Skip to main content
SpringerLink
Log in
Book cover

Quantum Correlations pp 1–59Cite as

Introduction

  • Chapter
  • First Online:

  • 529 Accesses

Part of the book series: Springer Theses ((Springer Theses))

Abstract

In this chapter, we provide a short but sufficient introduction to four topics that this thesis is built upon. It comprises of five sections on essential elements of convex geometry, the quantum resource theory formalism, quantum phase-space representation, and the notion of entropy in classical and quantum information. Importantly, the second section of this chapter deals purely with the mathematical concepts of algebraic and convex geometry that are necessary for a clear understanding of all other ideas provided. Respecting the fact that this is not generally easy for physicists to follow, we have done our best to bring the mathematical notions forward in an intuitive way with an eye on our language to be more sensible for physicists. Moreover, whenever possible, we have provided illustrations of the concepts.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    The situation can be even worse, i.e., that the existence of the convergence point generally depends on the chosen notion of convergence.

  2. 2.

    In fact, both the empty set and the \((-1,1)\) interval on itself are clopen sets.

  3. 3.

    Except the sequence which is constant except at finitely many points.

  4. 4.

    Note that, in a more general topological sense, this statement is not true because in that case interior points are defined using open sets rather than balls. This makes it possible to have interior points which are not limit points as in the example of discrete topological spaces.

  5. 5.

    Note that by no means n is not unique, as it depends on which subcover has been chosen.

  6. 6.

    Note that an affine transformation is not necessarily invertible, in the same way as a linear transformation on a vector space is not necessarily invertible. Here, by inverse image, we mean the domain on which the affine transformation is acting.

  7. 7.

    This is because points inside a convex set usually allow for various decomposition in terms of the boundary points that contradicts the definition of a face.

  8. 8.

    A monoid is simply a group without the necessity for containing inverse elements. In other words, it is the triplet \((\mathcal {S},\cdot ,e)\) such that (i) \(\mathcal {S}\) is closed under \(\cdot \); (ii) \(\cdot \) is associative; (iii) \(\mathcal {S}\) contains a unique identity element e with respect to \(\cdot \).

  9. 9.

    This measure is the (regularized) relative entropy of the resource [16].

  10. 10.

    Note that we have used PVMs only for simplicity reasons, and there is no restriction in extending the definition to the case where POVMs are considered.

References

  1. Ballentine LE (2000) Quantum mechanics: a modern development. World Scientific Publishing Co. Pte. Ltd

    Google Scholar 

  2. Muscat J (2014) Functional analysis: an introduction to metric spaces, hilbert spaces, and banach algebras. Springer International Publishing

    Google Scholar 

  3. Porteous IR (1969) Topological geometry. Van Nostrand Reinhold Company

    Google Scholar 

  4. Grünbaum B (2003) Convex polytopes. Springer International Publishing

    Google Scholar 

  5. Coppel WA (1998) Foundations of convex geometry. Cambridge University Press

    Google Scholar 

  6. Bengtsson I, Życzkowski K (2006) Geometry of quantum states: an introduction to quantum entanglement. Cambridge University Press

    Google Scholar 

  7. Horodecki M, Oppenheim J (2013) (Quantumness in the Context of) Resource theories. Int J Mod Phys B 27:1345019

    Article  ADS  MathSciNet  Google Scholar 

  8. Coecke B, Fritz T, Spekkens RW (2016) A mathematical theory of resources. Inf Comput 250:59

    Article  MathSciNet  Google Scholar 

  9. Brandão FGSL, Gour G (2015) Reversible framework for quantum resource theories. Phys Rev Lett 115:070503

    Article  ADS  MathSciNet  Google Scholar 

  10. Sperling J, Vogel W (2015) Convex ordering and quantification of quantumness. Phys Scr 90:074024

    Article  ADS  Google Scholar 

  11. Lami L, Regula B, Wang X, Nichols R, Winter A, Adesso G (2018) Gaussian quantum resource theories. Phys Rev A 98:022335. https://doi.org/10.1103/PhysRevA.98.022335. arXiv:1801.05450 [quant-ph]

  12. Genoni MG, Palma ML, Tufarelli T, Olivares S, Kim MS, Paris MGA (2013) Detecting quantum non-gaussianity via the wigner function. Phys Rev A 87:062104

