Riemannian-geometric generalizations of quantum fidelities and Bures-Wasserstein distance
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915798823469056 |
|---|---|
| author | Afham, A. Ferrie, Chris |
| author_facet | Afham, A. Ferrie, Chris |
| contents | We introduce a family of fidelities, termed generalized fidelity, which are based on the Riemannian geometry of the Bures-Wasserstein manifold. We show that this family of fidelities generalizes standard quantum fidelities such as Uhlmann-, Holevo-, and Matsumoto-fidelity and demonstrate that it satisfies analogous celebrated properties. The generalized fidelity naturally arises from a generalized Bures distance, the natural distance obtained by linearizing the Bures-Wasserstein manifold. We prove various invariance and covariance properties of generalized fidelity as the point of linearization moves along geodesic-related paths. We also provide a Block-matrix characterization and prove an Uhlmann-like theorem, as well as provide further extensions to the multivariate setting and to quantum Rényi divergences, generalizing Petz-, Sandwich-, Reverse sandwich-, and Geometric-Rényi divergences of order $α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04937 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Riemannian-geometric generalizations of quantum fidelities and Bures-Wasserstein distance Afham, A. Ferrie, Chris Quantum Physics Mathematical Physics We introduce a family of fidelities, termed generalized fidelity, which are based on the Riemannian geometry of the Bures-Wasserstein manifold. We show that this family of fidelities generalizes standard quantum fidelities such as Uhlmann-, Holevo-, and Matsumoto-fidelity and demonstrate that it satisfies analogous celebrated properties. The generalized fidelity naturally arises from a generalized Bures distance, the natural distance obtained by linearizing the Bures-Wasserstein manifold. We prove various invariance and covariance properties of generalized fidelity as the point of linearization moves along geodesic-related paths. We also provide a Block-matrix characterization and prove an Uhlmann-like theorem, as well as provide further extensions to the multivariate setting and to quantum Rényi divergences, generalizing Petz-, Sandwich-, Reverse sandwich-, and Geometric-Rényi divergences of order $α$. |
| title | Riemannian-geometric generalizations of quantum fidelities and Bures-Wasserstein distance |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2410.04937 |