Hello Michael,

as long as Alice doesn't want to say anything about her measurements, Bob will never find out whether she measured or not. Bob's density matrix remains unchanged.

It is about the mixed state analyzer in R. Werner's contribution to the Springer tracts in modern physics (Quantum information). He needs for this a telephone call of Alice, telling him the bases she measured in, to do this. This fixes the basis of the decomposition of rho.

If he already manipulated his states irreducibly, it is too late for any classical communication. Any reversible change (unitary) permits a classical communication, though.

No, not before they share the results which happens via a classical channel e.g. optic fiber.

The no-communication theorem is a no-go theorem from quantum (information) theory, which states that, during measurement on an entangled quantum system, it is not possible for one observer, by making a unread measurement (i.e. the results are not shared) on a subsystem of the total system, to communicate information to another observer. In other words: the description (density operator) for one subsystem A is not affected by an unread measurement on the other subsystem B, hereby A and B are entangled subsystems.

It seems, I was not specific enough in the formulation of the question. The answers above simply tell me that the no-communication theorem applies. But, there are certain assumptions in the formulation of this theorem... In addition to the original question, let us suppose that Alice and Bob communicated classically during the distribution stage. They both know the quantum state they share and can do any manipulations with their state during the distribution stage. But, when they distributed the state, classical communication is no longer possible. It seems for me that this case is not covered by the no-communication theorem. Can Alice now signal Bob her measurement?

Hello Michael,

as long as Alice doesn't want to say anything about her measurements, Bob will never find out whether she measured or not. Bob's density matrix remains unchanged.

It is about the mixed state analyzer in R. Werner's contribution to the Springer tracts in modern physics (Quantum information). He needs for this a telephone call of Alice, telling him the bases she measured in, to do this. This fixes the basis of the decomposition of rho.

If he already manipulated his states irreducibly, it is too late for any classical communication. Any reversible change (unitary) permits a classical communication, though.

 I agree with  Ilian Peruhov, Bob can not know if Alice measures her part or not befor receiving the classical data from Alice. Or as suggested  by  Andreas Osterloh, Bob should receive a call from Alice that she has done her masurements.

If Alice could know whether Bob measured or not his particle, SUPERLUMINAL communication were possible.  THIS is the "no go" beyond the impossibility of Alice to know what did or didn't Bob.

Our world does NOT permit superluminal communication. To put it simply, a world in which such communication is possible cannot exist. No difference would be between future and past, time would have no meaning.

In short, since the nature is consistent with itself, NO experiment, NO trick, can override this "no go".

Good luck !

Dear Michael.

Andreas, Ilian, and Sofia are correct.  Sofia's answer is in my view the most general as it even addresses the following extension to your question.  Lets bestow Bob the power to clone (not perfectly thats not allowed but imperfect cloning is allowed) his part of the state.  Let us further assume as you conveyed in your second message that Alice and Bob know the original state they share and Bob even knows what measurement Alice is going to do... he just doesn't know when. Is there a way for him to find this out without any communication by Alice?

If perfect cloning is possible then yes he can.  But perfect cloning is impossible.  It turns out that the goodness of the cloning procedure that Bob can do is intimately related to the no-signaling condition. The latter places an ultimate upper bound on the overall goodness of the copying procedure that Bob can perform.  In fact even if Bob has access to some yet undiscovered physical theory that is more powerful that quantum mechanics he will never be able  to clone his state better than what quantum mechanics allows, because quantum mechanics saturates the bound placed by no-signaling (people say that quantum mechanics is at the edge of the no-signaling condition).  Hence, unless Alice tells him otherwise, Bob can never know if Alice has perfomrmed her measurement or not.

The connection between no-signaling and cloning is a pretty fundamental one---particularly in the field of quantum cryptography.  As far as i know the most stringent criteria for security for QKD have been proven for precisely this type of attacks by the eavesdropper.  The literature on the issue is vast and I am fairly certain people have thought about weak measurement issue as well.  (I would be more than happy to supply some references if you need them).

Hope this helps. 

Dear Michael Siomau

There are many answers that are based on Copenhagen view, why not try realistic view.

In the realistic view both particles Alice and Bob are moving near each other, but in their own space which is the dynamic space time, while they seems to be moving away from each other from our frame because we are only observing  the projection of their paths and can not observe their real movements  in their frame because they have higher response to expansion of the universe. While our classical masses make us having low response to this expansion. Accordingly any effect that is happened to Alice is observed instantly by Bob due to their proximity in their frames.

See the following paper for more details

https://www.researchgate.net/publication/259996539_Solving_the_instantaneous_response_paradox_of_entangled_particles_using_the_time_of_events_theory

Article Solving the instantaneous response paradox of entangled part...