Parastatistics and a secret communication challenge

TL;DR

This work identifies a distinctive physical signature of $R$-parastatistics by formulating a secret-communication game that can only be won when paraparticles are present in the host topological phase. It provides two exactly solvable quantum-spin models in 2D and 3D where emergent paraparticles realize nontrivial $R$-matrices, and shows that participants can encode and transfer information solely through braiding-like exchanges of spatial positions. The winning strategy is shown to be robust against local noise and eavesdropping, offering an operational definition and experimental identity test for paraparticles. The authors also recast the construction in the symmetric fusion category framework, clarifying universal properties and identifying the 3D phases capable of supporting the game via a categorical criterion.

Abstract

One of the most unconventional features of topological phases of matter is the emergence of quasiparticles with exotic statistics, such as non-Abelian anyons in two dimensional systems. Recently, a different type of exotic particle statistics that is consistently defined in any dimension, called $R$-parastatistics, is also shown to be possible in a special family of topological phases. However, the physical significance of emergent parastatistics still remains elusive. Here we demonstrate a distinctive physical consequence of parastatistics by proposing a challenge game that can only be won using physical systems hosting paraparticles, as passing the challenge requires the two participating players to secretly communicate in an indirect way by exploiting the nontrivial exchange statistics of the quasiparticles. The winning strategy using emergent paraparticles is robust against noise, as well as the most relevant class of eavesdropping via local measurements. This provides both an operational definition and an experimental identity test for paraparticles, alongside a potential application in secret communication.

Figures (6)

• Figure 1: Illustration of the game process. Here we draw the 2D version for simplicity; in 3D, the circles become spheres. The two circles have radius $r_0$ defined by the players. During the game the Referees move the two circles along their respective paths to complete an exchange of positions. The two far separated points ${o},{s}$ are chosen by the players, while the two paths are chosen by the Referees.
• Figure 2: Illustration of the winning strategy using emergent paraparticles. ${o}$ and ${s}$ are chosen to be the two special points where a paraparticle can be locally created and measured, as described in properties (5) and (6). The dashed lines represent the paths traversed by the circles, along which the players have applied unitary operators $\hat{U}_{ij}$ in Eq. \ref{['eq:paraparticlemove']}.
• Figure S3: The 2D solvable spin model with open boundary conditions (figure adapted from Ref. wang2023para). Each black dot represents a 16 dimensional qudit, and each open circle represents a 64 dimensional qudit. Each colored triangle represents a 3-body interaction between qudits on its 3 vertices. In addition, there is an eight-body interaction term around every gray plaquette $\nu$ and every white plaquette $p$. Emergent paraparticles are created by matrix product string operators, an example of such an operator $\hat{\psi}_{i,a}^+$ is given in Eq. \ref{['eq:JWT_string_2D']}, which acts consecutively on all the purple dots on and near the purple string $\Gamma$.
• Figure S4: Properties of the emergent paraparticles in the 2D solvable model. (a) A two-particle state $|G;i^a j^b\rangle\equiv \hat{\psi}^+_{i,a}\hat{\psi}^+_{j,b}\ket{G}$. A paraparticle at position $i$ with internal state $a$ is created by $\hat{\psi}^+_{i,a}$, a string operator connecting site $i$ to the corner ${o}$. Importantly, the action of $\hat{\psi}^+_{i,a}$ on $\ket{G}$ does not depend on the shape of the string, implying that the Referees cannot detect the presence of such a string operator using any local measurement far from $i$ (e.g., any local measurements in the orange circles); (b) Paraparticles can also be locally created at ${s}$, using $\hat{U}^{\prime}_{{s},b}$, since the string operator $\hat{W}_{bc}$ in $\hat{\psi}^+_{i,a}$ in Eq. \ref{['eq:JWT_string_2D']} connecting ${o}$ and ${s}$ acts trivially on $\ket{G}$.
• Figure S5: Definition of the vertex term $\hat{A}_v$ and the plaquette term $\hat{B}_p$ in Eq. \ref{['eq:3DLGT']}.