Spectrum of the Reduced matrix's eigenvalues

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SUMMARY

The discussion centers on the spectrum of reduced density matrices and whether they can exhibit a continuum spectrum. It is established that pure density matrices possess a discrete spectrum. The conversation highlights that all compact operators have discrete eigenvalues, with zero as the only possible accumulation point. The query revolves around the classification of reduced density matrices as compact operators, which remains unresolved in the discussion.

PREREQUISITES
  • Understanding of density matrices in quantum mechanics
  • Knowledge of compact operators in functional analysis
  • Familiarity with eigenvalues and eigenvectors
  • Basic mathematical proficiency in operator theory
NEXT STEPS
  • Research the properties of compact operators in functional analysis
  • Study the implications of eigenvalue accumulation points in quantum mechanics
  • Explore the characteristics of reduced density matrices
  • Investigate the relationship between discrete and continuum spectra in quantum systems
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Quantum physicists, mathematicians specializing in functional analysis, and researchers studying the properties of density matrices in quantum mechanics.

fpaolini
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I would like to know if the density matrix spectrum is always discrete or if it is possible it has a continuum spectrum. It is clear that a pure density matrix has a discrete spectrum but it is not obvious in general.

I have heard that all compact operator has discrete eigenvalues and if it has an accumulation point it must be zero. It seems to me to be the case for reduced density matrix but as I am not a good mathematician I cannot see if a reduced matrix is or not a compact operator

Where could I find some discussion about that topic?
Thanks.
 
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