I have the impresssion that a more serious reference to the many papers in which J. Bell with admirable clarity discusses this problem should be useful. The essence which seems to me to be only vaguely touched in the debate is the fact that Quantum Mechanics is nonlocally causal, as proved by Bell himself.

I would also make reference to papers by Eberhard and by me and my collegues Rimini and Weber, where we present the rigorous mathematical proof the the reduction process of one of two entangled system cannot be used to send faster than light signals.

I am not sure that I have understood your question; it is more difficult than the no-communication theorem.  You mumble about an entangled state (EPR) where the momentum of the particle accessible to Alice is perfectly anti-correlated with the one accessible to Bob. So, if Bob measure a momentum p_b, he can say with certainty that Alice will measure p_a = -p_b (if she elects to measure the momentum of her particle).

So what? Such kind of correlations and predictions exist in classical physics also. But they cannot be used to instantly transfer new information from Bob to Alice; they only reveal information which where created before, and transported to Alice and Bob no faster than the speed of light.

The no-communication theorem says that quantum mechanics is not different from classical mechanics in the sense that local measurements cannot be used to instantly transform information between two parties, although the nature of the shared prior information it very different (to the extent that it challenges our "common-sense" notion of the nature of physical reality).

There is a good semi-popular account of all this: "Bertelman's socks and the nature of reality" by John Bell.

It is proposed that the quantum entanglement phenomenon can be used to synchronise remote clocks which is a form of communication between remote systems. There is an US patent:US 20140270802:Quantum synchronization for classical distributed systems that exploits just that.

In the patent specification we read:

"Quantum entanglement allows a single master clock as a non-local event distributed across all nodes of the distributed system, thereby creating a true simultaneous event and system wide synchronization" 

No one would have to prove that simultaneous is or is not equivalent to Einstein's synchronisation effect. If the device was ever built it would be quite easy.

Any proofs of that? like the one we have with slow clock transport synchronisation?

Wikipedia https://en.wikipedia.org/wiki/Criticism_of_the_theory_of_relativity#Superluminal_speeds

evades the question by:

Also quantum entanglement (denoted by Einstein as "spooky action at a distance"), according to which the quantum state of one entangled particle cannot be fully described without describing the other particle, does not imply superluminal transmission of information (see quantum teleportation), and it is therefore in conformity with special relativity.[B 16]*

The reference B 16 cannot be resolved into a readable peer reviewed document as it simply cannot be found.

Andrew> The reference B 16 cannot be resolved into a readable peer reviewed document as it simply cannot be found.

The no-communication theorem is discussed here:

https://en.wikipedia.org/wiki/No-communication_theorem

with references to some articles in peer-reviewed journals. With regard to the patent, the information is too vague for any opinion.

I don't think entanglement should be viewed as a spooky action at a distance. One way to think about it is in terms of the many-worlds interpretation, with each possible world developing in a causally consistent manner. For some reason, at some point a single one of all possible branches won out, expelling all other possibilities to our dreams,

To Dr. Olaussen. 1) Is it common pratique in scientific community, what a junior asks a professor to sign the manuscript to submit it to Physical Review Letters? It means, the professor reads the paper, makes minor changes and adds his name to the co-authors list. Could this be done? If it could be done, please contact me: [email protected], I need the signature. I can not reply to private messages in RG.

"One way to think about it is in terms of the many-worlds interpretation"

If all actions are predictable through the unchanging physical laws, I see no need in multi-worlds interpretation within the secular physics. The one world would be "more than enough" to construct the secular TOE, correct?

"about an entangled state (EPR) where the momentum of the particle accessible to Alice is perfectly anti-correlated with the one accessible to Bob. So, if Bob measure a momentum p_b, he can say with certainty that Alice will measure p_a = -p_b (if she elects to measure the momentum of her particle."

Please do not confuse the "spin" (of "quantum rotation") and the "4-momentum" (of motion throughout the cosmos). Is there the experimental paper, which proves, what the momentum-s are opposite? I doubted your formula p_a = -p_b. This is the only point of this post. Thank you.

Dear Prof. Martila,

                                  Let me add to this very interesting debate from our researches into quantum econophysical systems, some of which is purposefully meant to be private communications with select groups by collaboration groups, who therefore have dual objectives of multiplying their researches by quantum (read quantum mechanical information resonance) correlation leading to entanglement. This requires for example unidirectional transitivity stock and flow phenomenon or non Abelian vector products for example as in electronic computer information.

