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Given a channel $\Phi:\operatorname{Lin}(\mathbb{C}^n)\to\operatorname{Lin}(\mathbb{C}^m)$, we define its Holevo capacity as $$\chi(\Phi) = \sup_\eta \chi(\Phi(\eta)),$$ with the sup taken with respect to all ensembles $\eta\equiv (\eta_b)$, with $\eta_b\ge0$ and $\sum_b\operatorname{tr}(\eta_b)=1$, and we're using the shorthand notation $\chi(\Phi(\eta))\equiv (\Phi(\eta_b))_b$, meaning $\chi(\Phi(\eta))$ is the ensemble obtained applying $\Phi$ elementwise to the elements of $\eta$ (I'm mostly following Watrous' notation here).

What are examples where this quantity can be computed explicitly? As a bonus, it'd be great to have examples where one can compute also $\chi(\Phi^{\otimes n})$ for some $n$. Idea being to possibly get a more concrete and explicit idea on how using codes spanning multiple channel applications allows to pack information more efficiently.

glS
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1 Answers1

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One example is the phase damping channel, which is given by $\mathcal{N}(\rho) = (1-p)\rho + p Z \rho Z.$ The Holevo capacity is bounded above by the $\log$ of the dimension of the output space, which in this case is $1.$ For a lower bound, consider the uniform ensemble of two states $|0\rangle\langle0|$ and $|1\rangle\langle1|$, each occurring with probability $\frac{1}{2}.$ The Holevo capacity of this ensemble for the channel $\mathcal{N}$ is $$H\left(\sum_{i=1}^2 \frac{1}{2}\mathcal{N}(|i\rangle\langle i|)\right) - \sum_{i=1}^2 \frac12 H(\mathcal{N}(|i\rangle\langle i|)).$$ For the phase damping channel we have $\mathcal{N}(|i\rangle\langle i|) = |i\rangle\langle i|,$ so the lower bound becomes $H\left(\frac12\mathbb{1}\right),$ which is $1.$ This proves the Holevo capacity of the phase-damping channel is $1.$

This argument also works for the channel $\rho \mapsto (1-p)\rho + p X \rho X$.

For these two channels, and in fact all unital qubit channels, the Holevo capacity is known to be additive and so $\chi(\mathcal{N}^{\otimes n}) = n \chi(\mathcal{N})$.

HerrWarum
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