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

TL;DR

This paper introduces the $(n,k)$ Coset Monogamy Game, analyzing the maximum winning probabilities for players extracting complementary quantum information without communication, generalizing previous equal-information-size cases.

Contribution

It formulates a generalized $(n,k)$ game, derives a convex upper bound on winning probability based on the subspace rate, and proves optimality for unentangled strategies.

Findings

Derived a convex upper bound on winning probability as a function of $k/n$

Improved bounds over previous results for the case $k=n/2$

Established the achievability of the optimal bound for unentangled strategies

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.