What are the four bell states and how do they relate to entanglement swapping?

  • Context: Graduate 
  • Thread starter Thread starter StevieTNZ
  • Start date Start date
  • Tags Tags
    Measurements
Click For Summary

Discussion Overview

The discussion revolves around the four Bell states and their relationship to entanglement swapping, particularly in the context of photon polarization states. Participants explore the mathematical representation of these states and the implications of measuring certain photon pairs in specific bases.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether detecting photons 2 and 3 in specific states leads to projecting photons 1 and 4 into the same Bell state.
  • Another participant discusses the allowed bases in the 45/135 polarization and expresses uncertainty about how these calculations are performed.
  • A different participant proposes a GHZ state based on the detection of photons and suggests specific combinations in the 45/135 basis.
  • One participant introduces a shorthand notation for the polarization states and elaborates on the resulting terms when expanding the GHZ state, noting the cancellation and addition of terms based on the number of C and D terms.
  • Another participant speculates on the overall GHZ state after measuring in the Bell state and expresses uncertainty about the correct representation.
  • One participant reiterates the assumption about detecting specific states in the 45/135 basis and prompts further discussion on the reasoning behind this assumption.

Areas of Agreement / Disagreement

Participants express various hypotheses and calculations regarding the Bell states and GHZ states, but there is no consensus on the final representation or the implications of the measurements. Multiple competing views remain regarding the calculations and interpretations of the states.

Contextual Notes

Participants exhibit uncertainty about the calculations involved in determining the GHZ state and the implications of measuring in different bases. There are also unresolved mathematical steps and dependencies on specific definitions of the polarization states.

StevieTNZ
Messages
1,944
Reaction score
837
In entanglement swapping:

If we detect photons 2 and 3 in the following states |135>|45> or |45>|135> (accompanied by the appropriate coincidence counts), do we project photons 1 and 4 into the same bell-state (so photons 1 and 4 are also found in either |135>|45> or |45>|135>)?

How would we write this four photon GHZ state in terms of |V> and |H> polarisation? Would it be |H>|V>|H>|V> (+ OR -?) |V>|H>|V>|H>?

http://en.wikipedia.org/wiki/Bell_state
We have a list of the four bell states.
Taking the first, do we expand as:
|A>|B> = |0>|1> + |1>|0> ? Which would be |H>|V> + |V>|H>?
 
Physics news on Phys.org
Looking over it being |V>|H>|V>|H> - |H>|V>|H>|V>:

Would the following basis in 45/135 be the only ones allowed:
|45>|135>|45>|135>;
|135>|135>|45>|45>;
|135>|45>|135>|45>;
|45>|45>|135>|135>

In all honesty I don't know how these are calculated. Two H's expanded in |135> make a +, because they're both - 135.
Do we expand over |V>|H>|V>|H> then minus that result by what we get for |H>|V>|H>|V>? Or is it all done in one go?
 
Okay, I think I've figured it out. Appropriately detecting 2 and 3 in |45>|135> etc to create that bell state, and throwing 1 and 4 into the same bell state we get this GHZ state:
|V>|H>|H>|V> - |H>|V>|V>|H>

which allows the following combinations in 45/135 basis:
|45>|135>|45>|45>
|135>|135>|45>|135>
|135>|45>|135>|135>
|45>|45>|135>|45>

EDIT: corrected last |..> to say |45> rather than |135>
 
Let's write |45> as C for short and |135> as D.
So then |H\rangle=\frac{1}{\sqrt{2}}(C+D) and |V\rangle=\frac{1}{\sqrt{2}}(C-D)
Now |V_1\rangle|H_2\rangle|H_3\rangle|V_4\rangle - |H_1\rangle|V_2\rangle|V_3\rangle|H_4\rangle=
=\frac{1}{4}(C_1-D_1)(C_2+D_2)(C_3+D_3)(C_4-D_4)-\frac{1}{4}(C_1+D_1)(C_2-D_2)(C_3-D_3)(C_4+D_4)
That results in 32 terms where 8 term will cancel with other 8 terms and 8 terms will add with last 8 terms. So you will have 8 terms like that:
-D_1C_2C_3C_4+C_1D_2C_3C_4+C_1C_2D_3C_4-D_1D_2D_3C_4-C_1C_2C_3D_4+D_1D_2C_3D_4+D_1C_2D_3D_4-C_1D_2D_3D_4

As you can see these are terms with odd number of C and D. If you would swap - for + in the initial expression you would get terms with even number of C and D.
 
StevieTNZ said:
I would assume in this bell-state measuring in 45/135 basis, we would detect |45>|135> or |135>|45>
Why do you think so?
 

Similar threads

Undergrad Entanglement swapping in the context of Indivisible Stochastic Process
  • · Replies 2 ·
Replies
2
Views
1K
Graduate What happens when entangled photons are split into different paths?
  • · Replies 41 ·
2
Replies
41
Views
6K
Graduate Is delayed choice remote entanglement of photons derived from TDSE?
  • · Replies 58 ·
2
Replies
58
Views
5K
Graduate How to Measure Entangled Photons in the 45/135 Basis with Wave Plates and PBS?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad What is CV Bell State Measurement and How Does Entanglement Swapping Work?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad After entanglement swapping, how can we verify the new entanglement?
  • · Replies 5 ·
Replies
5
Views
2K
High School Why do photon polarization experiments show similar outcomes in different situations?
  • · Replies 11 ·
Replies
11
Views
3K
Undergrad What happens to entanglement inside a polarizer?
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Linear to Circular Polarisation Questions
  • · Replies 1 ·
Replies
1
Views
3K
High School Ways to understand the delayed entanglement swapping
  • · Replies 23 ·
Replies
23
Views
3K