Interpreting photon correlations from independent sources

[javisot20]

A question,

1- We start from 4 photons totally independent of each other, respecting MoE we entangle 1&2 and 3&4

2- We perform a swap between 2&3 producing non-local entanglement between 1&4, which also respects MoE

3-We delete all the information from the process


Can we know the entanglements now?
Can we know the entanglements at the beginning?

(I ask if by making new swaps and entanglements we can answer those 2 questions)

 

[Morbert]

A question,

1- We start from 4 photons totally independent of each other, respecting MoE we entangle 1&2 and 3&4

2- We perform a swap between 2&3 producing non-local entanglement between 1&4, which also respects MoE

3-We delete all the information from the process


Can we know the entanglements now?
Can we know the entanglements at the beginning?

(I ask if by making new swaps and entanglements we can answer those 2 questions)

Not sure what you mean by delete all information from the process. If you mean some quantum eraser event, that would be impossible as the registering of coincidences is irreversible. If you mean we throw away/don't look at the 4-fold coincidence result and hence don't discard any runs, then no correlations will be observed whether or not the quarter-wave plate was on or off.

 

[javisot20]

If you mean we throw away/don't look at the 4-fold coincidence result and hence don't discard any runs, then no correlations will be observed whether or not the quarter-wave plate was on or off.

In that context. The experimenter loses all data (and memory). Could I get that information with new swaps/entanglements?

 

[PeterDonis]

In that context. The experimenter loses all data (and memory). Could I get that information with new swaps/entanglements?

No, because all of the photons no longer exist after they are measured. If you throw away the information from those measurements, you've thrown away all the information there is about those photons.

 

[DrChinese]

1. No, I am not. The b and c labels are not irrelevant. They are vital. The BSM projects onto |Φ-〉 in the spatial modes b and c even though the H/V signatures are registered in b" and c". Careful, precise tracking of these modes makes clear the complementary relation between the BSM and the SSM, and why they cannot be interpreted as the same selection criteria even when the same 4-fold coincidence condition is used, as the time-evolution of the photons through the apparatus is different depending on the configuration of the quarter-wave plate.

2. No correlations appear that are canonically impossible according to me. You must have wires crossed. The BSM result that projects photons 2&3 onto the state |Φ-〉 in the spatial modes b and c will identify a subset of 1&4 measurements that exhibit correlations in all three mutually unbiased bases, without the need to insist on nonlocal influence. This is the entanglement swap.

3. A local interpretation of events is possible if the SSM and BSM choices amount to different selection criteria, as we can say the BSM or SSM themselves are not responsible for any of the correlations seen in 1&4. Instead, the identification of alternative subsets is what is responsible.

1. Their set up is very complicated, so either of us is to be forgiven if we get a technical point wrong.

But if you look again at the B and C plates (in the BiSA portion), and the related description, you will see that they do not ever change. I believe what you were referring to is changing is actually the electro optical modulators (EOMs). Those change settings to shift from SSM to BSM. So there IS a change in the apparatus as you say, we absolutely agree on that.

However, it has nothing to do with selection criteria at all. The EOMs change the phase of the photons relative to each other, allowing them to interference at the beam splitter (or not). That does not change the H/V polarization of the 2 & 3 photons. Of course, that change can cause an entanglement swap.


2. Just to be clear: there is no mathematical or statistical connection whatsoever between the right or left circular polarization of a photon and the horizontal or vertical polarization of that same photon. Similarly, there is no mathematical or statistical connection whatsoever between the initial right or left circular polarization of entangled photon 1 with the horizontal or vertical polarization of its partner 2.

It is canonical that the circular polarization of photon 1 is in a separable Product state with the vertical polarization of photon 2 after initial creation via PDC. Before any swap occurs, that places all four photons in a product state. There is no possibility of any statistical relationship in such a product state.

Surely you cannot disagree with this.


3. Again, this is quite impossible: Photons 1 and 4 are in a Product state UNLESS a swap occurs - that being a physical process which places the 1 and 4 photons into a Bell state (with Entangled State statistics). Selection criteria differences (which are primarily in your imagination) do not in any way change the above point 2. Why can't they?

If photons 1 and 4 are mutually distant, and never appear in a common area of spacetime, then a Local interpretation requires that they can never have anything other than a Product state relationship as they start life. After all, initially photon 2 is in a Product state with 1, and photon 3 is in a Product state with 4. Consequently: there can be no subset of 1 & 4 pairs that will ever display perfect correlation. And that is true regardless of any measurements performed on photons 2 & 3 (by your definition of locality). Photons 2 & 3 carry no useful information (when measured on the H/V basis) about the circular polarizations of photons 1 & 4. Therefore there is nothing to be gained about photons 1 & 4 from examination of photons 2 & 3.

Please tell me this much is clear. Everything measured is in a Product state with each other!

---

I will address your other points in a separate post.

 

[Morbert]

1. Their set up is very complicated, so either of us is to be forgiven if we get a technical point wrong.

But if you look again at the B and C plates (in the BiSA portion), and the related description, you will see that they do not ever change. I believe what you were referring to is changing is actually the electro optical modulators (EOMs). Those change settings to shift from SSM to BSM. So there IS a change in the apparatus as you say, we absolutely agree on that.

