Carrying an arbitrarily large amount of information using a single quantum particle

Theoretically speaking, a photon can travel arbitrarily long before it enters into a detector, resulting in a click. How much information can a photon carry? We study a bipartite asymmetric “two-way signaling” protocol as an extension of that proposed by Del Santo and Dakić. Suppose that Alice and Bob are distant from each other and each of them has an $n$-bit string. They are tasked to exchange the information of their local $n$-bit strings with each other, using only a single photon during the communication. It has been shown that the superposition of different spatial locations in a Mach-Zehnder (MZ) interferometer enables bipartite local encodings. We show that, after the travel of a photon through a cascade of $n$-level MZ interferometers in our protocol, the one of Alice and Bob whose detector clicks can access the other's full information of the $n$-bit string, while the other can gain one bit of information. That is, the wave-particle duality makes two-way signaling possible, and a single photon can carry an arbitrarily large (but finite) amount of information.

Figure 1

Optical implementation of the SD protocol. The photon is initially injected into the level-1 MZ interferometer. After traveling through the first $50:50$ beam splitter (BS1), the incident photon is half reflected and half transmitted in a coherent way. Alice and Bob can locally access the halves depicted in the red solid line and blue dashed line, respectively. For the local encoding at level 1, Alice inserts the $\pi$-phase modulator (PM A) if $x_1=0$ and does nothing if $x_1=1$; similarly, Bob inserts the $\pi$-phase modulators (PM B) if $y_1=0$ and does nothing if $y_1=1$. After the interference of these two coherent halves at the second $50:50$ beam splitter (BS2), the photon enters one of the two detectors [19].

Figure 2

Left: The optical details of a two-level evolution circuit. According to the local encoding at level 1, the leaving photon at BS2 is injected into one of the two fibers (F1 and F2), and enters one of these two MZ interferometers at level 2. Similarly, Alice (Bob) inserts PMs into these two MZ interferometers at level 2 if $x_2=1(y_2=1)$ and does nothing if $x_2=0(y_2=0)$. Right: The topological unfolding of the two-level circuit as a full two-level binary tree. The nodes therein denote the MZ interferometers, where the photon is spatially superposed, and the directed edges between nodes indicate the possible traveling paths of the photon.

Figure 3

The unfolding layout of the $n$-level circuit as a perfect $n$-level binary tree and the detectors. There are $2^{i-1}$ MZ interferometers at level $i$. The photon is initially injected into the level-1 MZ interferometer. Then the parity of the bit pair $(x_1,y_1)$ completely determines which one of the two level-2 MZ interferometers the photon will enter. Without loss of generality, let the photon go to the right MZ interferometer at level 2 if $x_1=y_1$, and the left MZ interferometer otherwise. More optical details are explained in Fig. 1. Similarly, the parity of $(x_2,y_2)$ determines the next target interferometers at level 3. This process is continued for a cascade of $n$ MZ interferometers. As a result, the light path of the photon completely depends on the $n$ bit pairs $(x_1,y_1)$, ..., $(x_n,y_n)$. Finally, the photon flies into one of the $2^n$ detectors, each of which is held by Alice (A) or Bob (B).

Figure 4

The time window in the $n=2$ case. (a) Assume that Alice holds the left two detectors ($D_1$ and $D_2$ in Fig. 2). Alice and Bob each performs the local encoding operation for $x_1$ at $t=0$ and for $y_1$ at $t=(d+\delta )/c$, respectively. Finally, the photon flies into one of the detectors accessible to Alice at time $(2d+\delta )/c\le \tau \le (2d+\delta )/c+\epsilon$. (b) In the same window $\tau$, Alice and Bob can only exchange some $x_j$ and $y_i$ by classical communication. (c) To finish the same task as in (a) using a classical information carrier, it requires a time window $3d/c\le {\tau }^{\prime }\le 3d/c+\epsilon$.

Figure 5

An effective circuit with two detectors. All the left paths have a time delay $\delta$, while the right paths have additional delays denoted by longer optical fibers.

Figure 6

The rate of success of our GSD protocol vs the number of circuit levels.