Schemes of transmission of classical information via quantum channels with many senders: Discrete- and continuous-variable cases

Superadditivity effects in the classical capacity of discrete multiaccess channels and continuous variable (CV) Gaussian MACs are analyzed. Several examples of the manifestation of superadditivity in the discrete case are provided, including, in particular, a channel which is fully symmetric with respect to all senders. Furthermore, we consider a class of channels for which input entanglement across more than two copies of the channels is necessary to saturate the asymptotic rate of transmission from one of the senders to the receiver. The five-input entanglement of Shor error correction codewords surpass the capacity attainable by using arbitrary two-input entanglement for these channels. In the CV case, we consider the properties of the two channels (a beam-splitter channel and a “nondemolition” quadratures coupling sum (XP) gate channel) analyzed in Czekaj et al. [Phys. Rev. A 82, 020302(R) (2010)] in greater detail and also consider the sensitivity of capacity superadditivity effects to thermal noise. We observe that the estimates of the amount of two-mode squeezing required to achieve capacity superadditivity are more optimistic than previously reported.

Figure 1

Additivity of the capacity regions for classical MACs. Capacity regions for channels ${\Phi }_I$ and ${\Phi }_{II}$ are presented, respectively, in panels (a) and (b). The capacity region of the parallel composition ${\Phi }_I\otimes {\Phi }_{II}$ of channels ${\Phi }_I$ and ${\Phi }_{II}$ is presented in panel (c) and it is given by the geometrical sum of capacities regions from panels (a) and (b) (see Ref. [6]).

Figure 2

Channel $\Phi$ from Sec. 3c. (a) Channel $\Phi$ working in parallel with identity channel $\mathcal{I}$; waved line denotes entangled state. (b),(c) Two modes of work of channel $\Phi$.

Figure 3

Lower bound for achieved capacity region for channel $\Phi$ from Sec. 3c working in parallel with the identity channel. Thick lines refer to the upper bound for geometrical sum of capacity regions of component channels.

Figure 4

The channel described in Sec. 3d with $n=5$ helper-senders and $n^{\prime }=3$ lines belonging to sender $A$. The message represented by the label $w=1,2,4$ is additionally sent to the receiver, which may be represented as a “flag” state $|w\rangle =|001,010,100\rangle$.

Figure 5

The parallel setup of channels described in Sec. 3d. The inputs of the channels used for transmission of entangled states are shown. The presented case consists of channels with two helper-senders $n=2$, each of which can send three particle entangled states $m=3$.

Figure 6

Equivalence ${\Phi }_w^{\otimes m}={\otimes }_i{\Gamma }_i$ on example of channel ${\Phi }_{w=\{1,1,2\}}^{\otimes 3}$. The dashed line delimits channel ${\Gamma }_1$; waved lines denote entangled states from by senders $B_1$ and $B_2$; labels with active helper-sender are in the circles.

Figure 7

Upper bound for the regularized capacity $C_A^{\left(m\right)}=\frac{1}{m}{C}_A\left({\Phi }^{\otimes m}\right)$ as a function of $m$—the number of channel copies the capacity is evaluated on—and the number of helper senders $n$ in the channel $\Phi$.

Figure 8

Gaussian channels exhibiting violation of locality rule on individual transmission rates. (a) Beam-splitter channel ${\Phi }_{\theta }$; (b) triple QND sum gate channel $\Phi$. The channels were first presented in Ref. [7].

Figure 9

Rates which are achievable by the different encoding schemes for the beam-splitter channel ${\Phi }_{\theta }$ as a function of the energy constraint $N_B$ for sender $B$. The rates are evaluated for the following values of $\theta$ and the energy constraints $N_A$ for sender $A$: (a) $\theta =\pi /4,N_A={10}^6$; (b) $\theta =0.2,N_A={10}^6$; (c) $\theta =0.5,N_A={10}^3$.

Figure 10

(a) Lines present bounds of the areas where $R_A^{\text{ent}}>R_A^{\text{prod-bound}}$. They are given by the condition $R_A^{\text{ent}}/R_A^{\text{prod-bound}}=1$. Lines refer to the cases of $N_A={10}^3,{10}^6,{10}^9$. For a given $N_A$ the superadditive region lies above the corresponding line. (b) The dotted line delimits the area where $R_A^{\text{ent}}>R_A^{\text{prod-bound}}$ for $N_A={10}^3$. Notice that the superadditive region is considered in a large scale of $N_B$. The figure shows that there is large window of parameters $N_B,\theta$ for which $R_A^{\text{ent}}>R_A^{\text{prod-bound}}$ can be observed.

Figure 11

Influence of thermal noise on capacity $R_A^{\left(1\right)}({\Phi }_{\theta }\otimes \mathcal{I})$ of the BS channel ${\Phi }_{\theta }$ working with parameters $\theta =0.25,N_A={10}^3$. (a) Dependence on the transitivity $T$ of the thermal noise channel, here $N_{\text{Th}}=0$; rates $R_A^{\left(1\right)}({\Phi }_{\theta }\otimes \mathcal{I})$ (solid) are compared with output entropy bound $R_A^{\text{prod-bound}}$ (dashed). (b) Dependency on the mean photon number $N_{\text{Th}}$ in the thermal state for $N_{\text{Th}}=0,2,10,20$ and an exemplary value $T=0.95$; here we provide output entropy bound $R_A^{\text{prod-bound}}$ only for $N_{\text{Th}}=0$ since additional thermal noise increases output entropy and makes the bound less tight.