However, it has nothing to do with selection criteria at all. The EOMs change the phase of the photons relative to each other, allowing them to interference at the beam splitter (or not). That does not change the H/V polarization of the 2 & 3 photons. Of course, that change can cause an entanglement swap.

I don't know if we disagree here or are just misreading each other. On page 14 of Ma et al, the time-evolution through the set up is shown for when swap = on. You can see that the evolution relates the detected signatures at the end of the run, in spatial modes b'' and c'' to the state that was in the incident spatial modes b and c. This is also made clear by Fig. 2 where we can see the ordering of b c b' c' b'' c''. We see from these evolution rules that Victor uses the signatures in b" and c" to resolve a BSM in b and c.

If we detect a coincidence in the same spatial mode b’’/c’’ (as shown in Fig. 2 of the main text) but with different polarization in the |𝐻〉/|𝑉〉 basis, photons 2 and 3 are projected onto |Φ + 〉 in spatial modes b and c
[...]
If we detect a coincidence in different spatial modes b’’ and c’’ but with same polarization in the |𝐻〉/|𝑉〉 basis, photons 2 and 3 are projected onto |Φ − 〉 in spatial modes b and c

Similarly, when the quarter wave plate is off, the signatures in b'' and c'' resolve an SSM in b and c.

As these measurements are complementary, we know that when the quarter-wave plate is on, the signatures in b" and c" cannot resolve the polarization of 2&3 in b and c. We can see this by applying Ma's time-evolution to HH and VV states
$$\begin{eqnarray*}&&|HH\rangle_{bc}\\
&&\xrightarrow{\text{BS 1}}i(|HH\rangle_{b'b'} + |HH\rangle_{c'c'})/\sqrt{2}\\
&&\xrightarrow{\text{EWPs & EOMS}}(|RR\rangle_{b'b'} + |LL\rangle_{c'c'})/\sqrt{2}\\
&&\xrightarrow{\text{BS 2}}i(|HV\rangle_{b''b''} - |HV\rangle_{c''c''} + |HH\rangle_{b''c''} - |VV\rangle_{b''c''})/2\end{eqnarray*}$$

$$\begin{eqnarray*}&&|VV\rangle_{bc}\\
&&\xrightarrow{\text{BS 1}}i(|VV\rangle_{b'b'} + |VV\rangle_{c'c'})/\sqrt{2}\\
&&\xrightarrow{\text{EWPs & EOMS}}(|LL\rangle_{b'b'} + |RR\rangle_{c'c'})/\sqrt{2}\\
&&\xrightarrow{\text{BS 2}}i(|HV\rangle_{b''b''} - |HV\rangle_{c''c''} - |HH\rangle_{b''c''} + |VV\rangle_{b''c''})/2\end{eqnarray*}$$
No 4-fold coincidence will resolve a separable state in spatial modes b and c when swap=on, as expected.

2. Just to be clear: there is no mathematical or statistical connection whatsoever between the right or left circular polarization of a photon and the horizontal or vertical polarization of that same photon. Similarly, there is no mathematical or statistical connection whatsoever between the initial right or left circular polarization of entangled photon 1 with the horizontal or vertical polarization of its partner 2.

It is canonical that the circular polarization of photon 1 is in a separable Product state with the vertical polarization of photon 2 after initial creation via PDC. Before any swap occurs, that places all four photons in a product state. There is no possibility of any statistical relationship in such a product state.

Surely you cannot disagree with this.

I don't understand the paragraph in bold. Photons 1&2 are prepared the entangled state Ψ-. This state can be represented in whatever basis we like (like the circular polarization of photon 1 and a linear polarization of photon 2) it is still an entangled state. What the state implies is if photons 1 and 2 are measured in the same basis, anticorrelation between outcomes will be observed. If photons 1 and 2 are measured in different, unbiased bases, no correlation will be observed.

3. Again, this is quite impossible: Photons 1 and 4 are in a Product state UNLESS a swap occurs - that being a physical process which places the 1 and 4 photons into a Bell state (with Entangled State statistics). Selection criteria differences (which are primarily in your imagination) do not in any way change the above point 2. Why can't they?

If photons 1 and 4 are mutually distant, and never appear in a common area of spacetime, then a Local interpretation requires that they can never have anything other than a Product state relationship as they start life. After all, initially photon 2 is in a Product state with 1, and photon 3 is in a Product state with 4. Consequently: there can be no subset of 1 & 4 pairs that will ever display perfect correlation. And that is true regardless of any measurements performed on photons 2 & 3 (by your definition of locality). Photons 2 & 3 carry no useful information (when measured on the H/V basis) about the circular polarizations of photons 1 & 4. Therefore there is nothing to be gained about photons 1 & 4 from examination of photons 2 & 3.

Yes, if a swap occurs, then photons 2 & 3 are projected onto Φ-, as are photons 1 & 4. If no swap occurs, no such projection occurs. We both agree with this.

Where we diverge is the interpretation of the projection. You interpret it as a very literal ontic event. But there are alternatives. E.g. The Copenhagen interpretation frames the projection as Victor updating the information he has about the outcomes of measurements on 1 and 4. The statistical ensemble interpretation frames the projection as the identification of a subensemble that will exhibit correlations between measurements on 1 and 4.

 

[Morbert]

I will address your other points in a separate post.

Feel free, but I suspect our core disagreement is section 3. from the last post.