In this work we study rank-one quantum games. In particular, we focus on the study of the computability of the entangled value $omega^*$. We show that the value $omega^*$ can be efficiently approximated up to a multiplicative factor of 4. We also stu
dy the behavior of $omega^*$ under the parallel repetition of rank-one quantum games, showing that it does not verify a perfect parallel repetition theorem. To obtain these results, we first connect rank-one games with the mathematical theory of operator spaces. We also reprove with these new tools essentially known results about the entangled value of rank-one games with one-way communication $omega_{qow}$. In particular, we show that $omega_{qow}$ can be computed efficiently and it satisfies a perfect parallel repetition theorem.
We show that Tsirelsons problem concerning the set of quantum correlations and Connes embedding problem on finite approximations in von Neumann algebras (known to be equivalent to Kirchbergs QWEP conjecture) are essentially equivalent. Specifically,
Tsirelsons problem asks whether the set of bipartite quantum correlations generated between tensor product separated systems is the same as the set of correlations between commuting C*-algebras. Connes embedding problem asks whether any separable II$_1$ factor is a subfactor of the ultrapower of the hyperfinite II$_1$ factor. We show that an affirmative answer to Connes question implies a positive answer to Tsirelsons. Conversely, a positve answer to a matrix valued version of Tsirelsons problem implies a positive one to Connes problem.
In this letter we show that the field of Operator Space Theory provides a general and powerful mathematical framework for arbitrary Bell inequalities, in particular regarding the scaling of their violation within quantum mechanics. We illustrate the
power of this connection by showing that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order $frac{sqrt{n}}{log^2n}$ when observables with n possible outcomes are used. Applications to resistance to noise, Hilbert space dimension estimates and communication complexity are given.
In this work we show that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order $sqrt{n}$ (up to a logarithmic factor) when observables with n possible outcomes are used. A central tool in th
e analysis is a close relation between this problem and operator space theory and, in particular, the very recent noncommutative $L_p$ embedding theory. As a consequence of this result, we obtain better Hilbert space dimension witnesses and quantum violations of Bell inequalities with better resistance to noise.
