Smooth manifold structure for extreme channels
| Autor: | Iten, Raban, Colbeck, Roger |
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| Rok vydání: | 2016 |
| Předmět: | |
| Zdroj: | Journal of Mathematical Physics 59, 012202 (2018) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1063/1.5019837 |
| Popis: | A quantum channel from a system $A$ of dimension $d_A$ to a system $B$ of dimension $d_B$ is a completely positive trace-preserving map from complex $d_A\times d_A$ to $d_B\times d_B$ matrices, and the set of all such maps with Kraus rank $r$ has the structure of a smooth manifold. We describe this set in two ways. First, as a quotient space of (a subset of) the $rd_B\times d_A$ dimensional Stiefel manifold. Secondly, as the set of all Choi-states of a fixed rank $r$. These two descriptions are topologically equivalent. This allows us to show that the set of all Choi-states corresponding to extreme channels from system $A$ to system $B$ of a fixed Kraus rank $r$ is a smooth submanifold of dimension $2rd_Ad_B-d_A^2-r^2$ of the set of all Choi-states of rank $r$. As an application, we derive a lower bound on the number of parameters required for a quantum circuit topology to be able to approximate all extreme channels from $A$ to $B$ arbitrarily well. Comment: 9 pages, v2: a few minor corrections to match journal version |
| Databáze: | arXiv |
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