Distributed quantum dense coding with two receivers in noisy environments

We investigate the effect of noisy channels in a classical information transfer through a multipartite state which acts as a substrate for the distributed quantum dense coding protocol between several senders and two receivers. The situation is qualitatively different from the case with one or more senders and a single receiver. We obtain an upper bound on the multipartite capacity which is tightened in the case of the covariant noisy channel. We also establish a relation between the genuine multipartite entanglement of the shared state and the capacity of distributed dense coding using that state, both in the noiseless and the noisy scenarios. Specifically, we find that, in the case of multiple senders and two receivers, the corresponding generalized Greenberger-Horne-Zeilinger states possess higher dense coding capacities as compared to a significant fraction of pure states having the same multipartite entanglement.

Figure 1

Schematic diagram of the DC protocol considered. An $(N+2)$-party quantum state, ${\rho }^{S_1{S}_2...S_N{R}_1{R}_2}$, is shared between $N$ senders, $S_1,S_2,...,S_N$, and two receivers, $R_1$ and $R_2$. We assume that after unitary encoding, the senders $S_1,S_2,...,S_r$ send their part to the receiver $R_1$, while the rest send their parts to the receiver $R_2$.

Figure 2

Plots of the quantities $\frac{a_1^2}{\sqrt{f\left(a_1\right)}}log\frac{1-\sqrt{f\left(a_1\right)}}{1+\sqrt{f\left(a_1\right)}}$ and $\frac{1-a_1^2}{\sqrt{g\left(a_1\right)}}log\frac{1-\sqrt{g\left(a_1\right)}}{1+\sqrt{g\left(a_1\right)}}$, which are respectively the left-hand and right-hand sides of Eq. (40), against $a_1$ and $\gamma$. The green (gray in print) surface represents the first, while the purple (dark in print) one is for the second expression. The intersection line (white line) is $a_1=\frac{1}{\sqrt{2}}$, for all $\gamma$. The base axes are dimensionless, while the vertical axis is in bits.

Figure 3

Noiseless case: how does a general four-qubit pure state compare with the gGHZ states? We randomly generate $5\times {10}^4$ four-qubit pure states uniformly with respect to the corresponding Haar measure, and their GGM is plotted as the abscissa, while ${\mathcal{B}}^{\text{LOCC}}$ is plotted as the ordinate. The red curved line represents the gGHZ states. Among the states generated randomly, $47.6\%$ (blue triangles) satisfy both the conditions in Result 2; $49\%$ (orange squares) violate either of the conditions, but still fall above the gGHZ line. Green circles represent $3.4\%$ states which violate the conclusion of Result 2. The line at abscissa equal to 2 corresponds to the capacity achievable without prior shared entanglement. The vertical axis is dimensionless, while the horizontal one is in bits.

Figure 4

Noiseless case: comparison between arbitrary four-qubit pure states and the gGHZ states, with constrained GGM. We randomly generate $5\times {10}^4$ four-qubit pure states uniformly with respect to the corresponding Haar measure, and their HV-GGM $\left({\mathcal{E}}^{\mathrm{HV}}\right)$ is plotted as the abscissa, while ${\mathcal{B}}^{\text{LOCC}}$ is plotted as the ordinate. The red curved line represents the gGHZ states. Among the states generated randomly, $0.7\%$ (green circles) states fall below the gGHZ line. Orange squares represent those states whose ${\mathcal{E}}^{\mathrm{HV}}$ is greater than $0.5$ (above the horizontal line)—they are very few in number, and constitute only $1.2\%$ of the total generated random states. The line at abscissa equal to 2 corresponds to the capacity achievable without prior shared entanglement. The vertical axis is dimensionless, while the horizontal one is in bits.

Figure 5

Fully correlated Pauli noise: the gGHZ states are again better than a significant fraction of states. We plot the GGM as the ordinate and ${\chi }_{\mathrm{noisy}}^{\mathrm{LOCC}}$ as the abscissa for $5\times {10}^4$ randomly generated four-qubit pure states uniformly with respect to the corresponding Haar measure for low (top panel) and high (bottom panel) full correlated Pauli noise. In the top panel, $q_0=0.93,q_1=0.01,q_2=0.02,\text{and}{q}_3=0.04$, while in the bottom panel, we choose $q_0=0.485,q_1=0.015,q_2=0.015,\text{and}{q}_3=0.485$. In the presence of high noise, almost all states are bounded by the four-qubit gGHZ states (red curved line). A significant fraction of the generated states lie above the gGHZ line even for low noise. It indicates that the gGHZ state is more robust against noise as compared to an arbitrary four-qubit pure state. The lines at abscissa equal to 2 correspond to the capacity achievable without prior shared entanglement. The vertical axis is dimensionless, while the horizontal one is in bits.