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For distinguishing quantum states sampled from a fixed ensemble, the gap inbipartite and single-party distinguishability can be interpreted as anonlocality of the ensemble. In this paper, we consider bipartite statediscrimination in a composite system consisting of $N$ subsystems, where eachsubsystem is shared between two parties and the state of each subsystem israndomly sampled from a particular ensemble comprising the Bell states. We showthat the success probability of perfectly identifying the state converges to$1$ as $N\rightarrow\infty$ if the entropy of the probability distributionassociated with the ensemble is less than $1$, even if the success probabilityis less than $1$ for any finite $N$. In other words, the nonlocality of the$N$-fold ensemble asymptotically disappears if the probability distributionassociated with each ensemble is concentrated. Furthermore, we show that thedisappearance of the nonlocality can be regarded as a remarkable counterexampleof a fundamental open question in theoretical computer science, called aparallel repetition conjecture of interactive games with two classicallycommunicating players. Measurements for the discrimination task include aprojective measurement of one party represented by stabilizer states, whichenable the other party to perfectly distinguish states that are sampled withhigh probability.

Bipartite discrimination of independently prepared quantum states as a counterexample to a parallel repetition conjecture