We study the nonlocal properties of states resulting from the mixture of an arbitrary entangled state ρ of two d-dimensional systems and completely depolarized noise, with respective weights p and 1 − p. We first construct a local model for the case in which ρ is maximally entangled and p at or below a certain bound. We then extend the model to arbitrary ρ. Our results provide bounds on the resistance to noise of the nonlocal correlations of entangled states. For projective measurements, the critical value of the noise parameter p for which the state becomes local is at least asymptotically log(d) larger than the critical value for separability.In 1964, Bell showed that some entangled states are nonlocal, in the sense that measurements on them yield outcome correlations that cannot be reproduced by a locally causal model [1]. This nonlocal character of entangled states may be demonstrated through the violation of Bell inequalities. All pure entangled states violate such an inequality, hence are nonlocal [2]. For noisy states, the picture is much subtler. Werner constructed in 1989 a family of bipartite mixed states, which, while being entangled, return outcome correlations under projective measurements that can be described by a local model [3]. This result has been extended to general measurements [4] and more parties [5]. Thus, while entanglement is necessary for a state to be nonlocal, in the case of mixed states it is not sufficient.Beyond these exploratory results, little is known about the relation between noise, entanglement, and quantum nonlocality. Understanding this relation, apart from its fundamental interest, is important from the perspective of Quantum Information Science. In this context, entanglement is commonly viewed as a useful resource for various information-processing tasks. Not all entangled states, however, are useful for every task: for example, quantum computation with slightly entangled states can be efficiently simulated on a classical computer [6], and bound entangled states are useless for teleportation [7]. For certain tasks, such as quantum communication complexity problems [8], or device-independent quantum key distribution [9], entangled states are useful only to the extent that they exhibit nonlocal correlations. Indeed, in these scenarios two (or more) distant observers, Alice and Bob, directly exploit the correlationsobtained by performing measurements M and N on a distributed entangled state ρ AB (in the above formula, M a and N b are the positive operators associated with the measurement outcomes a and b). If the entangled state ρ AB can be simulated by a local model, these correlations can be written aswhere λ denotes a shared classical variable distributed with probability measure µ, and P M (a|λ) and P N (b|λ) are the local response functions of Alice and Bob. For all practical purposes then, the entangled state ρ AB can be replaced by classical correlations, and so does not provide any improvement over what is achievable using classical resources [10]. In this work, we ...
