Undergrad Does quantum entanglement depend on the chosen basis?

[martinbn]

The notation on these slides don't make sense, as becomes clear if the dimension of the two Hilbert spaces are not the same.

Their notation does make sense. That's what the theorem they use says. https://en.wikipedia.org/wiki/Schmidt_decomposition

 

[martinbn]

Basis-dependent definitions:
https://arxiv.org/abs/quant-ph/0308043
https://arxiv.org/abs/quant-ph/0305023
https://arxiv.org/abs/quant-ph/0506099
http://arxiv.org/abs/1302.3509

Can you write down one definition, from these papers, that is basis dependent. I only leafed through the papers, but couldn't see anything of the sort.

 

[vanhees71]

Their notation does make sense. That's what the theorem they use says. https://en.wikipedia.org/wiki/Schmidt_decomposition

Ok, I misunderstood the statement in the slides.

 

[microsansfil]

Basis-dependent definitions:
https://arxiv.org/abs/quant-ph/0308043
https://arxiv.org/abs/quant-ph/0305023
https://arxiv.org/abs/quant-ph/0506099
http://arxiv.org/abs/1302.3509

Thank for the link. Now, just like Martinbn, I don't see where it is explicitly developed that an entangled pure state can be Basis-dependent.

The concept of entanglement is much more difficult to capture for a mixed state then for a pure state. Further, the mixed-state decomposition is not unique. Moreover, in realistic systems, a pure state inevitably falls into a mixed state.

If i understand the definition of entanglement given by vanhees71 : a pure state ρ ∈ H_A⊗H_B is entangled if and only if the reduced state ρ_A is not pure. Equivalently, ρ is separable if and only if ρ_A is pure. Here ρ_A is the reduced state defined as ρ_A≡Tr_Bρ. However, this definition says nothing about whether or not a entanglement state could be Basis-dependent.

/Patrick

 

[atyy]

Thank for the link. Now, just like Martinbn, I don't see where it is explicitly developed that an entangled pure state can be Basis-dependent.

The concept of entanglement is much more difficult to capture for a mixed state then for a pure state. Further, the mixed-state decomposition is not unique. Moreover, in realistic systems, a pure state inevitably falls into a mixed state.

If i understand the definition of entanglement given by vanhees71 : a pure state ρ ∈ H_A⊗H_B is entangled if and only if the reduced state ρ_A is not pure. Equivalently, ρ is separable if and only if ρ_A is pure. Here ρ_A is the reduced state defined as ρ_A≡Tr_Bρ. However, this definition says nothing about whether or not a entanglement state could be Basis-dependent.

It's probably clearer to say "measurement setup" than "basis". Anyway, the Sasaki et al reference is pretty close to what vanhees71 has been saying (as he himself noted). They give explicit examples in section V. You can also find comments in section 1.2.4 of https://arxiv.org/abs/1302.4654.

 

[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.