State after measurement in "Delayed choice for entanglement swapping"

Question

I'm reading this paper below, but it seems to me the conclusions are not correct.

https://arxiv.org/abs/quant-ph/9904042

Alice and Bob independently prepare the Bell state $\Psi^-$. They measure one qubit and send the other qubit to Eve. The joint state of the system is written in terms of the qubit-pair Eve now has on the right side of equation 6 (page 3).

That state, however, is the state assuming Alice and Bob have NOT measured a qubit. They then deduce probabilities of results if Eve performs Bell measurements on her qubit pair. It seems to me this is bogus: given that Alice & Bob measured and collapsed the state, how can we subsequently use the state before those measurements?

While writing this I realized the main jist of the paper may still work if we assume Alice/Bob measured "after" Eve, so Eve did measure on the state in equation 6. Maybe this is just a minor problem with how the concept was introduced? Or maybe I'm missing something bigger, "after" may be problematic here?

Answer 1

This has been experimentally verified in hundreds of experiments. Here are examples of well-known implementations by Zeilinger teams:

A. Experimental delayed-choice entanglement swapping (2012)

B. Experimental Nonlocality Proof of Quantum Teleportation and Entanglement Swapping (2002)

C. High-fidelity entanglement swapping with fully independent sources (2009)

There are several important points to make:

i) The order of execution ("after" or "before") of the Eve (or Victor) measurement has no observable affect on the outcomes. (Reference frame is of no consequence since order does not matter.) This is clearly demonstrated in the experimental implementations.

ii) In the A. reference, there is a common source for all 4 photons. But in the C. reference, the photons sources are fully independent and spatially separated.

iii) In fact, in that setup: the 1 & 4 photons which are tested by the CHSH inequality never exist in a common light cone and have no mutual interaction at all (assuming you consider c a hard and fast limit for an influence).

So your concept of "after" being the issue cannot be correct, since nothing changes regardless of how you define "after". And everything presented in the experiments fully conforms to standard QM and the presentation of Peres. Of course, Peres' paper is a primary reference for all of these.

So let's look at Peres' [6] which you mention. Yes, this is the state prior to a measurement by Eve. The decision to analyze everything from the perspective of what Eve's outcome is essentially arbitrary. That's because there is no difference due to the order of measurements. You could also arrange things so Eve's measurement occurs between Alice and Bob's, and again nothing changes. So the choice of looking at Eve's result first is a convenience. So why choose that? Once you learn Eve's Bell state (let's call those photons 2 & 3): photons 1 & 4 will violate the CHSH inequality. But as indicated in reference A., they will also be perfectly correlated in the EPR sense.

For the entanglement of Alice and Bob's photons (1 & 4) to be created, the photons Eve observes (2 & 3) must be indistinguishable. That means they must arrive together within a narrow time window. If one or the other of photon 2 or 3 are intentionally delayed by Eve, there will be no entanglement. This is demonstrated by obtaining Entangled State statistics or Separable (Product) State statistics as presented in reference A. figure 3 according to whether they are made distinguishable.

Also please note: while Peres' paper mentions (correctly) that there are 4 possible Bell states that may be observed, only 2 of the 4 can be experimentally identified using current technology. This limitation does not alter the conclusions.

---

And just to be clear about the Delayed Choice implementations in general: I would recommend the following roundup of Delayed Choice theory and experiment, Zeilinger is a co-author on this too.

D. Delayed-choice gedanken experiments and their realizations (2014)

Answer 2

You write:

Alice and Bob independently prepare the Bell state $\Psi^-$. They measure one qubit and send the other qubit to Eve. The joint state of the system is written in terms of the qubit-pair Eve now has on the right side of equation 6 (page 3).

In the paper in the paragraph before equation 6 the authors write (emphasis mine):

In the proposed experiment, two distant observers, conventionally called Alice and Bob, independently prepare two sets of singlets, whose states are denoted as $\Psi^-_A$ and $\Psi^-_B$. Alice and Bob keep one particle of each singlet and send the other particle to a third observer, Eve, who also arranges them in pairs (one from Alice and one from Bob). The three observers keep records specifying to which pair each particle belongs.

Alice and Bob each keep one particle and send the other to Eve. They do not measure it at that stage. The premise of your question contradicts the text of the paper you are citing. I advise you to learn to read more carefully.

Update. I will deal with the rest of the question:

It seems to me this is bogus: given that Alice & Bob measured and collapsed the state, how can we subsequently use the state before those measurements?

While writing this I realized the main jist of the paper may still work if we assume Alice/Bob measured "after" Eve, so Eve did measure on the state in equation 6. Maybe this is just a minor problem with how the concept was introduced? Or maybe I'm missing something bigger, "after" may be problematic here?

Alice and Bob may be spacelike separated from Eve at the time of their respective measurements in which case there is no fact of the matter about the order of the measurements. The results don't depend on the order of the measurements. It is helpful for calculations to have an order in which you're going to apply operations so you might as well apply the Alice and Bob operations first since it makes no difference to the result.

Another thing to note is that the word "collapse" doesn't appear anywhere in the paper: measurements are described by POVMs. Nothing about the results depends on collapse.

Collapse has a number of problems as a way of describing measurement. It doesn't describe repeated, continuous or unsharp measurements all of which appear in measurement theory and so it isn't compatible with how calculations are done in practice:

https://arxiv.org/abs/1604.05973

In many discussions there is no explicit equation of motion for collapse. which makes those discussions vague and difficult to test in detail. Theories that do have collapse built in explicitly don't currently explain the results of experiments on relativistic quantum systems, which are the vast bulk of real experiments:

https://arxiv.org/abs/2205.00568

In quantum theory without collapse when information is copied out of a system, such as a measurement instrument or record in a computer or lab notebook, quantum interference is suppressed: this is called decoherence

https://arxiv.org/abs/1911.06282

Decoherence explains why you don't diffract when you walk through a doorway without invoking collapse.

Bell correlations and other entanglement experiments such as the one above can be explained by decoherent systems carrying quantum information that can't be accessed by measurements on those systems alone: locally inaccessible information. The expectation values of the observables of the measurement result don't depend on the quantum information they contain so that information is shielded from decoherence. That information can only be accessed by comparing measurement results from different systems. The relevant correlations are created when the comparison takes place not at the time of the initial measurement:

https://arxiv.org/abs/quant-ph/9906007

https://arxiv.org/abs/1109.6223