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Maximum Likelihood Estimators of Quantum Probabilities

Mirko Navara, Jan Ševic

Abstract

Classical probability theory is based on assumptions which are often violated in practice. Therefore quantum probability is a proposed alternative not only in quantum physics, but also in other sciences. However, so far it mostly criticizes the classical approach, but does not suggest a working alternative. Maximum likelihood estimators were given very low attention in this context. We show that they can be correctly defined and their computation in closed form is feasible at least in some cases.

Maximum Likelihood Estimators of Quantum Probabilities

Abstract

Classical probability theory is based on assumptions which are often violated in practice. Therefore quantum probability is a proposed alternative not only in quantum physics, but also in other sciences. However, so far it mostly criticizes the classical approach, but does not suggest a working alternative. Maximum likelihood estimators were given very low attention in this context. We show that they can be correctly defined and their computation in closed form is feasible at least in some cases.
Paper Structure (11 sections, 1 theorem, 41 equations, 9 figures)

This paper contains 11 sections, 1 theorem, 41 equations, 9 figures.

Table of Contents

  1. Motivation of quantum probability
  2. Motivating example
  3. Basic notions
  4. Notation and formulation of the task
  5. Horizontal sums of classical models
  6. Motivating example continued
  7. Products
  8. Horizontal sums of quantum models
  9. Constructible lattices
  10. Non-constructible lattices
  11. Conclusion

Key Result

Lemma 1

Given numbers $n_1,\ldots,n_k\in[0,\infty]$, we look for probabilities $p_1,\ldots,p_k\in[0,1]$ such that $\sum_i p_i=1$ and is maximal. This task has a unique solution, which satisfies for all $i$; equivalently, the ratio is the same for all $i$.

Figures (9)

  • Figure 1: Greechie diagram of the motivating example
  • Figure 2: Greechie diagram of a part of the motivating example
  • Figure 3: Greechie diagram of the smallest non-constructible orthomodular lattice
  • Figure 4: Greechie diagram of a non-constructible orthomodular lattice for which MLE is feasible
  • Figure 5: Simplified diagram of the second step of MLE of the diagram from Fig. \ref{['f:Pi2']}
  • ...and 4 more figures

Theorems & Definitions (2)

  • Remark 1
  • Lemma 1