Cluster decomposition and EPR correlations

In summary, the conversation discusses the compatibility of Weinberg's 'cluster decomposition' principle with entangled states and EPR-type correlations. The cluster decomposition principle states that if multi-particle processes in distant laboratories are studied, then the S-matrix element for the overall process factorizes. However, this formulation is incompatible with EPR correlations. The correct form of the principle is that if the initial state can be factorized and the subsystems remain spatially separated, then the final state can also be factorized. This is a consequence of QFT and decoherence. Further discussions on this topic can be found in the paper "http://xxx.lanl.gov/abs/hep-th/0011258".
  • #1
metroplex021
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0
Does anyone know of a good discussion of the compatibility (or otherwise) of Weinberg's 'cluster decomposition' principle with the fact that entangled states yield distant but correlated measurements?
 
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  • #3
Salman2 said:
Here is a link to the topic from Weinberg's 1995 book--found on internet:

http://books.google.com/books?id=3w...page&q=weinberg cluster decomposition&f=false

Thanks for this - a useful link for future reference. But for now I'm looking specifically for discussion of CD and its relation to EPR-type correlations of distant measurement results, and to my knowledge, Weinberg doesn't discuss that here (which seems a strange oversight to me).
 
  • #4
The cluster decomposition principle is an interpretation of the factorization of the S matric for separated reaction :
if [tex]S_{\alpha_1,\beta_1}[/tex] corresponds to the amplitude for [tex]\alpha_1\rightarrow\beta_1[/tex]
and [tex]S_{\alpha_2,\beta_2}[/tex] corresponds to the amplitude for [tex]\alpha_2\rightarrow\beta_2[/tex]
then [tex]S_{\alpha_1\alpha_2,\beta_1\beta_2}=S_{\alpha_1,\beta_1}S_{\alpha_2,\beta_2}[/tex]
Each label indicates a specification for all particles in the initial (final) state, including momenta, spins, particle species, and anything else relevant to fully specify a particle state.

Now in an EPR-type experiment, we do not have independent reactions. We have only one final state which is not separable. Since we already cannot separate the state in QM, we have no reason to attempt to separate it in QFT and hope to get a sensible result. The problem is not with the cluster decomposition principle. The problem is with the thought experiment itself. It is generally necessary to assume that all relevant quantities are measured both in the prepared initial and the detected final state.
 
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  • #5
I think that the cluster decomposition principle (CDP) IN THE FORM EXPRESSED BY WEINBERG is wrong.

Indeed, in Sec. 4.3 Weinberg explicitly says:
"It is one of the fundamental principles of physics ... that experiments that are sufficiently separated in space have unrelated results."
...
"... the cluster decomposition principle states that if multi-particle processes ... are studied in N very distant laboratories, then the S-matrix element for the overall process factorizes."

Clearly, these statements formulated as such are incompatible with EPR correlations, and are therefore wrong.

Yet, it does not mean that CDP cannot be reformulated in a more careful way, such that it becomes compatible with EPR correlations. After all, a form of the CDP principle can be derived from QFT, and QFT is compatible with EPR correlations. Probably, CDP can be viewed as a consequence of QFT & decoherence, but I am not aware of any detailed discussion of that idea.
 
  • #7
humanino said:
The cluster decomposition principle is an interpretation of the factorization of the S matric for separated reaction :
if [tex]S_{\alpha_1,\beta_1}[/tex] corresponds to the amplitude for [tex]\alpha_1\rightarrow\beta_1[/tex]
and [tex]S_{\alpha_2,\beta_2}[/tex] corresponds to the amplitude for [tex]\alpha_2\rightarrow\beta_2[/tex]
then [tex]S_{\alpha_1\alpha_2,\beta_1\beta_2}=S_{\alpha_1,\beta_1}S_{\alpha_2,\beta_2}[/tex]
Each label indicates a specification for all particles in the initial (final) state, including momenta, spins, particle species, and anything else relevant to fully specify a particle state.

Now in an EPR-type experiment, we do not have independent reactions. We have only one final state which is not separable. Since we already cannot separate the state in QM, we have no reason to attempt to separate it in QFT and hope to get a sensible result. The problem is not with the cluster decomposition principle. The problem is with the thought experiment itself. It is generally necessary to assume that all relevant quantities are measured both in the prepared initial and the detected final state.
I would summarize and formalize it this way:
CDP says that if
1. the initial state (of spatially separated subsystems) can be factorized
and
2. the subsystems remain spatially separated all the time
then
the final state can also be factorized.

This is a correct form of CDP in QFT. But this is not the form explicitly stated by Weinberg.
 
  • #8
Demystifier said:
This is a correct form of CDP in QFT. But this is not the form explicitly stated by Weinberg.
There are quite some caveats with scattering theory alone, and one must assume that all relevant indices are measured or otherwise summed over. That is assumed when we say that we prepare an initial state. It is not sufficient to use half of the final state of an EPR-type experiment, which would be inseparable from another half somewhere else. In this situation, once the initial state (half final state of an EPR exp.) has been measured it becomes separated and the CDP applies.
 
  • #9
Yes, that is essentially what I had in mind when I mentioned decoherence in post #5.
 
  • #10
Thanks everyone for those replies. I'm glad it's not just me who thinks it's a bit odd. I'm going to keep looking for somewhere that Weinberg himself addresses (what seems to me) his strange wording of this principle. But any other refs would be really appreciated of course.
 

1. What is cluster decomposition in quantum mechanics?

Cluster decomposition is a fundamental principle in quantum mechanics that states that the expectation value of a product of observables in a quantum system can be written as the sum of expectation values of individual observables. This means that the behavior of a quantum system can be understood by studying the behavior of its individual components.

2. How does cluster decomposition relate to entanglement?

Cluster decomposition is closely related to the concept of entanglement in quantum mechanics. Entanglement refers to the phenomenon where two or more particles become linked in such a way that the state of one particle cannot be described without considering the state of the other particle. Cluster decomposition allows us to understand how entanglement arises in quantum systems and how it affects the behavior of these systems.

3. What are EPR correlations and their significance?

EPR correlations, also known as Einstein-Podolsky-Rosen correlations, refer to the correlations between two distant particles that are entangled with each other. These correlations were first proposed by Albert Einstein, Boris Podolsky, and Nathan Rosen to demonstrate the incompleteness of quantum mechanics. They have since been experimentally verified and are now recognized as a fundamental aspect of quantum mechanics.

4. How are EPR correlations related to cluster decomposition?

EPR correlations are closely related to cluster decomposition as they both involve the study of correlations between entangled particles. Cluster decomposition helps us understand the nature of these correlations and how they arise in quantum systems. It also allows us to make predictions about the behavior of entangled particles based on the properties of individual particles.

5. Can cluster decomposition and EPR correlations be used for quantum communication?

Yes, both cluster decomposition and EPR correlations have important applications in quantum communication. For example, cluster decomposition can be used to understand and improve the transmission of quantum information through noisy channels. EPR correlations, on the other hand, have been used to develop protocols for secure communication, such as quantum key distribution, which relies on the measurement of EPR correlations to ensure the security of transmitted information.

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