SKM, NH, SM

I have never been able to decide if Soumitra is making jokes or not with his outrageous statements (like above). I believe he must be.

Dear Dr. Mallick, I hope, I could be of help. I can not send a private replies in RG. My contact email is [email protected]

Be well,

Your Dmitri.

Dear Profs. Martila and Olaussen,

                                                             I would like to clarify further. Sorry, for not being clear. These are computer information systems with self energy and self information flow coupled using some kind of Econophysics Fine Structure Constant developed by us using Electronic Commerce experiments with capital market transactions and data flow and they worked quite successfully. Please refer to for e.g. our paper on Medical Physics of Health Insurance Field Theory on our Research Gate website which was part of our Millenium  Mathematics Prize solution for problem #3. There are additional references included. I would like to communicate privately also. The technology for word processing papers has become too hazardous and too technology dependant these days to keep up especially privately. Besides, technology does not diffuse to emerging markets as quickly as one would like to think. There are training and retraining costs for us for example and those deadly viruses, oh!. 

SKM, NH, SM

The statements made are  wrong: First of all, i non-relativistic quantum mechanics there's no such notion of superluminal communication, anymore than there is in non-relativistic classical mechanics.

The uncertainty principle does not affect the average values of quantities, it introduces relations among the higher moments. So, depending on the set up, it is meaningful to state that = - for instance. 

Entanglement simply means that P(A,B)≠P(A)P(B), where P(A,B) is the joint distribution and P(A) and P(B) are the individual distributions.

I have the impresssion that a more serious reference to the many papers in which J. Bell with admirable clarity discusses this problem should be useful. The essence which seems to me to be only vaguely touched in the debate is the fact that Quantum Mechanics is nonlocally causal, as proved by Bell himself.

I would also make reference to papers by Eberhard and by me and my collegues Rimini and Weber, where we present the rigorous mathematical proof the the reduction process of one of two entangled system cannot be used to send faster than light signals.

I am sorry, dear Remi. You asked, so I have deleted my "nonsense". But if I would be offended, I would gave you the funny link:

https://youtu.be/ffMnKoHPUKw

@ Remi: How do you *define* the tensor product of 2 operators? As far as I know and can see, the two expressions you give in (2) and (3) are simply identical. What am I missing?

But you wrote the density matrix in factorised form!

For a non-factorisable density matrix, (2) is of course the only meaningful definition. Since a non-factorisable density matrix can always be expressed as a linear combination of factorised density matrices,  (3) and its linear extension remain relevant.

Dear Remi, even the presidential election in USA is partly show. Therefore, the public in RG needs my sense of humor.

The way I understand the NCT is as follows: Alice and Bob share different elements of a quantum state. Now Alice performs a quantum operation on her part. This is much more general than just a unitary evolution. It means she is allowed to entangle her part at will with a local system, and then perform a measurement of arbitrary nature. But she is restricted to use only things in her part of the world.

Then the statement is that Bob, by observing his part of the state, cannot say what Alice did, or indeed whether she did anything at all.

What does one mean by this? Let rho_i be the initial density matrix. Then Alice performs a quantum operation (Q_A tensor identity), which means what I said above: she only can act on her part of the state. So the final state rho_f is given by

(Q_A tensor identity) (rho_i)

The point now is that

tr_1 rho_i = tr_1 rho_f

Thus, if I trace over Alice's degrees of freedom, I get the same state independently of whatever Alice may have done. That is algebra, following from the definition of quantum operation and the cyclicity of the trace. But the meaning of the partial trace is precisely this: what can Bob observe if he is limited to use observables of the type

identity tensor A_2

So the point is, when Bob is thus limited, he cannot possibly decide whether the actual state of the system is rho_i or rho_f.

The detailed algebraic proof can be found in the wikipedia article on NCT:

https://en.wikipedia.org/wiki/No-communication_theorem

Dear friends, the respectful professors. I must add to the original text the following: I AM WORKING WITHIN THE DAVID BOHM INTERPRETATION OF QM. In this interpretation not only the mathematical average

, but also the actual 4-momentums p are under consideration: the David Bohm theory is more general, accurate and rich, than the Copenhagen interpretation of N. Bohr.

To the friendly Remi, Thank you for the information, such ones do enrich our worldview. However, a Truth-Seeker must not rush with the final conclusion. The paper, you kindly provided, needs evaluation via the discussions. Latter were not conducted, because the paper dates only ~3 months. Hereby any "wishful thinking" must be "removed" from the paper "next edition".