    Google Scholar 

  13. Baumgratz T, Cramer M, Plenio MB (2014) Quantifying coherence. Phys Rev Lett 113:140401

    Article  ADS  Google Scholar 

  14. Plenio MB (2005) Logarithmic negativity: a full entanglement monotone that is not convex. Phys Rev Lett 95:090503

    Article  ADS  Google Scholar 

  15. Nielsen MA, Chuang IL (2011) Quantum computation and quantum information: 10th anniversary edition, 10th edn. Cambridge University Press, New York

    Google Scholar 

  16. Horodecki M, Oppenheim J, Horodecki R (2002) Are the laws of entanglement theory thermodynamical? Phys Rev Lett 89:240403

    Article  ADS  Google Scholar 

  17. Glauber RJ (2007) Quantum theory of optical coherence: selected papers and lectures. Wiley-VCH

    Google Scholar 

  18. Wigner E (1932) On the quantum correction for thermodynamic equilibrium. Phys Rev 40:749

    Article  ADS  Google Scholar 

  19. Vogel W, Welsch D-G (2006) Quantum optics. Wiley-VCH

    Google Scholar 

  20. Shahandeh F, Ringbauer M (2017) Optomechanical state reconstruction and nonclassicality verification beyond the resolved-sideband regime. arXiv:1709.01135 [quant-ph]

  21. Spekkens RW (2008) Negativity and contextuality are equivalent notions of nonclassicality. Phys Rev Lett 101:020401

    Article  ADS  MathSciNet  Google Scholar 

  22. Ferrie C, Emerson J (2008) Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations. J Phys Math Theor 41:352001

    Article  MathSciNet  Google Scholar 

  23. Ferrie C, Emerson J (2009) Framed Hilbert space: hanging the quasi-probability pictures of quantum theory. New J Phys 11:063040

    Article  Google Scholar 

  24. Hayashi M, Ishizaka S, Kawachi A, Kimura G, Ogawa T (2015) Introduction to quantum information science. Springer

    Google Scholar 

  25. Holevo A (2011) Probabilistic and statistical aspects of quantum theory. Scuola Normale Superiore Pisa

    Google Scholar 

  26. Ollivier H, Zurek WH (2001) Quantum discord: a measure of the quantumness of correlations. Phys Rev Lett 88:017901

    Article  ADS  Google Scholar 

  27. Rényi A (1961) On measures of entropy and information. In: Proceedings of the fourth berkeley symposium on mathematical statistics and probability: contributions to the theory of statistics, vol 1. University of California Press, p 547

    Google Scholar 

  28. Wehrl A (1976) Three theorems about entropy and convergence of density matrices. Reports Math Phys 10:159

    Article  ADS  MathSciNet  Google Scholar 

  29. Wehrl A (1978) General properties of entropy. Rev Mod Phys 50:221

    Article  ADS  MathSciNet  Google Scholar 

  30. Tsallis C (1988) Possible generalization of Boltzmann-Gibbs statistics. J Stat Phys 52:479

    Article  ADS  MathSciNet  Google Scholar 

  31. Hu X, Ye Z (2006) Generalized quantum entropy. J Math Phys 47:023502

    Article  ADS  MathSciNet  Google Scholar 

  32. Müller-Lennert M, Dupuis F, Szehr O, Fehr S, Tomamichel M (2013) On quantum Rényi entropies: a new generalization and some properties. J Math Phys 54:122203

    Article  ADS  MathSciNet  Google Scholar 

  33. Wilde MM, Winter A, Yang D (2014) Strong converse for the classical capacity of entanglement-breaking and hadamard channels via a sandwiched Rényi relative entropy. Commun Math Phys 331:593

    Article  ADS  Google Scholar 

  34. Rajagopal AK, Nayak AS, Devi ARU (2014) From the quantum relative Tsallis entropy to its conditional form: separability criterion beyond local and global spectra. Phys Rev A 89:012331

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Swansea University, Swansea, UK

    Farid Shahandeh

Authors
  1. Farid Shahandeh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Farid Shahandeh .

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Shahandeh, F. (2019). Introduction. In: Quantum Correlations. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-24120-9_1

Download citation

Publish with us

Policies and ethics

Search

Navigation