Witnessing latent time correlations with a single quantum particle

When a noisy communication channel is used multiple times, the errors occurring at different times generally exhibit correlations. Classically, these correlations do not affect the evolution of individual particles: a single classical particle can only traverse the channel at a definite moment of time, and its evolution is insensitive to the correlations between subsequent uses of the channel. In stark contrast, here we show that a single quantum particle can sense the correlations between multiple uses of a channel at different moments of time. Taking advantage of this phenomenon, it is possible to enhance the amount of information that the particle can reliably carry through the channel. In an extreme example, we show that a channel that outputs white noise whenever the particle is sent at a definite time can exhibit correlations that enable a perfect transmission of classical bits when the particle is sent at a superposition of two distinct times. In contrast, we show that, in the lack of correlations, a single particle sent at a superposition of two times undergoes an effective channel with classical capacity of at most 0.16 bits. When multiple transmission lines are available, time correlations can be used to simulate the application of quantum channels in a coherent superposition of alternative causal orders, and even to provide communication advantages that are not accessible through the superposition of causal orders.

Figure 1

A single quantum particle can travel through a transmission line at a superposition of two different moments of time $t_1$ (red) and $t_2$ (blue). Along the way, the particle experiences errors (yellow region), and the errors occurring at time $t_1$ are generally correlated with the errors occurring at time $t_2$. By taking advantage of these correlations, the errors can be mitigated or even completely removed.

Figure 2

A single particle can travel on a superposition of two different paths (red and blue), which traverse two transmission lines (top and bottom) at two moments of time $t_1$ and $t_2$. The errors occurring on successive uses of the same transmission line are correlated (yellow lines), so the particle experiences correlated errors across the two branches (red and blue) of the superposition. These time correlations are a resource that can be used to mimic the use of quantum channels in a superposition of orders, and even to achieve larger communication advantages.

Figure 3

A single particle is sent at a superposition of two times (red and blue dashed lines), through the same transmission line (green ovals). The green dotted line represents the correlations between random unitary processes ${\mathcal{U}}_m$ and ${\mathcal{U}}_n$ taking place with probability $p(m,n)$ at the two subsequent uses of the transmission line, respectively.

Figure 4

Performance in the transmission of a single particle through a correlated depolarizing channel. (a) Classical capacity in the lack of correlations. Without loss of generality, ${\varphi }_0=0$. The maximum capacity is 0.016 bits. (b) Lower bound to the classical capacity achieved with the correlated probability distribution $p(m,n)={\delta }_{n,\sigma \left(m\right)}/4$, where $\sigma$ is the permutation that exchanges 0 with 1, and 2 with 3. Without loss of generality, we set ${\varphi }_0={\varphi }_2=0$. The maximum lower bound is 1 bit.

Figure 5

A single particle is sent through a superposition of two paths (orange and blue dashed lines), each traversing two independent channels (green and red ovals), each of which exhibits time correlations between successive uses. The green and red dotted lines represent the correlations between the two subsequent uses of the same channel.

Figure 6

Performance in the transmission of a single particle through a network of correlated depolarizing channels, arranged as in Fig. 5. (a) Classical capacity in the lack of correlations. Without loss of generality, ${\varphi }_0=0$. The maximum capacity is 0.018 bits. (b) Lower bound to the classical capacity achieved with maximal correlations corresponding to the probability distributions $p_A(m,n)=p_B(m,n)={\delta }_{n,\sigma \left(m\right)}/4$, where $\sigma$ is the permutation that exchanges 0 with 1, and 2 with 3. Without loss of generality, ${\varphi }_0={\varphi }_2=0$. The maximum lower bound is 0.31 bits.

Figure 7

Blue: Maximum classical capacity in the absence of correlations, as a function of the dephasing parameter $s$. The maximum is computed over all realizations of the completely depolarizing channel, and is achieved by the random unitary realization with the choice of phases $\left\{{\varphi }_m\right\}$ that give the maximum capacity of 0.16 bits when $s=0$. Orange: Lower bound to the maximal classical capacity in the presence of correlations, as a function of the dephasing parameter $s$. The lower bound is computed by considering the correlated probability distribution $p(m,n)={\delta }_{n,\sigma \left(m\right)}/4$, where $\sigma$ is the permutation that exchanges 0 with 1, and 2 with 3, and ${\varphi }_0={\varphi }_1={\varphi }_2=0,{\varphi }_3=\pi /2$.

Figure 8

The left-hand side depicts a two-step correlated quantum channel $\mathcal{B}$ taking two input states on systems $S^{\left(1\right)}$ and $S^{\left(2\right)}$, in succession. The right-hand side shows the physical implementation of the two-step channel via two unitary channels ${\mathcal{W}}_1$ and ${\mathcal{W}}_2$ [51, 52] where the memory between the two uses of the channel is realized by an environment $E$, which is inaccessible to the communicating parties.

Figure 9

(a) Transmission of a single particle through a bipartite quantum channel $\tilde{\mathcal{B}}$ (green). (b) Transmission of a single particle through a two-step quantum channel $\tilde{\mathcal{B}}$ (green). In both cases, the particle is represented by a composite system $M\otimes C$, where $M$ represents the degrees of freedom used as the message, and $C$ represents the degrees of freedom used as the control. The isomorphism $\mathcal{U}$ converts the composite system $M\otimes C$ into the one-particle sector $(S^{\left(1\right)}\otimes \mathrm{Vac})\oplus (\mathrm{Vac}\otimes S^{\left(2\right)})$ of ${\tilde{S}}^{\left(1\right)}\otimes {\tilde{S}}^{\left(2\right)}$. The inverse map ${\mathcal{U}}^{†}$ converts the output state back into $M\otimes C$. For the applications in this paper, we take the input of the control system $C$ to be fixed in the state $\omega$ while the message system $M$ is accessible to the sender.

Figure 10

The channel $\mathcal{Z}(\tilde{\mathcal{A}},\tilde{\mathcal{B}})$, obtained by connecting two vacuum-extended two-step channels $\tilde{\mathcal{A}}$ (green) and $\tilde{\mathcal{B}}$ (red), such that the output of the first use of each channel is connected to the input of the second use of the other channel.

Figure 11

The superposition channel $\mathcal{S}\left[\mathcal{Z}(\tilde{\mathcal{A}},\tilde{\mathcal{B}})\right]$ of two two-step channels $\mathcal{A}$ and $\mathcal{B}$, specified by the vacuum extensions $\tilde{\mathcal{A}}$ (green) and $\tilde{\mathcal{B}}$ (red), where the alternative paths traverse the two correlated channels in the opposite order. For the applications in this paper, the input of the control system $C$ is fixed in the state $\omega$, while the message system $M$ is accessible to the sender.

Figure 12

Green: A plot of the classical capacity against ${\left|\right|F\left|\right|}_{\infty }$ for the channel ${\mathcal{C}}_{\omega ,F}$. Red: A plot of the classical capacity against ${\left|\right|F\left|\right|}_{\infty }^2$ for the channel ${\mathcal{C}}_{\omega ,F^2}$. In both cases $F={\sum }_{m=0}^3\frac{1}{4}{e}^{-i{\varphi }_m}{V}_m$ and is sampled over the phase parameters $\{{\varphi }_1,{\varphi }_2,{\varphi }_3\}$ with a numerical precision of $\pi /8$ for each parameter. We set ${\varphi }_0=0$ without loss of generality, as $F\rho F^{†}$ is invariant under the phase group $\text{U}\left(1\right)$. The classical capacity is here equal to the Holevo capacity (see Appendix pp2) and the Holevo capacity was calculated using the methods outlined in Appendix pp3.