Questions about HUP, entanglement and coherence

[San K]

HUP --> Heisenberg Uncertainty Principle

The below questions maybe a bit amateurish.

1. Can two photons be entangled on both position and momentum?...for a same time period

I think not because:

To entangle on momentum (or any property) we need a broad distribution of momentum for the photon. Same for position. However a photon cannot have a broad distribution for both (?).

2. why is a broad distribution (for the property on which the two photons will be entangled) needed for entanglement?

or in other words

why are we not able to violate bell inequalities when the distribution is narrow?

3. In a single particle, double-slit experiment, when we do which-way:

at that point do we have a broader distribution for momentum since the positions has been narrowed?

4. Most of the mystery lies in HUP it seems.
For example complimentarity is just an expression/corollary of HUP.

Position & momentum are entangled
Time & energy are entangled (?)

Would a dimension analysis, like the one below, throw some light on HUP?

Can one find similarities between position-momentum AND time-energy via dimensional analysis?

What position (meters) is to (kgs*(meters/sec) ---> position is to momentum

Time (sec) is to Energy (kg*(meters^2/sec) -----> Time is to energy

It does not seem to work. Dimension analysis does not shed any light on HUP.

4 b) are all complimentarities reducible to one complimentarity?

5. We say position and momentum are complimentary.

Since the velocity of the photon is always c, Is (relativistic) mass not constant?...in the case of a photon

Thus why can we not say position and velocity are complimentary?

6. when we narrow the position distribution, the momentum distribution must spread out.

Is there any analogy in classical mechanics?

7. For a single photon -- what does it mean to be incoherent (or coherent)?

I guess - for a single photon -- the wave-function traveling the various paths can be made incoherent and that would somehow make the single photon incoherent.

8. Coherency between two "self-coherent" photons means that they have a fixed phase relationship (?)

The photons are self coherent and photon A has a fixed phase relationship with photon B.

Does Coherency between two self-incoherent photons again mean there is a fixed phase relationship between photon A and photon B however photon A and photon B are "individually" self-incoherent?

9. In case of photon A being entangled with photon B:

the more entangled (and hence coherent?) photon A is with B

the less coherent it is with self.

In other words:

for A to be more coherent with B

A needs to be less coherent with itself...

what does being less coherent with itself mean?

 

[mfb]

1. Can two photons be entangled on both position and momentum?...for a same time period

Yes (the time does not matter here).

To entangle on momentum (or any property) we need a broad distribution of momentum for the photon. Same for position. However a photon cannot have a broad distribution for both (?).

It cannot have a narrow distribution for both, but broad distributions are easy to get.

2. why is a broad distribution (for the property on which the two photons will be entangled) needed for entanglement?

You need the possibility to get different measurement results somewhere to have something you could call "entangled".

why are we not able to violate bell inequalities when the distribution is narrow?

Bell inequalities can be shown with polarization alone, there is no need to have any relevant position/momentum distribution.

3. In a single particle, double-slit experiment, when we do which-way:
at that point do we have a broader distribution for momentum since the positions has been narrowed?

Broader than what? Not necessarily.

Position & momentum are entangled
Time & energy are entangled (?)

Not in the way entanglement is used in quantum mechanics.

Would a dimension analysis, like the one below, throw some light on HUP?

If the product of two quantities has the units of the Planck constant, it is usually connected to an uncertainty relation.

4 b) are all complimentarities reducible to one complimentarity?

What do you mean with "reducible"?

5. We say position and momentum are complimentary.
Since the velocity of the photon is always c, Is (relativistic) mass not constant?...in the case of a photon

Constant with respect to what?
Both momentum and energy can have some broad distribution.

Thus why can we not say position and velocity are complimentary?

Why do you think this should be possible?

6. when we narrow the position distribution, the momentum distribution must spread out.

Only if you reach the lower limit of the product of the uncertainties.

Is there any analogy in classical mechanics?

Uncertainty exists in classical mechanics, too. Entanglement does not.

7. For a single photon -- what does it mean to be incoherent (or coherent)?

The same as it means everywhere.

I guess - for a single photon -- the waves traveling the various paths can be made incoherent and that would somehow make the single photon incoherent.

Right.

8. Coherency between two "self-coherent" photons means that they have a fixed phase relationship (?)
The photons are self coherent and photon A has a fixed phase relationship with photon B.

'"self-coherent" photons'? Probably right, but I am not sure what you are asking here.

Does Coherency between two self-incoherent photons again mean there is a fixed phase relationship between photon A and photon B however photon A and photon B are "individually" self-incoherent?

How is this supposed to look like?

9. In case of photon A being entangled with photon B:
the more entangled (and hence coherent?) photon A is with B
the less coherent it is with self.
In other words:
for A to be more coherent with B
A needs to be less coherent with itself...
what does being less coherent with itself mean?

That does not make sense I think.

(I removed some empty lines in the quotes to make the post easier to read).

 

[San K]

well answered mfb. thanks for taking the time to respond.