Definition of bi-local measurement by Masanes et al.

[Sonderval]

Dear experts,
I'm currently working my way through the paper Masanes, Galley, Müller, https://arxiv.org/abs/1811.11060.
On page 3, they define what they call a bi-local measurement: If we have two systems a and b and we define an outcome probability function for some measurement f on system a and g on system b, the pair of measurements can be represented by a product
$$ (f \ast g) (\psi \otimes \phi) = f(\psi) g(\phi)$$
I find this very confusing because it seems to me to deny the possibility of entanglement: If the two states ##\psi## and ##\phi## are entangled (for example, two electrons entangled so that their spin is always the same), I think this statement does not hold anymore. (Probability for first electron to measure up could be 0.5, probability for second to measure down could also be 0.5, but combined probability would be zero.)
Probably I'm mis-interpreting something in the paper, but I have no idea where my mistake lies.
Any help is appreciated.

 

[atyy]

Would it make sense if a bi-local measurement is defined on unentangled (product) states, and that condition with the other postulates is sufficient to determine how a bi-local measurement behaves for entangled states?

Thus (?) to apply the bi-local measurement to an entangled state, one would write the entangled state as a sum of product states, then apply the definition to each product state.

Edit: If you look at Eq (16), it looks like the correct result for entangled states can be derived from their assumptions.

 

[Sonderval]

@atyy
Thanks. Yes, I suspect you're right and that this is what is more or less implied by the qualifier "local OPF", but at least to me it is not very clearly stated.