Selected references to topics in mathematics and quantum information theory

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

How to navigate these lists

I've tried to indicate the difficulty of the material using three levels: B Beginner M Medium A Advanced
Note that this is a rough and very subjective classification, and only meant to provide some rudimentary guidelines for how to navigate the material.

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!

Entanglement theory

Introductory articles and reviews

Bengtsson, Zyczkowski: Quantum entanglement (book chapter) B M
Brandao et al.: The Mathematics of Entanglement (lecture notes) B M
Friis et al.: Entanglement Certification − From Theory to Experiment (review article) B M
Gühne, Toth: Entanglement detection (review article) B M
Hein et al.: Entanglement in Graph States and its Applications (review article) B
Horodecki et al.: Quantum entanglement (review article) B M
Plenio, Virmani: An introduction to entanglement measures (review article) B

Entanglement measures

Christandl, Winter: "Squashed Entanglement" - An Additive Entanglement Measure (research article) M
Lancien et al.: Should Entanglement Measures be Monogamous or Faithful? (research article) M
Plenio: The logarithmic negativity: A full entanglement monotone that is not convex (research article) B
Vidal: Entanglement monotones (research article) M
Vidal, Werner: A computable measure of entanglement (research article) B M
Vollbrecht, Werner: Entanglement Measures under Symmetry (research article) B M

Bipartite entanglement

Doherty, Parrilo, Spedalieri: Distinguishing separable and entangled states (research article) M A
Doherty, Parrilo, Spedalieri: A complete family of separability criteria (research article) M A
Hayden, Leung, Winter: Aspects of generic entanglement (research article) M A
Horodecki, Horodecki: Reduction criterion of separability and limits for a class of protocols of entanglement distillation (research article) M
Horodecki, Horodecki, Horodecki: Separability of Mixed States: Necessary and Sufficient Conditions (research article) M
Horodecki, Horodecki, Horodecki: Mixed-state entanglement and distillation: is there a "bound" entanglement in nature? (research article) M
Rains: A semidefinite program for distillable entanglement (research article) A
Wolf: Partial Transposition in Quantum Information Theory (PhD thesis) B M A

Multipartite entanglement

Dür, Vidal, Cirac: Three qubits can be entangled in two inequivalent ways (research article) M
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Verstraete, Dehaene, De Moor: Normal forms and entanglement measures for multipartite quantum states (research article) A
Walter, Gross, Eisert: Multi-partite entanglement (book chapter) B
Walter et al.: Entanglement Polytopes: Multiparticle Entanglement from Single-Particle Information (research article) M A
Yang: A simple proof of monogamy of entanglement (research article) B

Haar measure and k-designs

Haar measure essentials

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

Exact k-designs

Dankert et al.: Exact and Approximate Unitary 2-Designs: Constructions and Applications M
Di Matteo: A short introduction to unitary 2-designs B
Webb: The Clifford group forms a unitary 3-design A
Zhu: Multiqubit Clifford groups are unitary 3-designs A

Decoupling in quantum information theory

Hayden: Decoupling: a building block for quantum information theory (tutorial) B
Dupuis et al: One-shot decoupling M

Random quantum channels and minimum output entropy

Brandao, Horodecki: On Hastings' counterexamples to the minimum output entropy additivity conjecture A
Fukuda, King, Moser: Comments on Hastings' additivity counterexamples A

Approximate designs

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Brandao, Harrow, Horodecki: Local random quantum circuits are approximate polynomial-designs M A
Brandao, Harrow, Horodecki: Efficient quantum pseudorandomness A
Harrow, Mehraban: Approximate unitary t-designs by short random quantum circuits using nearest-neighbor and long-range gates A
Hunter-Jones: Unitary designs from statistical mechanics in random quantum circuits M

High-energy physics and scrambling

Hayden, Preskill: Black holes as mirrors: quantum information in random subsystems (research article) B
Yoshida, Kitaev: Efficient decoding for the Hayden-Preskill protocol M
Cotler et al.: Chaos, complexity and random matrices M A
Liu et al.: Entanglement, quantum randomness, and complexity beyond scrambling M


Hayden, Leung, Winter: Aspects of generic entanglement M A
Aaronson et al.: Quantum pseudoentanglement M A
Collins et al: The Weingarten calculus A

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 articles, reviews, textbooks

