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Not All Probability Density Functions Are Tomograms.

Liubov A Markovich1,2,3, Justus Urbanetz1, Vladimir I Man'ko3,4

  • 1Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands.

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Summary

This study introduces tomographic probability density functions (pdfs) as a new way to represent quantum states, offering a classical probability approach for better state reconstruction, especially for complex, non-Gaussian states.

Keywords:
characteristic functionprobability distribution functionquantum state reconstructionsymplectic tomogram

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Area of Science:

  • Quantum Information Science
  • Quantum State Representation
  • Probability Theory

Background:

  • Quantum states are traditionally described by wave functions or density operators in Hilbert spaces.
  • Quasi-probability distribution functions, like the Wigner function, have limitations in direct analysis using classical probability tools.
  • Accurate quantum state representation is crucial for quantum information processing and metrology.

Purpose of the Study:

  • To explore the tomographic probability density function (pdf) as a complete and accurate representation of quantum states.
  • To identify the specific properties that distinguish valid quantum tomograms from general probability density functions.
  • To highlight the advantages of using tomograms for quantum state reconstruction, particularly for non-Gaussian states.

Main Methods:

  • Utilizing the framework of classical probability theory for the analysis of quantum state tomograms.
  • Investigating the mathematical conditions and constraints that pdfs must satisfy to be considered valid quantum tomograms.
  • Applying parametric and nonparametric density estimation methods to tomographic data.

Main Results:

  • Demonstrated that tomograms are true probability density functions (pdfs) capable of fully describing quantum systems.
  • Established that not all pdfs can serve as tomograms; specific quantum conditions must be met.
  • Showcased the utility of tomographic pdfs for enhanced state reconstruction, especially for multi-mode, non-Gaussian quantum states.

Conclusions:

  • Tomographic pdfs offer a powerful alternative to traditional quantum state descriptions, leveraging classical probability tools.
  • Understanding the 'quantum' nature of pdfs is essential for their application as tomograms.
  • This representation facilitates improved analysis and reconstruction of complex quantum states.