Sending classical information via three noisy channels in superposition of causal orders

In this work, we study the transmission of classical information through three completely depolarizing channels in superposition of different causal orders. We exploit the quantum 3-switch as a resource for classical communication. We perform a kind of quantum control that was not accessible to the previously treated two-channel case. The fine and full quantum control achieved using selected combinations of causal orders let us uncover several features: nonmonotonous behavior on the transmission of information with respect to the number of causal orders involved, and different values of the transmission of information depending on the specific combinations of causal orders considered. Our results are a stepping stone to assess the efficiency of coherent quantum control and optimize resources in the implementation of indefinite causal structures. Finally, we suggest an optical implementation using standard telecom technology to test our predictions.

Figure 1

Tripartite quantum switch. In the quantum switch with three parties, $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$, there are $\left(\genfrac{}{}{0pt}{}{3!}{m}\right)$ (with $m=1,2,...,6$ ) available superpositions of causal orders. Here we represent $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ as three different quantum channels, ${\mathcal{N}}_1$, ${\mathcal{N}}_2$, and ${\mathcal{N}}_3$, respectively. ${\rho }_c=\left|1\right.〉\left.〈1\right|$ encodes a causal order ${\mathcal{N}}_1\circ {\mathcal{N}}_2\circ {\mathcal{N}}_3$, i.e., ${\mathcal{N}}_3$ is applied first to the target system $\rho$; ${\rho }_c=\left|2\right.〉\left.〈2\right|$ encodes ${\mathcal{N}}_1\circ {\mathcal{N}}_3\circ {\mathcal{N}}_2$; ${\rho }_c=\left|3\right.〉\left.〈3\right|$ encodes ${\mathcal{N}}_2\circ {\mathcal{N}}_1\circ {\mathcal{N}}_3$; ${\rho }_c=\left|4\right.〉\left.〈4\right|$ encodes ${\mathcal{N}}_2\circ {\mathcal{N}}_3\circ {\mathcal{N}}_1$; ${\rho }_c=\left|5\right.〉\left.〈5\right|$ encodes ${\mathcal{N}}_3\circ {\mathcal{N}}_1\circ {\mathcal{N}}_2$; ${\rho }_c=\left|6\right.〉\left.〈6\right|$ encodes ${\mathcal{N}}_3\circ {\mathcal{N}}_2\circ {\mathcal{N}}_1$; and finally, $\rho ={\left|+\right.〉}_m{\left.〈+\right|}_m$, where ${\left|+\right.〉}_m=\frac{1}{\sqrt{m}}{\sum }_{k=1}^m\left|k\right.〉$ is a superposition of $m$ different causal orders. (a) For $m=6$ causal orders, there is only 1 combination to superimpose six definite causal orders. (b) For $m=5$ causal orders, there are $\left(\genfrac{}{}{0pt}{}{3!}{5}\right)=6$ combinations to superimpose five definite causal orders. (c) For $m=4$ causal orders, there are $\left(\genfrac{}{}{0pt}{}{3!}{4}\right)=15$ combinations to superimpose four definite causal orders. (d) For $m=3$ causal orders, there are $\left(\genfrac{}{}{0pt}{}{3!}{3}\right)=20$ combinations to superimpose three definite causal orders. (e) For $m=2$ causal orders, there are $\left(\genfrac{}{}{0pt}{}{3!}{2}\right)=15$ combinations to superimpose two definite causal orders. (f) For $m=1$ causal order, depending on ${\rho }_c$, we have $3!$ possibilities to combine the channels in a definite causal order. Panels (b) to (e) illustrate a single partial superposition among all possible combinations of causal orders of Table 3, see main text.

Figure 2

Superposition of $m$ causal orders. Holevo information ${\chi }_{\text{Q3S}}$ as the number of causal orders $m$ involved in ${\rho }_c$ is varied, for dimensions $d=2$ (blue) and $d=3$ (red) of $\rho$. The Holevo information can take two different values, ${\chi }^{\text{max}}$ and ${\chi }^{\text{min}}$, when there is a superposition of $m=2$, 3, and 4 causal orders, depending on the chosen causal orders in the superposition. ${\chi }^{\text{max}}$ is indicated by the top end of each color bar (for each $d=2$ and 3) and ${\chi }^{\text{min}}$ is given by the position of the clear horizontal line bar with a colored centered dot. The numerical values of ${\chi }^{\text{max}}$ and ${\chi }^{\text{min}}$ are given in Table 1. For a fixed number of $m$ causal orders, the Holevo information decreases as $d$ increases. Note that for superposition of two causal orders, ${\chi }^{\max }$ is equal to the value of Holevo information of two channels ${\chi }_{\text{Q2S}}$. See main text for explanation. The two triangles correspond to the values obtained in Ref. [14].

Figure 3

Optical proposal for the quantum $N$-switch channel. In both proposals, a frequency-delocalized single photon with $N!$ frequencies is injected as an input to a series of wavelength division multiplexers and demultiplexers (WDMs). Each frequency is used to route the order of operation of the channels. At the end of the quantum switch, the frequencies are coherently multiplexed into the output mode. All color lines represent optical links which connect the WDMs and the channels ${\mathcal{N}}_j$. (a) For the quantum 2-switch, if the frequency is on mode 1 (black), the order to apply the channels will be ${\mathcal{N}}_2\circ {\mathcal{N}}_1$. On the other hand, if the frequency is on mode 2 (blue), the order will be ${\mathcal{N}}_1\circ {\mathcal{N}}_2$. (b) For the quantum 3-switch, if the frequency is on mode 1 (black), the order to apply the channels will be ${\mathcal{N}}_3\circ {\mathcal{N}}_2\circ {\mathcal{N}}_1$. If the frequency is on mode 2 (blue), the order will be ${\mathcal{N}}_1\circ {\mathcal{N}}_3\circ {\mathcal{N}}_2$. If the frequency is on mode 3 (red), the order will be ${\mathcal{N}}_2\circ {\mathcal{N}}_1\circ {\mathcal{N}}_3$. If the frequency is on mode 4 (purple), the order will be ${\mathcal{N}}_2\circ {\mathcal{N}}_3\circ {\mathcal{N}}_1$. If the frequency is on mode 5 (yellow), the order will be ${\mathcal{N}}_3\circ {\mathcal{N}}_1\circ {\mathcal{N}}_2$. Finally, if the frequency is on mode 6 (green), the order will be ${\mathcal{N}}_1\circ {\mathcal{N}}_2\circ {\mathcal{N}}_3$. Sending, in both cases (a) and (b), a single photon in a superposition of modes (i.e. frequencies) will make it experience all causal orders simultaneously. Note that our optical proposal can be seen as the implementation for the architecture proposed in Ref. [11]. We propose frequency encoding using off-the-shelf and mature telecom components.