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We construct a hidden variable model for the EPR correlations using a Restricted Boltzmann Machine. The model reproduces the expected correlations and thus violates the Bell inequality, as required by Bell's theorem. Unlike most hidden-variable models, this model does not violate the locality assumption in Bell's argument. Rather, it violates measurement independence, albeit in a decidedly non-conspiratorial way.

In quantum theory, there can exist correlations between subsystems of a new kind that are absent in classical systems. These correlations are nowadays called "entanglement".

An entanglement measure is a functional on states quantifying the amount of entanglement across two subsystems (i.e. causally disjoint regions in the context of quantum field theory). A reasonable measure should satisfy certain general properties: for example, it should assign zero entanglement to separable states, and be monotonic under separable, completely positive maps ("LOCC-operations"). The v. Neumann entropy of the "reduced state" (to one of the subsystems) is one such measure if the state for the total system is pure. But for mixed states, it is not, and one has to consider other measures. In particular, one has to consider other measures if the subsystems have a finite non-zero distance.

In this talk I will present several good measures, and in particular analyze the « relative entanglement entropy", $E_R$, defined as the "distance" of the given state to the set of separable states, where "distance" is defined using Araki's relative entropy. I will show several features of this measure for instance: (i) charged states, where the relative entanglement entropy is related to the quantum dimension of the charge, (ii) vacuum states in 1+1 dimensional integrable models, (iii) general upper bounds for certain special regions in general CFTs in d dimensions, (iv) area law type bounds. I will also explain the relationship between $E_R$ and other entanglement measures, such as distillable entropy.

[Based on joint work with Jacobus Sanders.]