Help with Bell states and measurements

[Joao]

Hi everyone! Sorry for the bad english!

I'm trying to understand the "delayed choice entanglement swapping" (avaliable in https://arxiv.org/abs/1203.4834 ) and (long story short) , in the article we have pairs of photons that are entangled in the $\phi \pm$ Bell state and the photons polarization are measured in the Right/Left basis, Vertical/horizontal or plus/minus (I guess it's right handed or left handed circular polarization but I'm not sure).

So, I guess that it means:
In the right and left basis:
If we measure one photon of the pair as Right, the other one will also be measured as Right.
If we measure one photon of the pair as Left, the other one will also be measured as Left.

In the Vertical and Horizontal basis:
If we measure one photon as Vertical, the other one will also be measured as Vertical.
If we measure one photon as Horizontal, the other one will also be measured as horizontal.

In the plus and minus basis:
Now we have a difference in the $\phi +$ and in the $\phi -$
In $\phi +$, if we measure + in one photon, we will measure + in the other photon, and if we measure - in one photon, we will measure - in the other photon.
In $\phi -$, if we measure + in one photon, we will measure - in the other photon, and if we measure - in one photon, we will measure + in the other photon.

Please, am I right? Thanks!

 

[DrClaude]

Are you talking about what is said right after eq. (3)?

 

[Joao]

Are you talking about what is said right after eq. (3)?

Thanks a lot for the answer Doctor! Yes! I'm talking about that! More specifically, I'm trying to make sense of

"On the other hand, when Victor performs the separable-state measurement on photons 2 and 3 and does not swap entanglement, the correlation only exists in the (Horizontal and Vertical) |〉/|〉 basis and vanishes in the |+〉/|−〉(plus and minus) and |〉/|〉 (right and left) bases, as shown in Fig. 3B."

It's in the page 6, third paragraph.

What I'm trying to understand exactly is (sorry for my lack of capacity to Express better) is how the data collected by Alice and Bob are sorted, considering the data collected by Victor.

For example, let's suppose Alice and Bob have data that seems completely random to them, like this:
First pair
Alice: Right. Bob: Right

Second pair
Alice: Vertical. Bob: Vertical

Third pair
Alice: Right. Bob: Left

Forth pair
Alice: +. Bob: +

Fifth pair
Alice: right. Bob: right

Sixth pair
Alice: +. Bob: -

Seventh pair
Alice: horizontal. Bob: Horizontal

Eight pair
Alice: -. Bob: -.

Ninth pair
Alice: Right. Bob: left

Tenth pair
Alice: +. Bob: -.

By looking at this data, one can never tell with photons were entangled, nor what is the Bell state of the entanglement (if they were entangled). But if we look to Victor's choices, we could sort the data collected by Alice and Bob in different ways:

First pair
Alice: Right. Bob: Right
Victor's first pair: Horizontal and Horizontal.
Alice and Bob's photons weren't entangled.

Second pair
Alice: Vertical. Bob: Vertical
Victor's second pair: $\phi$ +
Alice and Bob's pair were entangled in $\phi$ +

Third pair
Alice: Right. Bob: Left
Victor's third pair: Vertical and Vertical
Alice and Bob's photons weren't entangled

Forth pair
Alice: +. Bob: +
Victor's forth pair: $\phi$ +
Alice and Bob's pair were entangled in $\phi$ +

Fifth pair
Alice: right. Bob: right
Victor's fifth pair: $\phi$ -
Alice and Bob's photon were entangled in $\phi$ -

Sixth pair
Alice: +. Bob: -
Victor's sixth pair: $\phi$ -
Alice and Bob's photon were entangled in $\phi$ -

Seventh pair
Alice: horizontal. Bob: Horizontal
Victor's seventh pair: vertical vertical
Alice and Bob's pair weren't entangled

Eight pair
Alice: -. Bob: -.
Victor's eight pair: horizontal horizontal
Alice and Bob's pair weren't entangled

Ninth pair
Alice: Right. Bob: left
Victor's ninth pair: vertical vertical
Alice and Bob's pair weren't entangled

Tenth pair
Alice: +. Bob: -.
Victors tenth pair: $\phi$ -
Alice and Bob's pair were entangled in $\phi$ -

Am I right?

Thanks again!