No. You cannot use entanglement for FTLC, and also, you already asked this. VTC as a duplicate of your previous nearly-identical question.

Entanglement has a lot of confusion as many "bad" articles proposed it and wrote about it. They blindly follow old rules of QM from the 1930s ... if you follow the rules you get bad conclusions. FTL has been refuted. Your error is unfortunately reading these amazing articles but not having the depth/experience to challenge them. There are quite a few articles that reject FTL .... but even these may use poor logic to reach that conclusion .... wikipedia has some references that refute FTL, start there.

@PhysicsDave I did. Months ago. "FTL has been refuted" is not exactly true. It has been refuted for local operations in (if I am not mistaken) discrete and finite Hilbert space. The proof is called the no-communication theorem. I have done my research. I wrote 11 pages preprint about. Now I want to know if I am right or wrong and no one can answer that question. All I am given instead of an answer is a circular argument about FTLC. The impossibility of FTLC is not a postulate of QM.

Hi @Andrzej, I think everything you wrote is correct, but why would that allow FTLC ? Also the idea seem very similar to "delayed choice quantum eraser" experiment

Hello @J.Delaney. Thank you for your question. To see this, we would need to calculate the reduced density matrix for photon 1 (the photon with the different polarization). However, it may be simpler to consider a single-photon detector placed behind a polarization filter that only allows diagonally polarized photons to pass ($|D\rangle = \frac{1}{\sqrt{2}}(|H\rangle + |V \rangle)$) In that setup, the probability of detecting the photon changes from 1/2 to 1 once the second photon loses the information about which slit it passed through.

No, I don't think so. I don't use postselection. It is more similar to induced coherence journals.aps.org/pra/abstract/10.1103/PhysRevA.44.4614 . Besides the above description is just a toy model. I don't use this setup in my article. In this topic, I wanted to focus on the question of unitarity. I am quite familiar with the way that FTLC propositions are being proven wrong. You can boil it down to unitarity or - in the broader picture - to local quantum operators and the completeness relation of Kraus operators.

@Andrzej The fact that a unitary operation on a full space induces a non-unitary operation on a subspace is quite trivial. I don't see how it is related to FTLC. You didn't write anything about it in your question so it is hard to understand what you mean

Yes, it is very trivial. My work is hardly revolutionary. I simply put together a few ideas. Of which way information erasure, induced coherence and global unitarity. As far as I know, no one has done that before. And I think this is a missing piece. Yes QM does allow for FTLC although we may not be able to ever build an experimental setup to actually see it. And, as you can see, this is somehow controversial.

@Andrzej There is a basic fact in QM that local operations on entangled systems do not communicate information (no matter what speed). Complicated experimental setups are usually just of way of hiding this simple truth behind confusing details. If you can't point to what exactly in your setup makes FTLC possible, then you probably just managed to confuse yourself

@J.Delaney I am confused. You have just admitted that "The fact that a unitary operation on a full space induces a non-unitary operation on a subspace is quite trivial.". And I agree. But now you are claiming: "QM that local operations on entangled systems do not communicate information". I assume you mean here "local unitary". And I thought we have just agreed that globally unitary operation might not be locally unitary operation.

@Andrzej "local operations" means that the systems are isolated from each other. The time evolution of an isolated system is unitary, so yes "local operations" means "local unitary". You can have non-unitary local time evolution only if the systems are not isolated, i.e. interacting with each other. But obviously interacting also means transferring information in the usual way.

@J.Delaney this is exactly the discussion I hoped for. Thank you for this discussion. Now to answer your question. Let me note that in your first response "I think everything you wrote is correct, but why would that allow FTLC" You do seem to share the same intuition that what I wrote in my post seems to be correct. And that is an example of globally unitary locally non unitary.

When it comes to the question of an isolated system: an isolated system is described by a wavefunction (one of the QM postulates) from which follows that a system which cannot be described with wavefunction but rather a density matrix $\rho$ such that $Tr(\rho^2) \neq 1$ is not isolated. It follows that a part of the entangled system is never isolated.

@Andrzej Those are different concepts. Isolated systems means that the full Hamiltonian can be written as $H = H_A + H_B$ , which implies that the sub-systems don't interact and evolve separately unitarily. The density matrix describes entanglement. Two systems can be isolated and entangled at the same time.

@J.Delaney I think that I agree. And I think we are talking about the same concept but in a different language. I guess I am stating that globally unitary but locally non-unitary would translate to $H \neq H_A \otimes H_B$. In other words, the Hamiltonian of the system which I have used as an example cannot be described as a product of Hamiltonians of the subsystems.

