A Sufficient Criterion for Divisibility of Quantum Channels
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910884941529088 |
|---|---|
| author | Ende, Frederik vom |
| author_facet | Ende, Frederik vom |
| contents | We present a simple, dimension-independent criterion which guarantees that some quantum channel $Φ$ is divisible, i.e. that there exists a non-trivial factorization $Φ=Φ_1Φ_2$. The idea is to first define an "elementary" channel $Φ_2$ and then to analyze when $ΦΦ_2^{-1}$ is completely positive. The sufficient criterion obtained this way -- which even yields an explicit factorization of $Φ$ -- is that one has to find orthogonal unit vectors $x,x^\perp$ such that $\langle x^\perp|\mathcal K_Φ\mathcal K_Φ^\perp|x\rangle=\langle x|\mathcal K_Φ\mathcal K_Φ^\perp|x\rangle=\{0\}$ where $\mathcal K_Φ$ is the Kraus subspace of $Φ$ and $\mathcal K_Φ^\perp$ is its orthogonal complement. Of course, using linearity this criterion can be reduced to finitely many equalities. Generically, this division even lowers the Kraus rank which is why repeated application -- if possible -- results in a factorization of $Φ$ into in some sense "simple" channels. Finally, be aware that our techniques are not limited to the particular elementary channel we chose. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_17103 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Sufficient Criterion for Divisibility of Quantum Channels Ende, Frederik vom Quantum Physics Mathematical Physics We present a simple, dimension-independent criterion which guarantees that some quantum channel $Φ$ is divisible, i.e. that there exists a non-trivial factorization $Φ=Φ_1Φ_2$. The idea is to first define an "elementary" channel $Φ_2$ and then to analyze when $ΦΦ_2^{-1}$ is completely positive. The sufficient criterion obtained this way -- which even yields an explicit factorization of $Φ$ -- is that one has to find orthogonal unit vectors $x,x^\perp$ such that $\langle x^\perp|\mathcal K_Φ\mathcal K_Φ^\perp|x\rangle=\langle x|\mathcal K_Φ\mathcal K_Φ^\perp|x\rangle=\{0\}$ where $\mathcal K_Φ$ is the Kraus subspace of $Φ$ and $\mathcal K_Φ^\perp$ is its orthogonal complement. Of course, using linearity this criterion can be reduced to finitely many equalities. Generically, this division even lowers the Kraus rank which is why repeated application -- if possible -- results in a factorization of $Φ$ into in some sense "simple" channels. Finally, be aware that our techniques are not limited to the particular elementary channel we chose. |
| title | A Sufficient Criterion for Divisibility of Quantum Channels |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2407.17103 |