Sending classical information through relativistic quantum channels

We investigate how special relativity influences the transmission of classical information through quantum channels by evaluating the Holevo bound when the sender and the receiver are in (relativistic) relative motion. By using the spin degrees of freedom of spin-$1/2$ fermions to encode the classical information, we show that, for some configurations, the accessible information in the receiver can be increased when the spin detector moves fast enough. This is possible by allowing the momentum wave packet of one of the particles to be sufficiently wide while the momentum wave packets of other particles are kept relatively narrow. In this way, one can take advantage of the fact that boosts entangle the spin and momentum degrees of freedom of spin-$1/2$ fermions to increase the accessible information in the former. We close the paper with a discussion of how this relativistic quantum channel cannot in general be described by completely positive quantum maps.

Figure 1

The behavior of the Holevo bound when Bob is at rest with respect to Alice as a function of the angle $\theta$, which characterizes the spin part of the second state, for the transmission of both two (full line) and four (dashed line) classical bits.

Figure 2

The Holevo bound $\chi \left({\tau }^{\prime }\right)$ as a function of $\alpha =-{\tanh }^{-1}v$ for different values of the normalized momentum $K_1=k_1/m$. We have fixed $W_0=0.05,$ $W_1=6,$ $\lambda =1/2,$ $K_0=1$, and $\theta =\pi /8$. We can see that although $\chi \left({\tau }^{\prime }\right)$ decreases with $\alpha$ initially, it begins to increase for larger values of the rapidity, becoming larger than the value for $\alpha =0$ when $K_1$ is large enough.

Figure 3

The Holevo bound $\chi \left({\tau }^{\prime }\right)$ as a function of $W_1$ for different values of the angle $\theta$ when $\alpha \to \infty$. We have fixed $K_0=1$, $K_1=10$, $W_0=0.05$, and $\lambda =1/2.$ When $\theta$ is small and $W_1$ is large enough, the accessible information always increases when Bob moves fast enough with respect to Alice. When $\theta$ is larger (in particular, when it is closer to $\pi /2$), the increase in $W_1$ always leads to a decrease in $\chi \left({\tau }^{\prime }\right)$ and therefore in the accessible information.

Figure 4

The Holevo bound $\chi \left({\tilde{\tau }}^{\prime }\right)$ as a function of $\alpha =-{\tanh }^{-1}v$ for different values of the angle $\theta$. We have chosen $K_0=1$, $K_1=50$, $W_0=0.05$, $W_1=6$, and ${\lambda }_1={\lambda }_2={\lambda }_3={\lambda }_4=1/4.$ For smaller angles ($\theta =0,$ $\theta =\pi /10$, and $\theta =\pi /8$), although $\chi \left({\tilde{\tau }}^{\prime }\right)$ initially decreases with $\alpha$, it eventually increases as the rapidity gets larger, becoming larger than its value for $\alpha =0$ when $\alpha \to \infty$.

Figure 5

${\Delta }_2\equiv \chi \left[\mathcal{E}\left(\tau \right)\right]-\chi \left(\tau \right)$ as a function of $\theta$ for different values of $\lambda$ when $\alpha \to \infty$. We have fixed $W_0=0.05,$ $W_1=6,$ $K_0=1$, and $K_1=50$. We can see that there are two ranges of angles in which ${\Delta }_2\ge 0$ and therefore the map $\mathcal{E}$ is not CP for angles within these ranges.

Figure 6

${\Delta }_4\equiv \chi \left[\mathcal{N}\left(\tilde{\tau }\right)\right]-\chi \left(\tilde{\tau }\right)$ as a function of $\theta$ for different values of ${\lambda }_1,{\lambda }_2,{\lambda }_3$ when $\alpha \to \infty$. We have fixed $W_0=0.05,$ $W_1=6,$ $K_0=1$, and $K_1=50$. We can see that there are two ranges of angles in which ${\Delta }_4\ge 0$ and therefore the map $\mathcal{N}$ is not CP for angles within these ranges.