] Nielsen, Chuang: Quantum Computation and Quantum Information (textbook) CS
• [OW15] O’Donnell, Wright: Quantum Computation and Information (lecture notes) CS
• [Pr22] Preskill: Quantum computation (lecture notes) Phys CS
• [Re22] Renes: Quantum Information Theory - Concepts and Methods (textbook) IT
• [SW10] Schumacher, Westmoreland: Quantum Processes, Systems, and Information (textbook) Phys IT
• [Wal18] Walter: Symmetry and Quantum Information (lecture notes) Math
• [Wat18] Watrous: The Theory of Quantum Information (textbook) Math
• [WP+12] Weedbrook et al.: Gaussian Quantum Information (review article) Phys
• [Wi19] Wilde: Quantum Information Theory (textbook) IT
• [Wo12] Wolf: Quantum Channels & Operations: Guided Tour (lecture notes) Math
• [Wo23] Wolf: Mathematical Introduction to Quantum Information Processing (lecture notes) Math

Entanglement theory

Introductory reading

• [BZ09] Bengtsson, Zyczkowski: Quantum entanglement (book chapter)
• [BCH+16] Brandão et al.: The Mathematics of Entanglement (lecture notes)
• [FVM+19] Friis et al.: Entanglement Certification − From Theory to Experiment (review article)
• [GT09] Gühne, Toth: Entanglement detection (review article)
• [HDE+06] Hein et al.: Entanglement in Graph States and its Applications (review article)
• [HHH+09] Horodecki et al.: Quantum entanglement (review article)
• [PV07] Plenio, Virmani: An introduction to entanglement measures (review article)

Entanglement measures

• [CW04] Christandl, Winter: “Squashed Entanglement” - An Additive Entanglement Measure
• [LMR25] Lami, Anna Mele, Regula: Computable entanglement cost under positive partial transpose operations
• [LDH+16] Lancien et al.: Should Entanglement Measures be Monogamous or Faithful?
• [Pl05] Plenio: The logarithmic negativity: A full entanglement monotone that is not convex
• [Vi00] Vidal: Entanglement monotones
• [VW02] Vidal, Werner: A computable measure of entanglement
• [VW01] Vollbrecht, Werner: Entanglement Measures under Symmetry

Bipartite entanglement

• [DPS02] Doherty, Parrilo, Spedalieri: Distinguishing separable and entangled states
• [DPS04] Doherty, Parrilo, Spedalieri: A complete family of separability criteria
• [HLW06] Hayden, Leung, Winter: Aspects of generic entanglement
• [HH99] Horodecki, Horodecki: Reduction criterion of separability and limits for a class of protocols of entanglement distillation
• [HHH96] Horodecki, Horodecki, Horodecki: Separability of Mixed States: Necessary and Sufficient Conditions
• [HHH98] Horodecki, Horodecki, Horodecki: Mixed-state entanglement and distillation: is there a “bound” entanglement in nature?
• [Ra02] Rains: A semidefinite program for distillable entanglement

Multipartite entanglement

• [DVC00] Dür, Vidal, Cirac: Three qubits can be entangled in two inequivalent ways
• [GT09] see above ↑
• [HDE+06] see above ↑
• [VDM03] Verstraete, Dehaene, De Moor: Normal forms and entanglement measures for multipartite quantum states
• [WGE16] Walter, Gross, Eisert: Multi-partite entanglement (book chapter)
• [WDG+13] Walter et al.: Entanglement Polytopes: Multiparticle Entanglement from Single-Particle Information
• [Ya06] Yang: A simple proof of monogamy of entanglement

Haar measure and k-designs

Introductory reading

• [Gl10] Gleason: Existence and uniqueness of Haar measure (lecture notes)
• [Wat18] Watrous: Permutation invariance and unitarily invariant measures (lecture notes)
• [Me24] Anna Mele: Introduction to Haar measure tools in quantum information (review article)

