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Quantum Information
Entangled Hbars

Tutorials on Quantum Information

Here we have collected links to materials which may help you either to find out what Quantum Information and Computing are all about, or else to learn in more depth about certain aspects of the field. As Mathematical Physicists we have focused on mathematical and physical aspects. So for the roots of the field in Computer Science, Cryptoanalysis, or experimental physics it is probably better to go to other sources. Tell us if you know of further texts, especially web accessible ones, which you feel might be helpful. Other comments (e.g. concerning errors or other reasons for not recommending any of the cited items) are also appreciated.

We roughly categorized our recommendations as follows:

Popular Introductions   Graduate Level   Courses   Fundamentals  

Popular Introductions

Various sites offer short non-technical explanations as to what this is all about. Try the collections at our own site, at Oxford, or at Los Alamos. A Web presentation of the whole topic was given in 1997 by Clare Warwick as a final year Computer Science project.

Many science magazines for the general public have carried introductory articles on quantum information. Don't expect too much from the following Web Sites, though: commercial organizations don't give away anything for free, so usually you only get appetizers, or tables of contents. Physics World (e.g.special issue (3/98)), Scientific American(Articles on teleportation@Innsbruck and NMR computing), The Economist, Physics Today, Nature, New Scientist, Science, American Scientist. You may also wish to browse the Physics News Updates supplied by the AIP (e.g. 98/388, 98/367, 97/310),

Quantum Information in General
A press release from the American Institute of Physics tries to say it in a few words.
Quantum Teleportation
The pages at IBM give you a flavour of how it all started. Slightly more extensive explanations at Oxford. Also read about the experiments in Innsbruck and at Caltech. At these sites you will also find further introductory texts, press releases, etc.
Quantum Cryptography
Short intros at Oxford or Los Alamos.
Quantum Computing
Non-technical expositions by Rieffel et al, Hayes, and Barenco et al. Here is an introduction focusing on the history of the subject, with a good collection of references.

Introductions at Graduate Level

The following sources at least require basic knowledge of quantum mechanics. There is much more on the net than we can list here. Good places to look for further sources are the "find" function of the LANL preprint archive, or local collections of papers.

Quantum Information in General
Perhaps look at the introductions to the courses below.
Focussing on general aspects and entanglement measures: Plenio and Vedral. Focussing on the uses of entanglement in computation: Ekert and Josza.
Quantum Cryptography
A quick route to the research literature (up to 1995) is this bibliography.
Quantum Computing
Basic tutorials by Sam Braunstein, Scarani, and Zalka. More extensive reviews by Andrew Steane, Vedral and Plenio.


Naturally, we recommend the local course (from the Winter term 1996/97, at Braunschweig and Osnabrück), of which there are some Lecture Notes on this server. Unfortunately, they are not very complete. In particular, they do not contain the definitions of entanglement measures and channel capacity. These will be treated in a forthcoming review paper.

Much more extensive material exists for the course Physics 229 at Caltech, which was given first by John Preskill in 97/98, and will be given this year in cooperation with Alexei Kitaev. The available course material also contains exercises. In the versions available so far, the formal definitions of channel capacities etc. are also still missing, but seem to be planned for this year. You may find Preskill's list of essential references useful.

Another course at this level was given by Umesh Vazirani in the Fall 1997 at Berkeley (Computer Science Dept). Notes and problem assignments are available on the net.

The transparencies of a short course given in May/June 1998 in Innsbruck by B. Schumacher are also quite interesting, but perhaps more suitable for experts who can supply the explanations missing on the transparencies for themselves.

Various further courses must exist in the meantime. Not too many appear to have left a trace in the net. Example: Special Topics in Quantum Computation (S. Lomonaco, UMBC Baltimore).


Before studying Quantum Information in depth you will have to get a good knowledge of basic quantum mechanics and its mathematical tools. Here are some textbooks, which a library for quantum information theory should have. Too bad several of the following are out of print!

Michael A. Nielsen, Isaac L. Chuang: Quantum Computation and Quantum Information (Cambridge University Press, 2000)
Josef Gruska: Quantum Computing (McGraw Hill, 2000)
Leslie Ballentine: Quantum mechanics (Prentice Hall 1990)
Modern general quantum mechanics text, without specific reference to quantum information. New edition announced at World Scientific.
Asher Peres: Quantum Theory: Concepts and Methods (Kluwer 1993)
By one of the pioneers of quantum information theory, frequently cited by researchers in the field.
E. Brian Davies: Quantum theory of open systems (Academic Press 1976)
Contains the basic mathematical apparatus for observables (positive operator valued measures), channels (completely positive superoperators) and continual measurement ("quantum stochastic processes" in his terminology). Strangely enough it was believed at the time that plain positivity (as opposed to complete positivity) would be enough to demand of a quantum operation. Consequently, the book is rather weak on complete positivity. So you should get:
Vern I. Paulsen: Completely bounded maps and dilations (Longman Scientific and Technical 1986)
Looks at complete positivity and all that, strictly from a mathematical perspective.
Vern I. Paulsen: Completely Bounded Maps and Operator Algebras (Cambridge University Press 2003)
Alexander S. Holevo: Probabilistic and statistical aspects of quantum theory (Academic Press 1976)
Another classic from the first period of quantum information theory, relating to estimation theory and the use of quantum channels for transmitting classical information.
Alexander S. Holevo: Statistical Structure of Quantum Theory (LNP M 67, Springer 2001)
Masanori Ohya and Denes Petz: Quantum entropy and its use (Springer 1993)
Key reference for entropic matters. Works with a general notion of channels between operator algebras. The operator algebraic language may be a bit tough when you only have a naive theoretical physics background, but it is useful, and indeed essential when you want to treat systems of infinitely many degrees of freedom. The following textbook would help you to upgrade:
Ola Bratteli and Derek W. Robinson: Operator algebras and quantum statistical mechanics, vol. I (Springer 1979)
Contains the essentials of C*-and W*-algebras in a concise form as needed in mathematical quantum theory.
Stephen Boyd and Lieven Vandenberghe: Convex Optimization (Cambridge University Press, 2004)
Also available online.
There is also a couple of volumes containing survey papers on various aspects of the field:
Gernot Alber et al.: Quantum Information (Springer/Telos 2001)
Dirk Bouwmesteester et al.: The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation (Springer 2000)
Gerd Leuchs, Thomas Beth et al.: Quantum Information Processing (Wiley-VCH 2003)
Hoi-Kwong Lo et al.: Introduction to Quantum Computation and Information (World Scientific 2001)

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