Speaking about entanglement I find it helpful to look at experiments.
In particular I like this one, even so it's far from more sophisticated experiments that are using state of the art equipment. That's because it goes into more basic details about entanglement experiment.
https://arxiv.org/abs/quant-ph/0205171
In particular for me equation (6) gives helpful insights:
[tex]|\psi_{DC}\rangle = \cos\theta_{l}|H\rangle_{s}|H\rangle_{i}+\exp[i\phi]\sin\theta_{l}|V\rangle_{s}|V\rangle_{i}[/tex]
Given that φ in the experiment can be smoothly adjusted in a quite straight forward way, that way one can experimentally change the state continuously between fully entangled state and completely "classical" product state.
For me it hints that entaglement can be dissected into local parts and a non-local part - a phase factor φ.

Is it just me or anybody else see this as insightful?

zonde said:

Given that φ in the experiment can be smoothly adjusted in a quite straight forward way, that way one can experimentally change the state continuously between fully entangled state and completely "classical" product state. For me it hints that entaglement can be dissected into local parts and a non-local part - a phase factor φ.

I have not read the paper. I just see your equation. It represents a rotation about the z-axis on the Bloch sphere, and every value of ϕ corresponds to a valid quantum state. The parameter ϕ is simply the relative phase between the first and second terms of the state vector, and it has nothing to do with the distinction between local and nonlocal properties.

anuttarasammyak said:

I have not read the paper. I just see your equation. It represents a rotation about the z-axis on the Bloch sphere, and every value of ϕ corresponds to a valid quantum state. The parameter ϕ is simply the relative phase between the first and second terms of the state vector, and it has nothing to do with the distinction between local and nonlocal properties.

Oh, but look at the paper. ϕ represents tilt of a quartz plate in pump beam that does not lead to any locally observable changes in either of the two arms of the experiment. While changes in other terms in contrast lead to observable consequences in local statistics.

zonde said:

tilt of a quartz plate in pump beam that does not lead to any locally observable changes in either of the two arms of the experiment

What is your basis for this claim? Please be specific about exactly what equations or text in the paper you are using.

zonde said:

Given that φ in the experiment can be smoothly adjusted in a quite straight forward way, that way one can experimentally change the state continuously between fully entangled state and completely "classical" product state.

No value of ##\phi## makes the state a classical product state. Perhaps you meant the ##\theta_l##? But if for some reason the ##\phi## is completely unknown, then the knowledge of the state is described by a mixed state, which under certain conditions can be interpreted as a classical statistical mixture.

PeterDonis said:

What is your basis for this claim? Please be specific about exactly what equations or text in the paper you are using.

I went over the paper. It seems the relevant statement is in one of the sources of the paper: https://arxiv.org/abs/quant-ph/9810003

ϕ is determined by the details of the phase-matching and the crystal thickness, but can be adjusted by tilting the BBO crystals themselves (but this changes the cones’ opening angles), by imposing a birefringent phase shift on one of the output beams, or by controlling the relative phase between the horizontal and vertical components of the pump light.

This statement that phase shift can be adjusted in one of the output beams indirectly indicates that there can't be locally observable changes as otherwise it would allow for FTL communication (you could send FTL signal by adjusting phase in one arm and observing local changes in the other arm).

Demystifier said:

No value of ##\phi## makes the state a classical product state. Perhaps you meant the ##\theta_l##? But if for some reason the ##\phi## is completely unknown, then the knowledge of the state is described by a mixed state, which under certain conditions can be interpreted as a classical statistical mixture.

Fair. Let me correct myself. When value of ##e^{i\phi}## is i or -i observed statistics will be identical to classical product state as interference between ##H_aH_b## and ##V_aV_b## polarization modes will disappear (given analyzers are measuring linear polarization).