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تسجيل مستخدم جديدA synchronous game for binary constraint systems
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Recently, W. Slofstra proved that the set of quantum correlations is not closed. We prove that the set of synchronous quantum correlations is not closed, which implies his result, by giving an example of a synchronous game that has a perfect quantum approximate strategy but no perfect quantum strategy. We also exhibit a graph for which the quantum independence number and the quantum approximate independence number are different. We prove new characterisations of synchronous quantum approximate correlations and synchronous quantum spatial correlations. We solve the synchronous approximation problem of Dykema and the second author, which yields a new equivalence of Connes embedding problem in terms of synchronous correlations.
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We introduce a game related to the $I_{3322}$ game and analyze a constrained value function for this game over various families of synchronous quantum probability densities.
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In a recent paper, the concept of synchronous quantum correlation matrices was introduced and these were shown to correspond to traces on certain C*-algebras. In particular, synchronous correlation matrices arose in their study of vario
We unify and consolidate various results about non-signall-ing games, a subclass of non-local two-player one-round games, by introducing and studying several new families of games and establishing general theorems about them, which extend a number of
known facts in a variety of special cases. Among these families are {it reflexive games,} which are characterised as the hardest non-signalling games that can be won using a given set of strategies. We introduce {it imitation games,} in which the players display linked behaviour, and which contains as subclasses the classes of variable assignment games, binary constraint system games, synchronous games, many games based on graphs, and {it unique} games. We associate a C*-algebra $C^*(mathcal{G})$ to any imitation game $mathcal{G}$, and show that the existence of perfect quantum commuting (resp. quantum, local) strategies of $mathcal{G}$ can be characterised in terms of properties of this C*-algebra, extending known results about synchronous games. We single out a subclass of imitation games, which we call {it mirror games,} and provide a characterisation of their quantum commuting strategies that has an algebraic flavour, showing in addition that their approximately quantum perfect strategies arise from amenable traces on the encoding C*-algebra. We describe the main classes of non-signalling correlations in terms of states on operator system tensor products.
The Data Processing Inequality (DPI) says that the Umegaki relative entropy $S(rho||sigma) := {rm Tr}[rho(log rho - log sigma)]$ is non-increasing under the action of completely positive trace preserving (CPTP) maps. Let ${mathcal M}$ be a finite dim
ensional von Neumann algebra and ${mathcal N}$ a von Neumann subalgebra if it. Let ${mathcal E}_tau$ be the tracial conditional expectation from ${mathcal M}$ onto ${mathcal N}$. For density matrices $rho$ and $sigma$ in ${mathcal N}$, let $rho_{mathcal N} := {mathcal E}_tau rho$ and $sigma_{mathcal N} := {mathcal E}_tau sigma$. Since ${mathcal E}_tau$ is CPTP, the DPI says that $S(rho||sigma) geq S(rho_{mathcal N}||sigma_{mathcal N})$, and the general case is readily deduced from this. A theorem of Petz says that there is equality if and only if $sigma = {mathcal R}_rho(sigma_{mathcal N} )$, where ${mathcal R}_rho$ is the Petz recovery map, which is dual to the Accardi-Cecchini coarse graining operator ${mathcal A}_rho$ from ${mathcal M} $ to ${mathcal N} $. In it simplest form, our bound is $$S(rho||sigma) - S(rho_{mathcal N} ||sigma_{mathcal N} ) geq left(frac{1}{8pi}right)^{4} |Delta_{sigma,rho}|^{-2} | {mathcal R}_{rho_{mathcal N}} -sigma|_1^4 $$ where $Delta_{sigma,rho}$ is the relative modular operator. We also prove related results for various quasi-relative entropies. Explicitly describing the solutions set of the Petz equation $sigma = {mathcal R}_rho(sigma_{mathcal N} )$ amounts to determining the set of fixed points of the Accardi-Cecchini coarse graining map. Building on previous work, we provide a throughly detailed description of the set of solutions of the Petz equation, and obtain all of our results in a simple self, contained manner.
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