@ Remi: I cannot follow you. In the proof of the NCT, it is assumed that Alice performs an arbitrary quantum operation on her part of the system. That is considerably more general than an unitary evolution, as it includes measurements of many sorts. However, it certainly does include unitary evolution of the type

U_A tensor identity

and the matrix rho is indeed treated by the same formalism you suggest: it is sandwiched between the unitary and its inverse. No implicit factorisation is assumed.

As to taking the partial trace: this is not a physical operation which changes the state. It is rather epistemological in nature. Bob can only observe the partial trace of the state with respect to Alice's Hilbert space: this simply reflects the fact that he has no access to observables involving Alice's degrees of freedom. This, however, is simply a given of the problem.

Under these circumstances, it is misleading to ask whether the partial trace is taken ``before'' or ``after'' the evolution. Rather, the system evolves due to Alice's interventions, but Bob, limited as he is to the partial trace over Alice's degrees of freedom, sees nothing of it.

Friends, if you will keep ignoring my call, I would delete current question. So, either you all visit my link below, or you should save your comments on a hard drive. To save them. Thank you.

https://www.researchgate.net/post/Is_lack_of_Human_Logic_proven_in_2016

Dear Professors Martila, Levraj and others,,

                              May I add something from Econophysics Systems (please refer to my paper on Haag Theorem application to Stock Markets on my RG page). You will see that I have used a matrix called the Arrow-Mallick -Planck Matrix which can factorise the state of each share (which is actually a microstructure brane world or a small company by itself) in the market (say profit or loss). This as you will find the proofs of the local (short run) and global (long run) dynamic equilibrium is the topological factorisation potential for the price quantity relationship which has to be uppersemicontinuous for it to be instantly or in the limit equilibrium force resultant of econophysics. There are vacuum potentials in the matric as well as spacextime potentials and the topological curvature of the Value Function or the Lagrangean or Hamiltonian is solved by the inner product of the pricexquantity by using a quantum mechanical aggregator technology. Statistical Econophysical Laws such as Bayes Rule or not Bayes Rule are solved in general by these Lagrangean or Hamiltonian Algorithms. Thus states can always be resolved by cardinal numbers even if they are not transitive as we have proved in the first part of our Millenium Mathematics Prize entry. The ordinal transformation of thse cardinal representations have been worked out using the utility field and the relativity measurement contained in the budget constraint. Hence Bayesian or not, cardinal or ordinal, transitive or oscillating, potentials give rise to quantum mechanical flows with some Alice and Bob unidirectional AOT convention in the Lattice. Just a small explanation out of my and our joint research.

SKM for SKM, NH, SM

Dear Prof. Soumitra K Mallick, please allow me to talk to you via emails. So, write the first letter to [email protected] . I hope, my letters would be of help. Please, please, please. Thank you!

All I can say is, I have nowhere in reference [9], which I have read, or in the wikipedia proof of the NCT, seen anything resembling your eq (12). So, as far as I can see, this is a straw man type argument. I remain unconvinced.

Indeed eq (12) is largely meaningless, since one cannot define rho_1 and rho_2 in a non-factorised case. I assume this is your main point. But this error is not to be foun din the actual proofs.

I'd vote for Dr. F. Leyvraz. I see now, what I was wrong. Dr. F. Leyvraz, I hope, what if you co-work with me, I would get more used to such beautiful theory as QM.

Remi@

I am puzzled by your eqn.4 in "The misuse ... II". A natural interpretation is to take the first line to mean |Psi12> and the second line to mean |Psi'12>. However, this interpretation is not consistent with your eqn.5, which is a factorized state

1/sqrt(2)(e-i theta |H1> + |V2>) | D2>.

I don't understand why you choose to use such vague formulation in eqn.4. I am also puzzled by your introduction of UPBS, since this is just the unit operator according to your expression in the itemized list below eqn.4.

Added:

One more thing is the change of notation from the first line of eqn.1 to the later expressions. How are |Hi> and |Vi> related to |0>i  and |1>i  (for i=1, 2)? And how exactly is |D2> related to these states?

Remi> to mean that this state evolves into that state

So it is just a way to define the evolution operator O1 O2? But the states on the two sides of the arrow are not directly related to |Psi12> and |Psi'12> as used later?

You imply in eqn.1 and eqn.3 that you start from an entangled state, denoted to as |Psi12> in eqn.3. But suddenly in eqn.5 |Psi12> has become a factorized state. Should  it not be given a new name? You indicate by words that there has been some action involved. Acting on which initial state? The action defined by eqn.4 cannot produce a factorized state from an entangled state, no matter how |D2> is related to |H2> and |V2> (which you never define in a precise way). I cannot get the mathematics you write to match with what you say is happening. 

