Control power of high-dimensional controlled dense coding

We quantitatively analyze and evaluate the control power in a high-dimensional noisy quantum channel for transmitting classical information. We calculate the control power of the controlled dense coding scheme in a four-dimensional extended Greenberger-Horne-Zeilinger (GHZ) class state channel. Different from controlled teleportation, there is no tradeoff between the control power and the classical capacity of the optimal four-dimensional GHZ state quantum channel. Only when the channel is noisy and the sender is allowed to gain some control authority can the tradeoff between the control power and the classical capacity be activated, and the overall control power for transmitting classical information will be enhanced by reducing the classical capacity of the high-dimensional quantum channel. This noise induced characteristic is very different from that of transmitting quantum information, and it may provide new ideas and methods to develop the resource theory of quantum channels.

Figure 1

Control power of four-dimensional channels as a function of the channel parameters to ${\lambda }_1$ and ${\lambda }_2$. (a) $P_B$ is the control power for the original CDC scheme in the case of ${\lambda }_2={\lambda }_4$. (b) $P_B^{\prime }$ is the control power for the decentralized CDC scheme in the case of ${\lambda }_2={\lambda }_4$. (c) The change $\mathrm{\Delta }{P}_B=P_B^{\prime }-P_B$ in the case of ${\lambda }_2={\lambda }_4$.

Figure 2

Control power of four-dimensional channels as a function of the channel parameters to ${\lambda }_1$ and ${\lambda }_2$. (a) $P$ is the control power for the original CDC scheme in the case of ${\lambda }_2={\lambda }_4$. (b) $P^{\prime }$ is the control power for the decentralized CDC scheme in the case of ${\lambda }_2={\lambda }_4$. (c) The change $\mathrm{\Delta }P=P-P^{\prime }$ in the case of ${\lambda }_2={\lambda }_4$.

Figure 3

Control power vs dimensional parameter $d$ ($16\ge d\ge 2$) in optimal-state channels. (a) The CP for the controlled teleportation scheme (CP here is defined according to fidelity and dimensionless). $P_{q_1}$ (black line with square) is the CP for controlled teleporting a $d$-dimensional qudit by using the standard $d$-dimensional GHZ state. $P_{q_2}$ (red line with circle) is the CP for controlled teleporting a $d$-dimensional qudit by using the standard $2^N$-dimensional GHZ state ($2^N>d>2^{N-1}$). (b) The controlled error rate for the controlled CDC scheme. $P_{c_1}$ (black line with square) is the CP for controlled transmitting $2{\log }_{2}d$ bits of classical information by using the standard $d$-dimensional GHZ state. $P_{c_2}$ (red line with circle) is the CP for controlled transmitting $2{\log }_{2}d$ bits of classical information by using the standard $2^N$-dimensional GHZ state ($2^N>d>2^{N-1}$).