Trade-off capacities of the quantum Hadamard channels

Coding theorems in quantum Shannon theory express the ultimate rates at which a sender can transmit information over a noisy quantum channel. More often than not, the known formulas expressing these transmission rates are intractable, requiring an optimization over an infinite number of uses of the channel. Researchers have rarely found quantum channels with a tractable classical or quantum capacity, but when such a finding occurs, it demonstrates a complete understanding of that channel’s capabilities for transmitting classical or quantum information. Here we show that the three-dimensional capacity region for entanglement-assisted transmission of classical and quantum information is tractable for the Hadamard class of channels. Examples of Hadamard channels include generalized dephasing channels, cloning channels, and the Unruh channel. The generalized dephasing channels and the cloning channels are natural processes that occur in quantum systems through the loss of quantum coherence or stimulated emission, respectively. The Unruh channel is a noisy process that occurs in relativistic quantum information theory as a result of the Unruh effect and bears a strong relationship to the cloning channels. We give exact formulas for the entanglement-assisted classical and quantum communication capacity regions of these channels. The coding strategy for each of these examples is superior to a naïve time-sharing strategy, and we introduce a measure to determine this improvement.

Figure 1

Plot of (a) the CQ trade-off curve and (b) the CE trade-off curve for a $p$-dephasing qubit channel for $p=0$, $0.1$, $0.2,\dots ,0.9$, $1$. The trade-off curves for $p=0$ correspond to those of a noiseless qubit channel and are the rightmost trade-off curve in each plot. The trade-off curves for $p=1$ correspond to those for a classical channel, and are the leftmost trade-off curves in each plot. Each trade-off curve between these two extremes beats a time-sharing strategy, but these two extremes do not beat time sharing.

Figure 2

Plot of (a) the CQ trade-off curve and (b) the CE trade-off curve of a $1\to N$ cloning channel for $N=1$, $2$, $3$, $5$, $8$, $12$, and $24$. The trade-off curves for $N=1$ correspond to those for the noiseless qubit channel and are the rightmost trade-off curves in each plot. In both plots, proceeding left from the $N=1$ curve, we obtain trade-off curves for $N=2$, $3$, $5$, $8$, $12$, and $24$ and notice that they all beat a time-sharing strategy by a larger relative proportion when $N$ increases.

Figure 3

The figure plots the CQE capacity region for a $1\to 10$ cloning channel. It features three distinct surfaces. The first is the flat vertical plane that arises from the bound $R+2Q⩽\log {}_2(\frac{N+1}{N})+1$, which is the entanglement-assisted classical capacity of a $1\to N$ cloning channel. The plane extends infinitely upward because we can always achieve these rate triples simply by wasting entanglement. The second surface is that below and to the left of the plane, formed by combining the CE trade-off curve with the inverse of superdense coding, as described in Sec. 3c. The final surface is that below and to the right of the plane, formed by combining the CQ trade-off curve with the inverse of entanglement distribution, as described in Sec. 3c.

Figure 4

Plot of the (a) CQ trade-off curve and (b) the CE trade-off curve of an Unruh channel for $z=0$, $0.2$, $0.4$, $0.6$, $0.8$, and $0.95$. The trade-off curves for $z=0$ correspond to those for the noiseless qubit channel and are the rightmost trade-off curves in each plot. In both plots, proceeding left from the $z=0$ curve, we obtain trade-off curves for $z=0.2$, $0.4$, $0.6$, $0.8$, and $0.95$ and notice that they all beat a time-sharing strategy by a larger relative proportion as $z$ increases.

Figure 5

The figure plots the CQE capacity region for an Unruh channel with acceleration parameter $z=0.95$. It features three distinct surfaces. The first is the flat vertical plane that arises from the bound $R+2Q⩽{\mathcal{C}}_{\mathrm{EAC}}$, where ${\mathcal{C}}_{\mathrm{EAC}}$ is the entanglement-assisted classical capacity of an Unruh channel. The plane extends infinitely upward because we can always achieve these rate triples simply by wasting entanglement. The second surface is that below and to the left of the plane, formed by combining the CE trade-off curve with the inverse of superdense coding, as described in Sec. 3c. The final surface is that below and to the right of the plane, formed by combining the CQ trade-off curve with the inverse of entanglement distribution, as described in Sec. 3c.

Figure 6

The figures plot (a) the relative gain $G_{\mathrm{CQ}}$ for the CQ trade-off curve and (b) the relative gain $G_{\mathrm{CE}}$ for the CE trade-off curve for the dephasing channel, the cloning channel, and the Unruh channel. The figures plot these relative gains as a function of the dephasing parameter $p\in [0,1]$ for the $p$-dephasing qubit channel (bottom horizontal axis), as a function of the acceleration parameter $z\in [0,1]$ for the Unruh channel (bottom horizontal axis), and as a function of the number of clones$\hspace{0.5em}N$ for the $1\to N$ cloning channel (top horizontal axis). The plot on the left demonstrates that the relative gain $G_{\mathrm{CQ}}$ for the cloning channels is best as $N$ increases, while the Unruh channel features an improved relative gain over a dephasing channel for large accelerations. The plot on the right features different behavior—the relative gain $G_{\mathrm{CE}}$ of the dephasing channel outperforms that for the Unruh channel if we consider the parameters $p$ and $z$ on equal footing, in spite of their drastically different physical interpretations.