Optimal Strategies for Winning Certain Coset-Guessing Quantum Games

TL;DR

The paper analyzes a coset-guessing game where a $2m$-qubit coset state is split between Bob and Charlie after Alice reveals a subspace $W$. It derives a tight upper bound $p_m = \Theta(2^{-m})$ on the probability that Bob and Charlie simultaneously guess the coset parameters, and constructs an explicit optimal strategy based on a Hadamard-CNOT encoding circuit that realizes a CSS-code–like representation of the coset states. The key insight is that quantum information primarily correlates Bob and Charlie’s responses rather than enhancing their marginal guessing rates, with entanglement either locally removable (reducing to a classical game) or managed via Bell-pair decompositions to preserve optimal joint success. The work connects coset-state monogamy to CSS codes and provides circuit-level constructions that may inform CSS-code implementations and related quantum information tasks, while offering a foundation for broader asymptotic analyses and potential cryptographic implications.

Abstract

In a recently introduced coset guessing game, Alice plays against Bob and Charlie, aiming to meet a joint winning condition. Bob and Charlie can only communicate before the game starts to devise a joint strategy. The game we consider begins with Alice preparing a 2m-qubit quantum state based on a random selection of three parameters. She sends the first m qubits to Bob and the rest to Charlie and then reveals to them her choice for one of the parameters. Bob is supposed to guess one of the hidden parameters, Charlie the other, and they win if both guesses are correct. From previous work, we know that the probability of Bob's and Charlie's guesses being simultaneously correct goes to zero exponentially as m increases. We derive a tight upper bound on this probability and show how Bob and Charlie can achieve it. While developing the optimal strategy, we devised an encoding circuit using only CNOT and Hadamard gates, which could be relevant for building efficient CSS-coded systems. We found that the role of quantum information that Alice communicates to Bob and Charlie is to make their responses correlated rather than improve their individual (marginal) correct guessing rates.

Figures (2)

• Figure 1: Examples of coset states for $m = 1$.
• Figure 2: An illustration of the considered coset guessing game.

Theorems & Definitions (19)

• theorem 1
• proof
• lemma 1
• lemma 2
• lemma 3
• proposition 1
• proof
• proposition 2
• proof
• proposition 3