Exact k-designs

• [DCE+09] Dankert et al.: Exact and Approximate Unitary 2-Designs: Constructions and Applications
• [DiM14] Di Matteo: A short introduction to unitary 2-designs
• [We16] Webb: The Clifford group forms a unitary 3-design
• [Zh17] Zhu: Multiqubit Clifford groups are unitary 3-designs

Decoupling in quantum information theory

• [Hay11] Hayden: Decoupling: a building block for quantum information theory (tutorial)
• [DBW+14] Dupuis et al: One-shot decoupling

Random quantum channels and minimum output entropy

• [BH10] Brandao, Horodecki: On Hastings’ counterexamples to the minimum output entropy additivity conjecture
• [FKM10] Fukuda, King, Moser: Comments on Hastings’ additivity counterexamples

Approximate designs

• [BHH16a] Brandao, Harrow, Horodecki: Local random quantum circuits are approximate polynomial-designs
• [BHH16b] Brandao, Harrow, Horodecki: Efficient quantum pseudorandomness
• [DCE+09] see above ↑
• [HM23] Harrow, Mehraban: Approximate unitary t-designs by short random quantum circuits using nearest-neighbor and long-range gates
• [HJ19] Hunter-Jones: Unitary designs from statistical mechanics in random quantum circuits
• [LL24] LaRacuente, Leditzky: Approximate Unitary k-Designs from Shallow, Low-Communication Circuits
• [SHH24] Schuster, Haferkamp, Huang: Random unitaries in extremely low depth

Approximate design constructions have been at the center of attention in the past years, which resulted in a flurry of new results. A somewhat comprehensive overview can be found in my own paper with Nick on this topic, and the concurrent work by Schuster et al.

High-energy physics and scrambling

• [HP07] Hayden, Preskill: Black holes as mirrors: quantum information in random subsystems
• [YK17] Yoshida, Kitaev: Efficient decoding for the Hayden-Preskill protocol
• [CHJ+17] Cotler et al.: Chaos, complexity and random matrices
• [LLZ+18] Liu et al.: Entanglement, quantum randomness, and complexity beyond scrambling

Miscellaneous

• [HLW06] see above ↑
• [ABF+24] Aaronson et al.: Quantum pseudoentanglement
• [CMN22] Collins et al: The Weingarten calculus

Quantum channel capacities

Note: This material is mainly concerned with quantum channel capacities in the finite-dimensional setting, which is the setting I focus on in my research. There is a wealth of literature for quantum channel capacities in the continuous variable setting that I may add in the future. In the meantime, Section V in this review article on Gaussian quantum information and a slightly older review article on Gaussian quantum channels both give an excellent overview.

Introductory reading

• [Hol20] Holevo: Quantum channel capacities (review article)
• [KW20] see above ↑
• [Smi10] Smith: Quantum Channel Capacities (review article)
• [Wi19] see above ↑

Coding theorems

• [BSS+99] Bennett et al: Entanglement-Assisted Classical Capacity of Noisy Quantum Channels
• [HMW+08] Hayden et al.: A decoupling approach to the quantum capacity
• [KW20] see above ↑
• [SW97] Schumacher, Westmoreland: Sending classical information via noisy quantum channels
• [Wi19] see above ↑

Additivity

• [BDS97] Bennett, DiVincenzo, Smolin: Capacities of Quantum Erasure Channels
• [BSS+99] see above ↑
• [DS05] Devetak, Shor: The capacity of a quantum channel for simultaneous transmission of classical and quantum information
• [Ki01] King: Additivity for a class of unital qubit channels
• [Ki03] King: The capacity of the quantum depolarizing channel
• [LLS18a] Leditzky, Leung, Smith: Quantum and private capacities of low-noise channels
• [LLS+23a] Leditzky et al.: The platypus of the quantum channel zoo
• [Sh02] Shor: Additivity of the Classical Capacity of Entanglement-Breaking Quantum Channels
• [Sh04] Shor: Equivalence of Additivity Questions in Quantum Information Theory
• [Sm08] Smith: The private classical capacity with a symmetric side channel and its application to quantum cryptography
• [Wa12] Watanabe: Private and Quantum Capacities of More Capable and Less Noisy Quantum Channels

