Does quantum entanglement depend on the chosen basis?

In summary: Please correct me if I am wrong.In the first example the observables under consideration are either polarization in 0 and 90-degree direction (relative to an arbitrary direction in the plane perpendicular to the photons' momenta) vs. polarization in ##\pm 45^{\circ}##. In the latter case the state is obviously entangled, while it's not in the former.This is what I meant.
  • #36
vanhees71 said:
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?
 
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  • #37
atyy said:
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.
atyy said:
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.
 
  • #38
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.
 
<h2>1. What is quantum entanglement?</h2><p>Quantum entanglement is a phenomenon in which two or more particles become connected in such a way that the state of one particle is dependent on the state of the other, regardless of the distance between them.</p><h2>2. How does quantum entanglement work?</h2><p>Quantum entanglement occurs when two or more particles are created or interact in a way that their properties become correlated. This means that measuring the state of one particle will instantly determine the state of the other particle, even if they are separated by large distances.</p><h2>3. Does quantum entanglement depend on the chosen basis?</h2><p>Yes, quantum entanglement does depend on the chosen basis. The basis refers to the set of properties or measurements used to describe the state of a particle. The entanglement between two particles can only be observed if the chosen basis is compatible with the entangled state of the particles.</p><h2>4. Why is the chosen basis important in quantum entanglement?</h2><p>The chosen basis is important in quantum entanglement because it determines which properties of the particles are entangled. If the chosen basis is not compatible with the entangled state of the particles, the entanglement will not be observed.</p><h2>5. Can quantum entanglement be used for communication?</h2><p>No, quantum entanglement cannot be used for communication. While entangled particles can instantaneously affect each other's states, this effect cannot be used to transmit information faster than the speed of light. This is due to the no-communication theorem in quantum mechanics.</p>

1. What is quantum entanglement?

Quantum entanglement is a phenomenon in which two or more particles become connected in such a way that the state of one particle is dependent on the state of the other, regardless of the distance between them.

2. How does quantum entanglement work?

Quantum entanglement occurs when two or more particles are created or interact in a way that their properties become correlated. This means that measuring the state of one particle will instantly determine the state of the other particle, even if they are separated by large distances.

3. Does quantum entanglement depend on the chosen basis?

Yes, quantum entanglement does depend on the chosen basis. The basis refers to the set of properties or measurements used to describe the state of a particle. The entanglement between two particles can only be observed if the chosen basis is compatible with the entangled state of the particles.

4. Why is the chosen basis important in quantum entanglement?

The chosen basis is important in quantum entanglement because it determines which properties of the particles are entangled. If the chosen basis is not compatible with the entangled state of the particles, the entanglement will not be observed.

5. Can quantum entanglement be used for communication?

No, quantum entanglement cannot be used for communication. While entangled particles can instantaneously affect each other's states, this effect cannot be used to transmit information faster than the speed of light. This is due to the no-communication theorem in quantum mechanics.

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