The Schmidt rank for the commuting operator framework
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916294578667520 |
|---|---|
| author | van Luijk, Lauritz Schwonnek, René Stottmeister, Alexander Werner, Reinhard F. |
| author_facet | van Luijk, Lauritz Schwonnek, René Stottmeister, Alexander Werner, Reinhard F. |
| contents | In quantum information theory, the Schmidt rank is a fundamental measure for the entanglement dimension of a pure bipartite state. Its natural definition uses the Schmidt decomposition of vectors on bipartite Hilbert spaces, which does not exist (or at least is not canonically given) if the observable algebras of the local systems are allowed to be general C*-algebras. In this work, we generalize the Schmidt rank to the commuting operator framework where the joint system is not necessarily described by the minimal tensor product but by a general bipartite algebra. We give algebraic and operational definitions for the Schmidt rank and show their equivalence. We analyze bipartite states and compute the Schmidt rank in several examples: The vacuum in quantum field theory, Araki-Woods-Powers states, as well as ground states and translation invariant states on spin chains which are viewed as bipartite systems for the left and right half chains. We conclude with a list of open problems for the commuting operator framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_11619 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Schmidt rank for the commuting operator framework van Luijk, Lauritz Schwonnek, René Stottmeister, Alexander Werner, Reinhard F. Quantum Physics Mathematical Physics Operator Algebras In quantum information theory, the Schmidt rank is a fundamental measure for the entanglement dimension of a pure bipartite state. Its natural definition uses the Schmidt decomposition of vectors on bipartite Hilbert spaces, which does not exist (or at least is not canonically given) if the observable algebras of the local systems are allowed to be general C*-algebras. In this work, we generalize the Schmidt rank to the commuting operator framework where the joint system is not necessarily described by the minimal tensor product but by a general bipartite algebra. We give algebraic and operational definitions for the Schmidt rank and show their equivalence. We analyze bipartite states and compute the Schmidt rank in several examples: The vacuum in quantum field theory, Araki-Woods-Powers states, as well as ground states and translation invariant states on spin chains which are viewed as bipartite systems for the left and right half chains. We conclude with a list of open problems for the commuting operator framework. |
| title | The Schmidt rank for the commuting operator framework |
| topic | Quantum Physics Mathematical Physics Operator Algebras |
| url | https://arxiv.org/abs/2307.11619 |