Discrete dynamics in the set of quantum measurements
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866915079198343168 |
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| author | Rico, Albert Życzkowski, Karol |
| author_facet | Rico, Albert Życzkowski, Karol |
| contents | A quantum measurement, often referred to as positive operator-valued measurement (POVM), is a set of positive operators $P_j=P_j^†\geq 0$ summing to identity, $\sum_jP_j=\mathbb{1}$. This can be seen as a generalization of a probability distribution of positive real numbers summing to unity, whose evolution is given by a stochastic matrix. We describe discrete transformations in the set of quantum measurements by {\em blockwise stochastic matrices}, composed of positive blocks that sum columnwise to identity, using the notion of {\em sequential product} of matrices. We show that such transformations correspond to a sequence of quantum measurements. Imposing additionally the dual condition that the sum of blocks in each row is equal to identity, we arrive at blockwise bistochastic matrices (also called {\em quantum magic squares}). Analyzing their dynamical properties, we formulate our main result: a quantum analog of the Ostrowski description of the classical Birkhoff polytope, which introduces the notion of majorization between quantum measurements. Our framework provides a dynamical characterization of the set of blockwise bistochastic matrices and establishes a resource theory in this set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_05835 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Discrete dynamics in the set of quantum measurements Rico, Albert Życzkowski, Karol Quantum Physics Mathematical Physics A quantum measurement, often referred to as positive operator-valued measurement (POVM), is a set of positive operators $P_j=P_j^†\geq 0$ summing to identity, $\sum_jP_j=\mathbb{1}$. This can be seen as a generalization of a probability distribution of positive real numbers summing to unity, whose evolution is given by a stochastic matrix. We describe discrete transformations in the set of quantum measurements by {\em blockwise stochastic matrices}, composed of positive blocks that sum columnwise to identity, using the notion of {\em sequential product} of matrices. We show that such transformations correspond to a sequence of quantum measurements. Imposing additionally the dual condition that the sum of blocks in each row is equal to identity, we arrive at blockwise bistochastic matrices (also called {\em quantum magic squares}). Analyzing their dynamical properties, we formulate our main result: a quantum analog of the Ostrowski description of the classical Birkhoff polytope, which introduces the notion of majorization between quantum measurements. Our framework provides a dynamical characterization of the set of blockwise bistochastic matrices and establishes a resource theory in this set. |
| title | Discrete dynamics in the set of quantum measurements |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2308.05835 |