Holevo: Quantum channel capacities (review article) B
Khatri, Wilde: Principles of Quantum Communication Theory: A Modern Approach (textbook) B M A
Smith: Quantum Channel Capacities (review article) B
Wilde: Quantum Information Theory (textbook) B M A

Coding theorems

Bennett et al: Entanglement-Assisted Classical Capacity of Noisy Quantum Channels (research article) M
Hayden et al.: A decoupling approach to the quantum capacity (research article) M
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Schumacher, Westmoreland: Sending classical information via noisy quantum channels (research article) B M
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Bennett, DiVincenzo, Smolin: Capacities of Quantum Erasure Channels (research article) B
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Devetak, Shor: The capacity of a quantum channel for simultaneous transmission of classical and quantum information (research article) B M
King: Additivity for a class of unital qubit channels (research article) B
King: The capacity of the quantum depolarizing channel (research article) B M
Leditzky, Leung, Smith: Quantum and private capacities of low-noise channels (research article) M
Leditzky et al.: The platypus of the quantum channel zoo (research article) M A
Shor: Additivity of the Classical Capacity of Entanglement-Breaking Quantum Channels (research article) M
Shor: Equivalence of Additivity Questions in Quantum Information Theory (research article) M
Smith: The private classical capacity with a symmetric side channel and its application to quantum cryptography (research article) B
Watanabe: Private and Quantum Capacities of More Capable and Less Noisy Quantum Channels (research article) M


Bausch, Leditzky: Error Thresholds for Arbitrary Pauli Noise (research article) A
DiVincenzo, Shor, Smolin: Quantum Channel Capacity of Very Noisy Channels (research article) M A
Fukuda, King, Moser: Comments on Hastings' Additivity Counterexamples (research article) M A
Leditzky, Leung, Smith: Dephrasure channel and superadditivity of coherent information (research article) M
Leditzky et al.: Generic nonadditivity of quantum capacity in simple channels (research article) M A
Leung et al.: Maximal Privacy Without Coherence (research article) M A
Siddhu: Entropic singularities give rise to quantum transmission (research article) M
Smith, Smolin: Degenerate Quantum Codes for Pauli Channels (research article) A
Smith, Yard: Quantum Communication With Zero-Capacity Channels (research article) M

Representation theory and its applications in quantum information theory

General representation theory

Fulton, Harris: Representation theory: A first course (textbook) B M
Serre: Linear representations of finite groups (textbook) M
Teleman: Representation theory (lecture notes) M

Representation theory of the symmetric group

Audenaert: A digest on representation theory of the symmetric group (survey, copy available on request) M
Christandl: The structure of bipartite quantum states - Insights from group theory and cryptography (PhD thesis) B M
James: The representation theory of the symmetric groups (textbook) M
Zhao: Young Tableaux and the representations of the symmetric group (student article) B
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Representation theory of Lie groups and Lie algebras

Čap: Lie algebras and representation theory (lecture notes) B M
Čap: Lie groups (lecture notes) B M
Carter, Segal, MacDonald: Lectures on Lie Groups and Lie Algebras (textbook) B M
Itzykson, Nauenberg: Unitary Groups: Representations and Decompositions (review article) B M
Procesi: Lie Groups - An approach through Invariants and Representations (textbook) M
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Schur-Weyl duality

Harrow: Applications of coherent classical communication and the Schur transform to quantum information theory (PhD thesis) B M
Walter: Symmetry and Quantum Information (lecture notes) B
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Selected applications of representation theory in quantum information theory

Christandl et al.: One-and-a-half quantum de Finetti theorems (research article) A
Christandl, Koenig, Renner: Post-selection technique for quantum channels with applications to quantum cryptography (research article) M
Christandl, Schuch, Winter: Entanglement of the antisymmetric state (research article) A
Gross, Nezami, Walter: Schur-Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations (research article) A
Haah et al.: Sample-optimal tomography of quantum states (research article) A
Harrow: The Church of the Symmetric Subspace (review article) B
Keyl, Werner: Optimal Cloning of Pure States, Judging Single Clones (research article) M
Keyl, Werner: Estimating the spectrum of a density operator (research article) M
Leditzky: Optimality of the pretty good measurement for port-based teleportation (research article) M
Mozrzymas et al.: Optimal Port-based Teleportation (research article) A
O'Donnell: Learning and Testing Quantum States via Probabilistic Combinatorics and Representation Theory (survey) B
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