Happy that J Delaney knows the nuts and bolts of QM to help you out. I am not at this level. In general, the EM field is never discussed in these experiments/reports.... they all assume the photon acts independently of the apparatus field .... all apparatuses have modes (allowed paths) for EM energy (photons) in the apparatus field. The field decides long in advance of the photon what is possible.

@PhysicsDave, this is what worries me. In my simplified model, it looks like I can circumvent the no-communication theorem. However, once I consider all interactions—such as how I produce photons (squeezed vacuum), their interactions with optical elements, dispersion, lenses, and so on—it's possible that, in the full picture, the Hamiltonian is separable. If that’s the case, then the corresponding unitaries would also be separable, making FTLC impossible. But that level of analysis goes far beyond what is typically done in quantum optics.

At this point, I cannot prove that a globally unitary operation must also be locally unitary. Such a proof (applicable to any Hilbert space—discrete, continuous, finite, or infinite) would, of course, amount to a complete refutation of FTLC.

QM is pretty cool, in the 1930s there was great discussion, in the 1940s all the best scientists got pulled into WWII and afterwards into the nuclear age. QM was left to a lot of academics and interpretations ... not so good. Today we have many physicist pursuing research dollars and FTL is very sexy so be warned. I'm a senior guy, I'm impressed by your depth in QM (far beyond mine) but my advice is take QM with some skepticism ... the future needs nuclear energy, electric cars/batteries, global warming science, etc ... QM is old stuff.

FTL haș already taken some of your time ... and you will not be very popular if you disprove it once and for all .... some will appreciate your work, some not. In general society needs physicists and society is pretty happy whatever they are working on ...

@Andrzej That's what I meant by too many details obscuring the truth. You don't need to analyze every small component of an engine in order to know that total energy is conserved. Equivalently, you know from first principles that an isolated system must evolve unitarily. Conversely if it doesn't, that it must be in interaction (i.e. communicating) with another system

@J.Delaney, what you’re describing is exactly what I’m claiming. Two subsystems of an entangled state are not isolated (even if they are spatially separated), and in principle, there is nothing in quantum mechanics that would prevent communication. Let me put it another way: $$\hat U_{AB} |\phi \rangle = |\phi' \rangle $$ where $ |\phi \rangle$ is the initial state I described earlier, and $ |\phi' \rangle $ is the state after free-space propagation. Now, suppose we assume that $\hat U_{AB}$ describes two isolated systems.

That would mean $$\hat U_{AB} = \hat I_A \otimes \hat U_B$$ where $\hat I_A$ acts on the first photon polarization state and $\hat U_B$ acts on the spatial mode of the second photon. Under this assumption, $\hat U_B |\text{slit 1} \rangle = | \text{distant} \rangle$ and $U_B |\text{slit 2} \rangle = | \text{distant} \rangle$. This leads to a contradiction because $\hat U_B$ would not be unitary. We know $\langle \text{slit 1} | \text{slit 2} \rangle = 0 $ but $\langle \text{slit 1} |\hat U_B^{\dagger} \hat U_B |\text{slit 2} \rangle = 1 $ and unitary operation must preserve inner product.

In other words $\hat U_{AB} \neq \hat U_A \otimes \hat U_B \Rightarrow \hat H_{AB} \neq \hat H_A \otimes \hat H_B$

@Andrzej That's right, so you just showed that $A$ and $B$ are interacting with each other. which means they can't be arbitrarily far apart ...

In practice? I would bet you are right. According to QM? There is nothing preventing both systems from being light-years apart... @PhysicsDave thank you for your advice and kind words.

@Andrzej Assuming that systems can interact that way is equivalent to assuming that your Hamiltonian contains non-local interaction terms. In a sense you are right that QM by itself does not prevent this, but we know that QM is just the classical limit of a relativistic QFT, in which all interactions are local ... if you allow for non-local interactions, then it follows pretty much by definition that you will get FTLC

@J.Delaney, that’s why I believe my results are both trivial and non-controversial. At the same time, however, they contribute to our understanding of locality, causality, and FTLC in the framework of quantum mechanics.

@Andrzej Yes. Hopefully this discussion contributed a bit as well !

Assuming your FTL signaling device is attempting to communicate from Alice to Bob using entangled pairs: there is no more likelihood you send a signal one direction versus the other (Bob to Alice). Regardless, it should be obvious that any coherence you could possibly induce on one side has absolutely no effect on the other. (Entangled photons are not coherent.)

@DrChinese could you be more specific? With what exactly do you not agree? With the approximation stating that at a large enough distance from both slits, we wouldn't be able to distinguish through which slit did the photon pass?