A quantum cloning game with applications to quantum position verification

TL;DR

The optimal winning probability of a quantum cloning game, in which separate collaborative parties receive a classical input, is provided and it is shown that it decays exponentially when the game is played several times in parallel.

Abstract

We introduce a quantum cloning game in which $k$ separate collaborative parties receive a classical input, determining which of them has to share a maximally entangled state with an additional party (referee). We provide the optimal winning probability of such a game for every number of parties $k$, and show that it decays exponentially when the game is played $n$ times in parallel. These results have applications to quantum cryptography, in particular in the topic of quantum position verification, where we show security of the routing protocol (played in parallel), and a variant of it, in the random oracle model.

Figures (4)

• Figure 1: Schematic representation of the $k$-party quantum cloning game, where $\left|\Psi\right\rangle_{RP_x}=\left|\Phi^+\right\rangle_{RP_x}$. If $\left|\Psi\right\rangle_{RP_x}$ is arbitrary, this represents a $\Psi$-QCG$_k$.
• Figure 2: Schematic representation of the $n$-fold parallel repetition of the $2$-party quantum cloning game.
• Figure 3: Schematic representation of the $(H,n)$-routing QPV protocol. If $r_0$ is an empty bit string, and $x=r_1$, this figure represents the $n$-parallel repetition of the routing QPV protocol. The time arrow is represented by $t$.
• Figure 4: Schematic representation of a generic attack to the $(H,n)$-routing protocol (and in particular, to the routing protocol)

Theorems & Definitions (13)

• Definition 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 5
• Proposition 6
• Proposition 7
• Theorem 8
• Definition 9
• Lemma 10