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Cited by 1,357Quantum Information and CryptographyQuantum Mechanics and ApplicationsQuantum Computing Algorithms and Architecture

Geometry of Quantum States: An Introduction to Quantum Entanglement

Bengtsson, Ingemar, Zyczkowski, Karol·2006·Cambridge University Press eBooks
DOI 10.1017/cbo9780511535048
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Abstract

Quantum information theory is at the frontiers of physics, mathematics and information science, offering a variety of solutions that are impossible using classical theory. This book provides an introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. After a gentle introduction to the necessary mathematics the authors describe the geometry of quantum state spaces. Focusing on finite dimensional Hilbert spaces, they discuss the statistical distance measures and entropies used in quantum theory. The final part of the book is devoted to quantum entanglement - a non-intuitive phenomenon discovered by Schrödinger, which has become a key resource for quantum computation. This richly-illustrated book is useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied.

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  • Any linear optical element, either deterministic or stochastic, is equivalent to a composite device made of at most four nondepolarizing elements in parallel.

    “An important theorem in classical polarization optics states that any linear optical element ͑either deterministic or stochastic͒ is equivalent to a composite device made of at most four nondepolarizing elements in parallel ͓32͔.”

    From Linear optics and quantum maps (2007) · cites this paper as [32]

  • The concentration of measure effect of the gate fidelity implies that as n increases, the measure of the set of states which produce a fidelity close to the minimum yet far from the average is exponentially small in n.

    “A result that may be useful in this direction of research is the 'concentration of measure effect' of the gate fidelity which implies that as n increases, the measure of the set of states which produce a fidelity close to the minimum yet far from the average is exponentially small in n [41, 42].”

    From Characterizing quantum gates via randomized benchmarking (2012) · cites this paper as [42]

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Authors
Bengtsson, I., & Zyczkowski, K.
Affiliations
Stockholm University (SE)
ORCID
Bengtsson, Ingemar
Venue
Cambridge University Press eBooks
Publisher
Cambridge University Press
Published
2006-05-11
DOI
10.1017/cbo9780511535048
OpenAlex
W360650912
Cited by
1,357
References
0
Citations by year
2026: 50 · 2025: 82 · 2024: 65 · 2023: 53 · 2022: 64 · 2021: 75 · 2020: 81 · 2019: 74 · 2018: 66 · 2017: 62 · 2016: 96 · 2015: 83 · 2014: 81 · 2013: 90 · 2012: 60
FWCI
20.23
Type
book
Language
en
Open access
closed
Licence
www.cambridge.org/core/terms (unspecified, 1204 d)
Locations
Cambridge University Press eBooks (publishedVersion)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
Keywords
quantum information, quantum state space, finite-dimensional Hilbert space, statistical distance measures, quantum entropy, quantum entanglement, quantum computing

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