English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

For a positive integer $d$ and a unitary representation$\rho:G\rightarrow\mathrm{U}(d)$ of a compact group $G$, the twirling channelfor this representation is the linear mapping $\Phi: M_d\rightarrow M_d$defined as $\Phi(X)=\int_{G}\mathrm{d}\mu(g)\,\rho(g)X\rho(g^{-1})$ for every$X\in M_d$, where $\mu$ is the Haar measure on $G$. Such channels are examplesof mixed-unitary channels, as they are in the convex hull of the set of unitarychannels of a fixed size. By Carath\'eodory's theorem, these channels canalways be expressed as a finite linear combination of unitary channels. Weconsider the mixed-unitary rank twirling channels---which is the minimum numberof distinct unitary conjugations required to express the channel as a convexcombination of unitary channels---and show that the mixed-unitary rank of everytwirling channel is always equal to its Choi rank, both of which are equal tothe dimension of the von Neumann algebra generated by the representation.Moreover, we show how to explicitly construct minimal mixed-unitarydecompositions for these types of channels and provide some examples.

Twirling channels have minimal mixed-unitary rank