Quantum Measurement Trees, II: Quantum Observables as Ortho-Measurable Functions and Density Matrices as Ortho-Probability Measures

• URL: http://arxiv.org/abs/2509.22617v1
• Date: Fri, 26 Sep 2025 17:43:51 GMT
• Title: Quantum Measurement Trees, II: Quantum Observables as Ortho-Measurable Functions and Density Matrices as Ortho-Probability Measures
• Authors: Peter J. Hammond,
• Abstract summary: Given a quantum state in the finite-dimensional space $ Cn $, the range of possible values of a quantum observable is usually identified with the discrete spectrum of eigenvalues of a corresponding Hermitian matrix.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Given a quantum state in the finite-dimensional Hilbert space $ \C^n $, the range of possible values of a quantum observable is usually identified with the discrete spectrum of eigenvalues of a corresponding Hermitian matrix. Here any such observable is identified with: (i) an ``ortho-measurable'' function defined on the Boolean ``ortho-algebra'' generated by the eigenspaces that form an orthogonal decomposition of $ \C^n $; (ii) a ``numerically identified'' orthogonal decomposition of $ \C^n $. The latter means that each subspace of the orthogonal decomposition can be uniquely identified by its own attached real number, just as each eigenspace of a Hermitian matrix can be uniquely identified by the corresponding eigenvalue. Furthermore, any density matrix on $ \C^n $ is identified with a Bayesian prior ``ortho-probability'' measure defined on the linear subspaces that make up the Boolean ortho-algebra induced by its eigenspaces. Then any pure quantum state is identified with a degenerate density matrix, and any mixed state with a probability measure on a set of orthogonal pure states. Finally, given any quantum observable, the relevant Bayesian posterior probabilities of measured outcomes can be found by the usual trace formula that extends Born's rule.

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