Existence of a three-sender quantum simultaneous decoder

Let $x,y,z\rightarrow\rho_{x,y,z}$ be a cccq quantum multiple access channel, with Senders 1, 2, and 3 accessing inputs $x$, $y$, and $z$, respectively. Let $p_X$, $p_Y$, and $p_Z$ be input distributions, and independently generate product-distributed codebooks $\{X^n(k)\}_{k=1}^K$, $\{Y^n(l)\}_{l=1}^L$, and $\{Z^n(m)\}_{m=1}^M$. Set

$$R_{1}=\frac{1}{n}\log_{2}K+\delta,\qquad R_{2}=\frac{1}{n}\log_{2}L+\delta,\qquad R_{3}=\frac{1}{n}\log_{2}M+\delta,$$

where $\delta>0$, and suppose

$$\begin{aligned} R_{1} &\leq I(X;B|YZ)_\rho, & R_{2} &\leq I(Y;B|XZ)_\rho, & R_{3} &\leq I(Z;B|XY)_\rho,\\ R_{1}+R_{2} &\leq I(XY;B|Z)_\rho, & R_{1}+R_{3} &\leq I(XZ;B|Y)_\rho, & R_{2}+R_{3} &\leq I(YZ;B|X)_\rho,\\ R_{1}+R_{2}+R_{3} &\leq I(XYZ;B)_\rho. \end{aligned}$$

Here the Holevo information is evaluated on

$$\rho^{XYZB}=\sum_{x,y,z}p_X(x)p_Y(y)p_Z(z)|x\rangle\langle x|^X\otimes|y\rangle\langle y|^Y\otimes|z\rangle\langle z|^Z\otimes\rho_{x,y,z}^B.$$

Existence of a three-sender quantum simultaneous decoder. There exists a decoding POVM $\{\Lambda_{l,m,k}\}_{l,m,k}$ such that, for every $\epsilon>0$ and sufficiently large $n$, the expected average probability of error satisfies

$$\mathbb{E}\left\{\frac{1}{KLM}\sum_{k,l,m}\operatorname{Tr}\left[(I-\Lambda_{k,l,m})\rho_{X^n(k),Y^n(l),Z^n(m)}\right]\right\}\leq\epsilon,$$

where the expectation is over the three random codebooks. This conjectured decoder would extend simultaneous decoding from two-sender quantum multiple access channels to the three-sender case needed for the Han–Kobayashi achievable rate region; its status is not resolved in the supplied source.