Limits on nonlocal correlations from the structure of the local state space

Abstract

The outcomes of measurements on entangled quantum systems can be nonlocally correlated. However, while it is easy to write down toy theories allowing arbitrary nonlocal correlations, those allowed in quantum mechanics are limited. Quantum correlations cannot, for example, violate a principle known as macroscopic locality, which implies that they cannot violate Tsirelson's bound. This work shows that there is a connection between the strength of nonlocal correlations in a physical theory, and the structure of the state spaces of individual systems. This is illustrated by a family of models in which local state spaces are regular polygons, where a natural analogue of a maximally entangled state of two systems exists. We characterize the nonlocal correlations obtainable from such states. The family allows us to study the transition between classical, quantum, and super-quantum correlations, by varying only the local state space. We show that the strength of nonlocal correlations -in particular whether the maximally entangled state violates Tsirelson's bound or not -depends crucially on a simple geometric property of the local state space, known as strong self-duality. This result is seen to be a special case of a general theorem, which states that a broad class of entangled states in probabilistic theories -including, by extension, all bipartite classical and quantum statescannot violate macroscopic locality. Finally, our results show that there exist models which are locally almost indistinguishable from quantum mechanics, but can nevertheless generate maximally nonlocal correlations.

An information-theoretic principle implies that any discrete physical theory is classical

“…Very illustrative examples are given by the polygon models 10 : these are abstract state spaces where O A is a regular polygon, so they are special kinds of discrete theories. As the whole situation can be drawn in only three dimensions, the polygon models provide examples for which we can give a picture (Fig.…”

“…3). To see the interplay of states and effects in such a low-dimensional example, it is useful to represent effects as vectors in the same space as the states 10 . To evaluate an effect at some state, one simply takes the scalar product of the state and the vector representing the effect.…”

“…First, equation (10) implies that u A ðTðoÞÞ ¼ f ðoÞ ¼ 0 for all o 2 F f (c.f. the definition (9) of F f ).…”

“…Now we look at the shape mismatch. If condition (b) is violated (that is, convðF f [ F f Þ 6 ¼ aff ðF f [ F f Þ \ O A ), then for every linear map that satisfies equations (10) and (11), there is a state r such that TðrÞ = 2 O 1 A (that is, TðrÞ is not a state). Therefore, such a T cannot be a transformation.…”

“…7). Equation (10) implies that the impossible face F f is mapped to the zero-vector, while equation (11) means that the certain face F f is left invariant. In the case of the pentagon illustrated in Fig.…”