Remi> Are these style or substance questions, Kare?

Since you don't really specify what you do, it is impossible to say. But I strongly suspect it is substance; it definitely is style also.

Remi> Please look at the mathematics

The way I interpret your mathematics, it is wrong in a rather trivial manner (which is why I try to find out if you mean something else). 

Remi> what is to stop one transforming a non-factorisable state into a factorisable state?

You cannot do it by local unitary operations. I.e. unitary operations of the form U1 U2, because such operations preserve the Schmidt numbers (see section 2.5 in Nielsen and Chuang). The quantum measurement process, which is impossible to understand, can transform entangled states to factorized states.

Remi> Here we've transformed the basis of the second photon from HV to the number (Fock) basis.

You claim you do. But you never show the calculations.

If you start with the state

(|H>1 |V>2 + |V>1 |H>2)/sqrt(2)

and apply the unitary operations indicated by eqn.4, with UPBS as defined in the itemized list below eqn.4, you end up with the state

(e-i(theta+pi/4) U1 |H1> |V2> + ei pi/4 U1 |V1> |H2>)/sqrt(2)

which has Schmidt numbers (1/sqrt(2), 1/sqrt(2)). No local change of bases can change this to a factorized state, which has Schmidt numbers (1, 0).

As far as I can see, the proof of the NCT, say in Ghirardi et al's 's paper, which is not that general, but quite elementary, hence easy to discuss (Reference: Lett. Nuovo Cimento 27 No 10 (1980) p. 293). The proof of the theorem is between eqs (2) and (7), this last being the theorem.

(2) is the setup: the initial state consists of a general density matrix between the *two* systems U and V. This rho_(UV) does *not* factorise. This matrix has an additional tensor products with two apparatuses. These are distant and do factorise: we assume that there are no quantum correlations between apparatuses, since these are presumably classical.

Now the authors define two unitary operators, U_(UA) and U_(VB). These are assumed to commute, and the former only involves the apparatus A which it entangles with the total system UV, whereas the second only involves the apparatus B, which it similarly entangles with UV. If you think that A and B are spatially separated, and that measurements are performed on different parts of the common state, the hypothesis of commutation is reasonable.

If you apply only U_(UA) you get (3),if you apply both, you get (4).

Now in (5) they compute the expectation value of an observable which lives in the Hilbert space of measuring apparatus A in the case (4). There are three lines: first a definition, then plugging (4) into the definition, and finally cyclic invariance of the trace.

Then comes the crucial observation: U_(VB) commutes with both U_(UA) by definition as well as with L_A, since the latter lives on the Hilbert space of the A apparatus. As a consequence U_(VB) disappears, and the theorem follows

@ Remi: I do not get you. It is not Nature that applies U_(UA) or U_(VB) but (Wo)Man in her/his role as experimenter: an unitary operation in which a compound quantum system is evolved together with a measuring apparatus is a measurement (although perhaps a somewhat naive model of one)

@Remi. I am following the debate. Who is meant under "can you attach...." in the latter comment, I am sorry for my mental disabilities.

I predict the soon end of the debate. The apathy arrived.

Dear Remi, Here goes. As to ``the cyclical nature of the trace, it is a fact: tr(AB)=tr(BA), whether A and B commute or not:

tr(AB) = sum_i (AB)_(ii) = sum_(ij)A_(ij)B_(ji)

tr(BA) = sum_i (BA)_(ii) = sum_(ij)B_(ij)A_(ji)

so that the two are indeed equal.

Note that, when A, B and C do not commute, it follows from what I just said, that we have

tr(ABC) = tr(CAB), but in general not tr(ABC) = tr(BAC)

Best wishes,

Francois

I am sorry, I lost the sense of who is winning this scientific debate.

Remi> Physics is always a test of one's intuition, gained through experiment.

It is your privilege to practise quantum physics by intuition.

Remi> teoretians/applied mathematicians taking the mathematics literally.

Yeah, that is really terrible.

Remi> Kare, ahh! I See what you are getting at. Yes, dropped constant at equation

No, you did not see. But it is no use pointing.

“When I write a formula,’ Humpty Dumpty said in rather a scornful tone, ‘it means just what I choose it to mean — neither more nor less.’ " 

Not so fast Remi, not so fast (faster than light, that is)!

