Winning Rates of $(n,k)$ Quantum Coset Monogamy Games

TL;DR

The paper investigates the coset-monogamy trade-off when two parties must extract complementary information from a random coset state, generalizing prior equal-information settings to a subspace rate $R=k/n$. It derives a convex, information-theoretic upper bound on the winning rate, showing $\omega(R) \le 2^{-rac{1}{2}R^*}$ with $R^* = \min\{R,1-R\}$, by applying operator-norm techniques to a structured sum over subspaces, aided by a novel collection of orthogonal subspace permutations. It also fully characterizes the optimal unentangled strategies, giving $p_{opt}^{u}(n,k)=\frac{1}{N}\sum_{m=0}^k 2^{m^2} \binom{n-k}{m}_2 \binom{k}{m}_2 2^{-m} = O(2^{- ext{min}\{k,n-k\}})$ and showing achievability with a simple deterministic scheme and a GHZ-like eigenstate. Together, these results reveal that the optimal winning probability decays exponentially with $n$ for all $R$, extend previous $R=1/2$ bounds, and illuminate the role of monogamy-of-entanglement constraints in coset-state based cryptographic scenarios, while also noting limitations for cryptographic uncloneability.

Abstract

We formulate the $(n,k)$ Coset Monogamy Game, in which two players must extract complementary information of unequal size ($k$ bits vs. $n-k$ bits) from a random coset state without communicating. The complementary information takes the form of random Pauli-X and Pauli-Z errors on subspace states. Our game generalizes those considered in previous works that deal with the case of equal information size $(k=n/2)$. We prove a convex upper bound of the information-theoretic winning rate of the $(n,k)$ Coset Monogamy Game in terms of the subspace rate $R=\frac{k}{n}\in [0,1]$. This bound improves upon previous results for the case of $R=1/2$. We also prove the achievability of an optimal winning probability upper bound for the class of unentangled strategies of the $(n,k)$ Coset Monogamy Game.

Figures (1)

• Figure 1: A representation of the game dynamics for the $(n,k)$ Coset Monogamy Game. Alice chooses parameters $W \in \text{Gr}_2(n,k)$, $x,z \in \mathbb{F}_2^n$ uniformly at random and prepares the coset state $\ket{W_{x,z}}$. She sends the coset state over the quantum channel $\Phi$ to Bob and Charlie before publicly broadcasting the subspace choice $W \in \text{Gr}_2(n,k)$. Bob and Charlie measure their subsystems with POVMs to determine guesses for $x$ and $z$ up to cosets of $W$ and $W^\perp$, respectively.

Theorems & Definitions (11)

• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 1
• proof
• Theorem 2