Does quantum entanglement depend on the chosen basis?

[zonde]

First of all, "subsystem" can also refer to two (compatible) quantities for a single particle (as in the example of the SG experiment, where one spin component, usually $\sigma_z$, and position are compatible observables),

I'm not sure I understand your definition of "subsystem". In SG experiment you measure only position. If spin is another "subsystem" how do you even perform it's measurement?

 

[martinbn]

It's probably clearer to say "measurement setup" than "basis".

Well, the thread is about whether the definition of entanglement depends on the choice of basis.

You can also find comments in section 1.2.4 of https://arxiv.org/abs/1302.4654.

A dissertation at the Budapest University of Technology and Economics is hardly the definitive reference. Nevertheless, in section 1.2.4 they do not discuss dependence on basis. They discuss the factorization of the Hilbert space into a product of two spaces. My guess is that you are confusing that with the choice of basis.

 

[vanhees71]

Well, I think that the confusion is due to the fact that there seem to be the two slightly different definitions of "entanglement". Having read a bit in the nice review paper by the Horodecki family,

https://doi.org/10.1103/RevModPhys.81.865
https://arxiv.org/abs/quant-ph/0702225

I come to the conclusion that indeed the stronger assumption that a state is considered on-entangled (separable) iff the state doesn't factorize into a direct product, $\hat{\rho} = \hat{\rho}_A \otimes \hat{\rho}_B$ (for the case that one consideres the factorization in only two distinct subsystems (bipartite entanglement)). This is a basis-independent statement, but it's not a simple task to figure out for a given state whether it's entangled or not (except for a pure state for the entire system as discussed in #30). The Horodeckis seem to be the experts having investigated this question in great detail and for the most general cases. So the above cited RMP article seems to be pretty authorative.

BTW I'd say that a dissertation from a well-respected university can be considered a definitive reference since it's usually as much peer reviewed (if not with more care) as a usual peer-reviewed paper in a well-respected scientific journal. In the case of the dissertation quoted above, this is the more credible since it seems to be based on several publications in peer-reviewed well-respected scientific journals.