No wise fish would go anywhere without a tortoise :=D

They first do U1 tensor 1, then 1 tensor U2. This amounts to doing U1 tensor U2, so I am not sure what you say amounts to a significant difference.

Isn't the communication between the Physical Sciences and the Economic Sciences (with doping in both via engineering sciences) established by the commonality of the Mathematical and "Logical" postulates. So as long as they do and somehow satisfy the result of stylised or experimental data we hope to be be developing satisfying science, enough to invent new tools which are statically successful with or without failures.

SKM, NH, SM

Remi, the Prof. Mallick is the author in the famous Nature, isn't it?

So far I cannot follow your notation, and as to the mathematical issue, I insist that

U1 tensor identity

multiplied by

identity tensor U2

is equal to

U1 tensor U2

as working out the components in any consistent basis will readily show.

Dear remi, Just a short calculation to show what I mean. With the index forest involved, I found it far more pleasant to generate a PDF.

I have tried to put parentheses enough to avoid ambiguity. On the other hand, U1 tensor U2 is an MN by MN matrix, and so are U1 tensor identity and identity tensor U2, so the two latter can be multiplied, in the ordinary sense, and can (and in fact do) yield the former. If I form

(A tensor B) C

it means that C is an MN by MN matrix, or possibly a rectangular MN by P matrix, so that

A tensor (BC)

is meaningless, since I cannot take the product of an N by N matrix, namely B, with an MN by P matrix, namely C.

I do not fully understand your question. However, the issue is always the same: you should have a clear idea of what you mean by the quantities you use, and you should check whether they make sense algebraically. Defining objects in the most accurate manner possible is the best way to start a calculation and make sure one does not get into trouble.

Dear Professor Cornwall,

                                             Isn't it the case that you can use geometrically the parallelogram law or algebraically Kroenecker Deltas to multiply or divide tensors like any vector. But we have always used an objective function usually a mathematical-statistical Tychonoff type of scaling function to evaluate such factor products. We don't want to rise higher up in terms of applications.

SKM for SKM, NH, SM

Dear Remi:

Quite generally, one has

(A1 tensor A2)(B1 tensor B2) = (A1 B1) tensor (A2 B2)

as can be shown using a computation entirely similar to that performed in my blurb. All you do is replace the identities by general matrices and look at the consequences.

Some of these facts may be easier to understand in function spaces: if the space of functions in the variable x describes one (one-dimensional) system and similarly the spcace of functions in the variable y describe another, than the compound system is described by functions of the type f(x, y). The space of the functions in two variables is the tensor product of the two spaces of functions in one variable. Operators such as d/dx, which in an obvious sense ``only act on x'' are of the type

d/dx tensor identity

whereas operators of the type d/dy are of the type

identity tensor d/dy

That these commute, and that identities of the type shown above hold, should then be clear.

Dear Prof. Cornwall and Martilla,

                                                           I just want to add a remark that in all sciences "stationarity" is an objective the system is required to acquire dynamically for any experiment or analysis to be carried out. This one can do by building in very preliminary machinery like voltage stabilisers or circuit breakers. Such mechanical or electrical or possibly both only make the system and the subsequent quantum or probabilistic dynamics more robust. If you care to look at my paper on Stock Exchanges and by-pass of Haag's no go theorem you will see that mathematically such a gadget has been introduced by way of mathematical modelling to get meaningful stock market equilibrium. Thanks to the Radner & Rothschild paper in the Bell Journal, they had missed out on the structural assumption, I have introduced it to get advantage. Profs. Radner and Rothschild are aware of that. Left to Nature an evolutionary biological system reaches a genetic stable equilibrium precisely because of the Nature imposed penalty of extinction, isn't it so. So why not communicate at preliminary stages ? It's communication afterall not extinction.

SKM for SKM, NH, SM

Remi: "If the forum may permit me, can I ask does a lossy detector collapse the wavefunction - even for those times it didn't register - just because it is in the way of the particle stream?" It is not the consequence of the math. We should look it in Nature.

Dear Prof. Martila and others,

                                                    Algorithmic solutions can also be valid Natural Solutions especially if it is communicated through financial networks (Mallick (2007) on Research Gate say) because the problem is shown to be repeated every two periods so the random environment becomes mathematically transitive (Mallick, Hamburger and Mallick (2016) solution of Millenium Prize problem of P vs. NP) => over an infinite horizon of arrow of time the wavefunction of the economy does not collapse sequentially.

SKM for SKM, NH, SM