🤖 AI Summary
This paper investigates the optimal coordinated guessing problem for Bob and Charlie in a coset guessing quantum game under no real-time communication: Alice prepares a $2m$-qubit entangled state, distributes the first and last $m$ qubits to Bob and Charlie respectively, and announces one classical parameter; the players must each guess two hidden parameters, winning only if both guesses are correct. We derive, for the first time, a tight exponential upper bound $2^{-m}$ on the winning probability and prove its achievability. We reveal that the quantum advantage stems from establishing response correlations—not from enhancing individual guessing accuracy. Furthermore, we design a local encoding circuit using only CNOT and Hadamard gates that efficiently maps EPR pairs into the CSS code space, achieving globally optimal performance. Our results demonstrate that, under bounded classical coordination, the optimal strategy is fully implementable via local operations alone, and the exponential decay rate $2^{-m}$ is information-theoretically optimal and cannot be improved.
📝 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 an optimal strategy, we devised an encoding circuit using only CNOT and Hadamard gates, which builds CSS codes from EPR pairs using only local operations. 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.