Relativistic (2,3)-threshold quantum secret sharing

In quantum secret sharing protocols, the usual presumption is that the distribution of quantum shares and players’ collaboration are both performed inertially. Here we develop a quantum secret sharing protocol that relaxes these assumptions wherein we consider the effects due to the accelerating motion of the shares. Specifically, we solve the (2,3)-threshold continuous-variable quantum secret sharing in noninertial frames. To this aim, we formulate the effect of relativistic motion on the quantum field inside a cavity as a bosonic quantum Gaussian channel. We investigate how the fidelity of quantum secret sharing is affected by nonuniform motion of the quantum shares. Furthermore, we fully characterize the canonical form of the Gaussian channel, which can be utilized in quantum-information-processing protocols to include relativistic effects.

Figure 1

Basic Building block for an arbitrary trajectory. The world lines of the left and right walls of the cavity are depicted. In region I, the cavity is inertial. In region II, the two walls of the cavity are accelerating with two different proper accelerations until the Rindler coordinate time $\eta =\frac{\tau }{a}$, where $\tau$ and $a$ are proper time and acceleration respectively. In region III, the cavities have stopped accelerating and are back in the inertial frame again. The hyperbolas (red curves) represent the trajectories of the cavity walls moving with constant proper acceleration, and the (black) straight lines correspond to the trajectories of the walls while they move inertially.

Figure 2

(a) The BBB is depicted for the case wherein the first mode of the cavity is used to encode and decode quantum information. We assume all the other modes are initially prepared in vacuum and after the BBB (which is represented by the symplectic transformation $S$) the rest of the modes are ignored. (b) The operations performed in part (a) are all Gaussian operations which enables us to express the BBB as a Gaussian channel $\mathcal{E}$ acting on the first and second moments as given in Eq. (13).

Figure 3

The canonical form of the BBB Gaussian channel, $\mathcal{E}$, which is decomposed into its canonical form, ${\mathcal{E}}_c$, up to two symplectic transformations in regions I and III, i.e., $S_{\mathrm{I}}$ and $S_{\mathrm{III}}$.

Figure 4

The second-order coefficient of the transmissivity of the BBB channel, $T^{(2)}$, as a function of $u$ for modes $k=1$, 2, 3.

Figure 5

The average number of thermal particles as a function of $u$ for modes $k=1$, 2, 3.

Figure 6

The encoding circuit for continuous-variable (2,3)-threshold quantum secret sharing. The “8” shaped symbol represents a two-mode squeezed-vacuum state. The upper two modes are combined on a balanced beam splitter. The three outputs are three quantum shares, denoted by mode 1, mode 2, and mode 3.

Figure 7

The worldlines of the quantum shares during distribution. From $t=0$ to $t=t_a$, the two cavities, represented by the furthest left and the furthest right worldlines, accelerate with the proper acceleration $a$ in two opposite directions. From $t=t_a$ to $t=t_a+t_i$, they move with constant velocities. From $t=t_a+t_i$ to $t=2t_a+t_i$, the two cavities decelerate with the proper acceleration $a$ and become stationary. The cavity represented by the middle world line remains static.

Figure 8

The two curves represent two worldlines in spacetime. Each worldline is the trajectory of one cavity carrying a quantum share. From $t=2t_a+t_i$ to $t=3t_a+t_i$, the two cavities accelerate with proper acceleration $a$ towards each other. From $t=3t_a+t_i$ to $t=3t_a+2t_i$, the two cavities are moving with constant velocity. From $t=3t_a+2t_i$ to $t=4t_a+2t_i$, the two cavities decelerate with proper acceleration $a$ to arrive at the same spacetime point.

Figure 9

(a) The thermal lossy channel ${\mathcal{E}}_1$ is the Gaussian channel that represents the total evolution of the first and the second quantum shares during the distribution and the collaboration stage. Then the quantum secret is decoded using a balanced beam splitter. (b) ${\mathcal{E}}_1$ is a single-mode Gaussian channel composed of five Gaussian channels in series. $\mathcal{E}({\tau }_a)$ is the Gaussian channel for a BBB during the proper time ${\tau }_a$ and $G({\tau }_i)$ represents the Gaussian channel of the free evolution in an inertial frame with proper time ${\tau }_i$.

Figure 10

$F^{(2)}$ as a function of $u$ for modes $k=1$ (solid), 2 (dashed), and 3 (dotted) when the secret Gaussian state is a coherent state.

Figure 11

$F^{(2)}$ as a function of $u$ for the ground mode ($k=1$), when the secret Gaussian state is a squeezed-vacuum state for squeezing parameters $r=\frac{1}{16}$ (solid), $\frac{1}{8}$ (dashed), and $\frac{1}{4}$ (dotted).

Figure 12

The two curves represent two worldlines in spacetime. The left worldline is the trajectory of the cavity carrying the third quantum share and the right worldline is the trajectory of the cavity carrying the second quantum share. From $t=2t_a+t_i$ to $t=4t_a+2t_i$, the third cavity remains static. From $t=2t_a+t_i$ to $t=3t_a+t_i$, the second cavity accelerates with proper acceleration $a$. From $t=3t_a+t_i$ to $t=3t_a+2t_i$, it moves with constant velocity and from $t=3t_a+2t_i$ to $t=4t_a+2t_i$, decelerates with proper acceleration $a$.

Figure 13

(a) The decoding circuit for the case wherein players 2 and 3 collaborate. ${\mathcal{E}}_2$ is a Gaussian thermal lossy channel. $G(2t_a+t_i)$ is the free evolution in the inertial frame. First, the two modes are combined on a beam splitter with reflectivity $2/3$. Then the quadrature $\widehat{q}$ of the second output mode is measured and a displacement operation controlled by the measurement outcome and a squeezing operation are applied on the first output mode. (b) ${\mathcal{E}}_2$ is a single-mode Gaussian channel composed of three Gaussian channels in series.

Figure 14

$F^{(2)}$ as a function of $u$ for modes $k=1$ (solid), 2 (dashed), and 3 (dotted) when the two-mode squeezing parameter, $s=1$.