Subadditivity of the minimum output entropy and superactivation of the classical capacity of quantum multiple access channels

We study subadditivity of the minimum output entropy ($H_{\text{min}}$) of quantum multiple access channels (MACs). We provide an example of violation of the additivity theorem for $H_{\text{min}}$ known in classical information theory. Our result is based on a fundamental property of $\text{MACs}$, i.e., independence of each sender. The channels used in the example can be constructed explicitly. On the basis of subadditivity of $H_{\text{min}}$ we also provide an example of extremal superadditivity (superactivation) of the classical capacity region of MACs.

Figure 1

The general schema of the channels ${\Gamma }_A$ and ${\Gamma }_B$. $X_i$ are $d$-qubit input lines. $M_{\text{I}}$ and $M_{\text{II}}$ are measurements. Its result is denoted by $m_{\text{I}}$ and $m_{\text{II}}$, respectively. $f_{\mathrm{opt}}$ is a classical randomness extractor with properties described by Theorem 1.

Figure 2

The parallel setup of the channels ${\Gamma }_A$ and ${\Gamma }_B$. Dashed lines depict entanglement of the inputs in the case of $|{\Psi }_{0,0}\rangle$ transmission through ${\Gamma }_A\otimes {\Gamma }_B$.

Figure 3

Construction of the channel $\mathrm{\Gamma ̃}$. The solid lines represent the qubit lines and the dashed lines represent the bit lines. The bit lines control CNOTs performed on the output of the channel.