Extension of the Watanabe-Sagawa-Ueda uncertainty relation for measurement errors to infinite-dimensional systems
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912674215886848 |
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| author | Nogami, Ryosuke |
| author_facet | Nogami, Ryosuke |
| contents | We extend the Watanabe--Sagawa--Ueda (WSU) uncertainty relations for measurement errors to infinite-dimensional systems. The original WSU formulation provided a definition of measurement errors with a clear physical interpretation based on quantum estimation theory, but was restricted to finite-dimensional systems, excluding important observables such as position and momentum. Using pseudo-inverse forms of positive-semidefinite forms, we develop a framework for classical and quantum estimation theory for models whose parameter space is the set of full-rank states on an infinite-dimensional Hilbert space, and derive classical and quantum Cramér--Rao inequalities. We extend the WSU measurement errors to both bounded and unbounded operators, and derive corresponding error-error uncertainty relations. The resulting uncertainty relation inequalities are stronger than the original WSU bound due to an improved derivation method. Our results provide a theoretical framework for applying estimation-based uncertainty relations to observables with continuous values in infinite-dimensional systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_09309 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Extension of the Watanabe-Sagawa-Ueda uncertainty relation for measurement errors to infinite-dimensional systems Nogami, Ryosuke Quantum Physics We extend the Watanabe--Sagawa--Ueda (WSU) uncertainty relations for measurement errors to infinite-dimensional systems. The original WSU formulation provided a definition of measurement errors with a clear physical interpretation based on quantum estimation theory, but was restricted to finite-dimensional systems, excluding important observables such as position and momentum. Using pseudo-inverse forms of positive-semidefinite forms, we develop a framework for classical and quantum estimation theory for models whose parameter space is the set of full-rank states on an infinite-dimensional Hilbert space, and derive classical and quantum Cramér--Rao inequalities. We extend the WSU measurement errors to both bounded and unbounded operators, and derive corresponding error-error uncertainty relations. The resulting uncertainty relation inequalities are stronger than the original WSU bound due to an improved derivation method. Our results provide a theoretical framework for applying estimation-based uncertainty relations to observables with continuous values in infinite-dimensional systems. |
| title | Extension of the Watanabe-Sagawa-Ueda uncertainty relation for measurement errors to infinite-dimensional systems |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2305.09309 |