Non-additivity

• [BL21] Bausch, Leditzky: Error Thresholds for Arbitrary Pauli Noise
• [DWS98] DiVincenzo, Shor, Smolin: Quantum Channel Capacity of Very Noisy Channels
• [FKM10] see above ↑
• [LLS18b] Leditzky, Leung, Smith: Dephrasure channel and superadditivity of coherent information
• [LLS23b] Leditzky et al.: Generic nonadditivity of quantum capacity in simple channels
• [LLS+14] Leung et al.: Maximal Privacy Without Coherence
• [Si21] Siddhu: Entropic singularities give rise to quantum transmission
• [SS07] Smith, Smolin: Degenerate Quantum Codes for Pauli Channels
• [SY08] Smith, Yard: Quantum Communication With Zero-Capacity Channels

Representation theory and its applications in quantum information theory

Introductory reading

• [EGH+08] Etingof et al.: Introduction to representation theory (textbook/lecture notes)
• [FH04] Fulton, Harris: Representation theory: A first course (textbook)
• [Se77] Serre: Linear representations of finite groups (textbook)
• [Te05] Teleman: Representation theory (lecture notes)

Representation theory of the symmetric group

• [Au06] Audenaert: A digest on representation theory of the symmetric group (survey, copy available on request)
• [Ch06] Christandl: The structure of bipartite quantum states - Insights from group theory and cryptography (PhD thesis)
• [JK84] James, Kerber: The representation theory of the symmetric group (textbook)
• [Zh10] Zhao: Young Tableaux and the representations of the symmetric group (student article)
• [FH04] see above ↑

Lie groups and Lie algebras

• [Ca16] Čap: Lie algebras and representation theory (lecture notes)
• [Ca24] Čap: Lie groups (lecture notes)
• [CSM95] Carter, Segal, MacDonald: Lectures on Lie Groups and Lie Algebras (textbook)
• [Hu72] Humphreys: Introduction to Lie Algebras and Representation Theory (textbook)
• [IN66] Itzykson, Nauenberg: Unitary Groups: Representations and Decompositions (review article)
• [Pr07] Procesi: Lie Groups - An approach through Invariants and Representations (textbook)
• [Se01] Serre: Complex Semisimple Lie Algebras (textbook)
• [Se64] Serre: Lie Algebras and Lie Groups (textbook)
• [Ch06] see above ↑
• [FH04] see above ↑

Schur-Weyl duality

• [Ha05] Harrow: Applications of coherent classical communication and the Schur transform to quantum information theory (PhD thesis)
• [Wal18] see above ↑
• [Ch06] see above ↑

Selected applications of representation theory in quantum information theory

• [CKM+07] Christandl et al.: One-and-a-half quantum de Finetti theorems
• [CKR09] Christandl, Koenig, Renner: Post-selection technique for quantum channels with applications to quantum cryptography
• [CSW12] Christandl, Schuch, Winter: Entanglement of the antisymmetric state
• [GNW1] Gross, Nezami, Walter: Schur-Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations
• [HHJ+17] Haah et al.: Sample-optimal tomography of quantum states
• [Ha13] Harrow: The Church of the Symmetric Subspace (review article)
• [KW99] Keyl, Werner: Optimal Cloning of Pure States, Judging Single Clones
• [KW01] Keyl, Werner: Estimating the spectrum of a density operator
• [Le22] Leditzky: Optimality of the pretty good measurement for port-based teleportation
• [MSS+18] Mozrzymas et al.: Optimal Port-based Teleportation
• [OD17] O’Donnell: Learning and Testing Quantum States via Probabilistic Combinatorics and Representation Theory (survey)
• [Ch06] see above ↑
• [Ha05] see above ↑
• [Wal18] see above ↑

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