Gleason-Type Theorem for Projective Measurements, Including Qubits: The Born Rule Beyond Quantum Physics

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• 作者：F. De Zela
• 关键词：Born rule ; Gleason’s theorem ; Quantum probability
• 刊名：Foundations of Physics
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：46
• 期：10
• 页码：1293-1306
• 全文大小：509 KB
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Physics
  Quantum Physics
  Relativity and Cosmology
  Biophysics and Biomedical Physics
  Mechanics
  Condensed Matter
  
• 出版者：Springer Netherlands
• ISSN：1572-9516
• 卷排序：46

文摘

Born’s quantum probability rule is traditionally included among the quantum postulates as being given by the squared amplitude projection of a measured state over a prepared state, or else as a trace formula for density operators. Both Gleason’s theorem and Busch’s theorem derive the quantum probability rule starting from very general assumptions about probability measures. Remarkably, Gleason’s theorem holds only under the physically unsound restriction that the dimension of the underlying Hilbert space \(\mathcal {H}\) must be larger than two. Busch’s theorem lifted this restriction, thereby including qubits in its domain of validity. However, while Gleason assumed that observables are given by complete sets of orthogonal projectors, Busch made the mathematically stronger assumption that observables are given by positive operator-valued measures. The theorem we present here applies, similarly to the quantum postulate, without restricting the dimension of \(\mathcal {H}\) and for observables given by complete sets of orthogonal projectors. We also show that the Born rule applies beyond the quantum domain, thereby exhibiting the common root shared by some quantum and classical phenomena.