Quantum Measurement Trees, II: Quantum Observables as Ortho-Measurable Functions and Density Matrices as Ortho-Probability Measures
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912609016479744 |
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| author | Hammond, Peter J. |
| author_facet | Hammond, Peter J. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_22617 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum Measurement Trees, II: Quantum Observables as Ortho-Measurable Functions and Density Matrices as Ortho-Probability Measures Hammond, Peter J. Quantum Physics Mathematical Physics Quantum Theory 81 (Primary) 81P13 (Secondary), Probability Theory 60 (Primary) 60A05 (Secondary) 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. |
| title | Quantum Measurement Trees, II: Quantum Observables as Ortho-Measurable Functions and Density Matrices as Ortho-Probability Measures |
| topic | Quantum Physics Mathematical Physics Quantum Theory 81 (Primary) 81P13 (Secondary), Probability Theory 60 (Primary) 60A05 (Secondary) |
| url | https://arxiv.org/abs/2509.22617 |