Understanding Tsirelson's Problem: Comparing Two Models

[Bashyboy]

Hello,

I am currently reading about Tsirelson's problem, and have a few questions. I hope you can give me your pardon, if they seem ill-posed or misinformed,

Here is what I understand so far--or at least I think I understand! Tsirelson's problem seems to be about whether two mathematicals models yield the same correlation functions. The physical situation being modeled is that of two independent observers (Alice and Bob) conducting measurements on the same physical system, which is commonly modeled using a tensor product of Hilbert spaces, each factor corresponding to an observer conducting measurements on a physical system being measured. The other model used is some bigger Hilbert space and the assumption that all the observables of Alice commute with all of Bob's.

My first question is, how equivalent are these two models, irrespective of whether they yield the same correlation functions? It would seem that they have to agree in some respects otherwise we would reject one model as incorrect.

My next question is, what if the system consists of entangled particles, or does "independent observers" exclude this possibility? If there is no entanglement, why wouldn't the two observers measure the same things? If they did make the same exact measurements, would that imply they have the same Hilbert spaces...yes? no?

Again, please be gracious of my ignorance. I would really like to understand this problem. Thank you!

 

[eljose79]

Thank you for your interest in Tsirelson's problem and for your questions. I am happy to clarify some of the points you have mentioned.

To answer your first question, the two models used in Tsirelson's problem are not equivalent. In fact, they are fundamentally different in their assumptions and mathematical representations. The model using the tensor product of Hilbert spaces is based on the concept of local realism, where the measurements made by Alice and Bob are independent and do not influence each other. On the other hand, the model using a single, larger Hilbert space assumes the presence of non-local correlations between the measurements made by Alice and Bob. These two models are not interchangeable and cannot be compared in terms of equivalence.

As for your second question, the concept of independent observers does not exclude the possibility of entanglement. In fact, entanglement is an essential aspect of quantum mechanics and is present in many physical systems. However, in the context of Tsirelson's problem, the two observers are considered independent in the sense that their measurements are not influenced by each other. This means that even if the system consists of entangled particles, the measurements made by Alice and Bob are still considered independent.

If the two observers make the exact same measurements, it does not necessarily imply that they have the same Hilbert spaces. The mathematical representation of a Hilbert space is not unique, and there can be different Hilbert spaces that yield the same measurement outcomes. Therefore, having the same measurements does not necessarily imply the same Hilbert spaces.

I hope this helps clarify some of your questions. If you have any further inquiries, please do not hesitate to ask. Thank you for your interest and I wish you all the best in understanding Tsirelson's problem.