Channel capacity enhancement with indefinite causal order

Classical communication capacity of a channel can be enhanced either through a device called a “quantum switch” or by putting the channel in a quantum superposition. The gains in the two cases, although different, have their origin in the use of a quantum resource, but is it the same resource? Here this question is explored through simulating large sets of random channels. We find that quantum superposition always provides an advantage, while the quantum switch does not: it can either increase or decrease communication capacity. The origin of this discrepancy can be attributed to a subtle combination of superposition and noncommutativity.

Figure 1

Quantum circuit representing the quantum switch. Target system ${\rho }_{in}^t$ passes through ${\mathcal{C}}_0\circ {\mathcal{C}}_1$ and ${\mathcal{C}}_1\circ {\mathcal{C}}_0$ in a superposition determined by control qubit $c$. (a) The control is in the reduced state ${|0\rangle \langle 0|}^c$ and the target passes only through ${\mathcal{C}}_0\circ {\mathcal{C}}_1$. (b) The control is in the reduced state ${|+\rangle \langle +|}^c$ and the target passes through a balanced superposition of ${\mathcal{C}}_0\circ {\mathcal{C}}_1$ and ${\mathcal{C}}_1\circ {\mathcal{C}}_0$.

Figure 2

Quantum circuit representing the one-pass superposition of ${\mathcal{C}}_0$ and ${\mathcal{C}}_1$. Target system ${\rho }_{in}^t$ passes through ${\mathcal{C}}_0$ and/or ${\mathcal{C}}_1$ in a superposition determined by control qubit ${\rho }^c$. (a) The control is in the reduced state ${|0\rangle \langle 0|}^c$ and the target passes only through ${\mathcal{C}}_0$. (b) The control is in the reduced state ${|+\rangle \langle +|}^c$ and the target passes through a balanced superposition of ${\mathcal{C}}_0$ and ${\mathcal{C}}_1$.

Figure 3

Comparison of Holevo capacities of the quantum switch and the one-pass superposition. Each point corresponds to a couple of randomly generated quantum channels. Holevo capacities are estimated numerically.

Figure 4

Comparison of Holevo capacities of the quantum switch and the one-pass superposition for different implementations of 50 randomly generated channels. Each line corresponds to 200 different implementations of channel $\mathcal{C}$ due to the freedom in the Kraus decomposition. $\chi (\mathcal{C}⊞\mathcal{C})$ depends on the implementation, while $\chi (\mathcal{C}\bowtie \mathcal{C})$ is independent of the implementation.

Figure 5

Comparison of Holevo capacities of self-switch and self-superposition of 1000 random channels. Holevo capacity always increases in the latter case but no such regularity exists in the former. The $+$ sign stands alternatively for $\bowtie$ or $⊞$.

Figure 6

Comparison of Holevo capacities of the quantum switch and the one-pass superposition of a random channel $\mathcal{C}$ combined with a totally depolarizing channel $\mathcal{N}$. Each point corresponds to a random channel $\mathcal{C}$. Dotted lines are linear functions with the slopes 1 and 1/2. The $+$ sign stands alternatively for $\bowtie$ or $⊞$.

Figure 7

Relative advantage of the quantum switch $\chi (\mathcal{C}\bowtie \mathcal{C})/\chi \left(\mathcal{C}\right)$ as a function of $Q\left(\mathcal{C}\right)$, degree of commutativity of the Kraus decomposition, on a set of 1000 random channels. While the quantum switch can gain or lose communication capacity depending on $\mathcal{C}$, it does so more often at higher values of $Q\left(\mathcal{C}\right)$.

Figure 8

Advantage $\chi (\mathcal{C}\bowtie \mathcal{C})/\chi (\mathcal{C}\circ \mathcal{C})$ as a function of $Q\left(\mathcal{C}\right)$, the degree of commutativity of the Kraus decomposition. If the Kraus operators nearly commute, then the Holevo capacities are similar. If they do not commute, then the effect of the indefinite causal order gets stronger.