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Quantum Mechanics According to Martians: Density Matrix Theory

  • Bernard Zygelman
Chapter

Abstract

We discuss the density matrix approach introduced by John von Neumann to quantum mechanics. We illustrate how it is applied in the calculation of expectation values and Born probabilities. The postulates of quantum mechanics, according to the Copenhagen interpretation, are summarized from the vantage point of the density operator framework. We learn how density operators are used to discern coherent, or pure, state ensembles from statistical mixtures of pure states. It is shown that, for entangled states, the traced density operator to a lower dimensional Hilbert space, results in reduced density operators that describe a mixed state. The Schmidt decomposition theorem and the Schmidt number, which measures the degree of entanglement of quantum states, is introduced and discussed. We define and illustrate the concept of von Neumann entropy.

References

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    Istvan Hargittai, Martians of Science: Five Physicists Who Changed the Twentieth Century, Oxford U. Press, 2006CrossRefGoogle Scholar
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    William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd edn. Cambridge University Press, 2007zbMATHGoogle Scholar
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    Michael E. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information, Cambridge U. Press, 2011zbMATHGoogle Scholar
  4. 4.
    Claude Shannon, The Bell System Technical Journal, 27, 379–423,623–656, 1948Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Bernard Zygelman
    • 1
  1. 1.Department of Physics and AstronomyUniversity of NevadaLas VegasUSA

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