12th UK Foundations of Physics — Nonlocal Bohmian Field Theory & Holism

0:00 Thank you. The point is a good one. It has an answer, and it also has an expectation of the field. I might have an opportunity to go back to that. The new part of the work is number four, where we've actually applied the cause of the interpretation of boson fields to the mark's end of the dichromaton, and we have delayed choice configuration, and solved the initial value problem. But I suspect it would be more interesting, rather than going through a lot of mathematics, to kind of go through the background and mention the complementarity. So I'll start by describing the one of age choice experiment in the Mark Zender configuration, and then comment on complementarity and how it responds. and very briefly, the answer given by Jung's non-relativistic interpretation, and then spend a bit more time on the last part. I'm going to begin with two configurations. We know from Bohr's principle of complementarity that to define complementary concepts require mutually exclusive experimental arrangements. And this is something that Bohr emphasized throughout. So, for example, if I take a Mark Zender and Sperometer, let me just put this in my head. Our beam of light comes in this way. Should I point it from here? Light comes in here, this is the beam splitter. Splits both ways, there's a mirror and a mirror. Goes through a phase shifter, comes to the second beam splitter, and then combines and interferes. I put the phase shifter in, it doesn't need to be there, just for more general mathematical interest. But with the phase zero, if you take the phase shifter away and set the phase to zero, then the feature will be that the D-beam is completely extinguished. So that reveals interference. And we only get a single C-beam emerging. That's for cycle zero. So in this configuration, my

2:30 two detectors here, I will observe only counts in the C-beam and revealing interference effect. In the second configuration, if I place the detectors before beam splitter number two, then clearly I'm going to detect which path the light has taken. So in this description, I have a which path and I have a particle description. So those are my two mutually exclusive experiment configurations. But part would be that the interesting thing that Wheeler added to make this is a very choice experiment, is we have the same configuration, but he allowed the detectors to be swingable. So we can swing them the interference position, or the which path position. And we can do this at the last instance. So for example, whatever physics has gone on here, the light beam has come through, it splits, or it goes one way, whatever it's going to do. And at the very last minute, after the physics has happened here, we change the configuration. So the issue that is raised is, what about the historical development of the physical process? Is history changed? Does history come into creation at the time that we detect? Those are the issues that are raised. If we begin by accepting the position that complementary concepts have physical significance, that in a configuration with interference we really do have a wave, or in a configuration of which path we really have particles, and if you have that position, there are two possibilities. Either we change history at the time of measurement, or we create history at the time of measurement. emphasized one point from Complementarity, that no phenomena is a phenomena until it is an observed phenomenon. So whether he suggested that history would have been created or not, I don't know. In his paper he doesn't explicitly state. Or whether he simply meant that we couldn't discuss the experiment until the final measurement was complete. What does Bohr tell us? What would Bohr's response be to this situation? Bohr himself, I think, was very clear. And I've just summarized some pieces of complementarity which are relevant to this particular discussion.

5:00 One, pairs of complementary concepts require mutually exclusive experimental configurations for their definition. Number two, classical concepts are abstractions to aid thought, and that to attach physical significance to these concepts is physically meaningless. You know that Bohr had in mind that in your particular experiment which defined both of these past ways, but cannot attach physical significance to these concepts, and similarly for particle configurations and complementary concepts in general. And the experimental arrangement itself should be viewed as a whole not further analyzable. Summarizing his view, the idea that we could have this. We have experimental configuration, we have experimental results, but to describe the outline of physical reality is impossible. Now if we take this point of view, then clearly there is no question or anything in the wheeler configuration. Let me just put it up again. We have the experiment. The physics that goes on here doesn't change. Whatever is going on, we don't describe. The only time it becomes relevant to apply the concepts of particle and weight are only after the final measurements have been taken. So complementarity explains this in a natural way. The price to be paid, however, is that we give up the description of underlying physical reality. Now the question, is it really necessary for us to do that? Do we have to give up the description of underlying physical reality? I think Bohm in 1952 suggested that it wasn't. I will only sketch the answer. In the Bohm model, we have that particles, electrons, quantum objects are particles only. They have an electron as a particle, a neutron as a particle, etc. But associated with a particle are two fields, the r and the s field, which come from the wave function. If you take the quantum state, equate it to r e to the is over h bar, then are two real fields that co-determine each other. They co-determine each other in the sense that you can derive an equation of motion from the R field and obtain

7:30 trajectories, or you can break from the S field. And both carry parallel information in that sense. So you can, so they co-determine each other in that way. In the Bohr model, the particle will go one way only. The interference will be explained by the superposition of the RMS fields. Now, the part, the main part now is, well, I want to essentially give a description of this experiment. Lentology will be different according to the causal interpretation of boson fields. This answers a wheeler-deway choice kind of experiment by providing a causal model from the end. And this doesn't matter which way you arrange your detectors at the last instant. The physics stays the same throughout. In the non-notifistic case we have a picture of particles with two RNS fields. Now what is the picture with the boson field? Picture begins when we notice that second quantization occurs by cheating the field functions and the conjugate momentous operators satisfying the equal time computation relationship. Now if you notice this is exactly equivalent to writing a field Schrodinger equation. The equation differs from the ordinary Schrodinger equation because it's a functional equation. It's nonlinear, it is non-local, but the solution is a functional. We may notice that although the Hamiltonians that go with the equation are invariant, covariant, the equation itself is not. It's a Schrodinger equation, it's not covariant. One implication that has relativity is not clear, but certainly the answers that it gives are fully consistent with relative complexity at the statistical level. Okay, so our solutions are functional, and if you take the modulus of the square of the functional, field configuration, as our earlier speaker was pointing out. So we have, well, just a picture of a classical field with a number of possible forms and it's given by the probability squared of the wave function. You might notice that you can always approximate a functional as a function of many variables. If you divide space up into cells and put a field function in each of the cells,

10:00 you can approximate your functional as a function of many variables. And this suggests by analogy with the many body problem, the many body weight function, that we have non-locality is an inherent part of the field. So when we have a classical picture of the field, an essential quantum element is that if you do something with the field at one point, you can affect it in a distant place. So the field is non-locally change in general. And in that way, it differs from a classical theory. The rough steps, we can do this, make the substitution, psi-cazari into the iso over h-bar substitutions equation, and then we end up with two equations, continuity equation and the Hamilton-Jacobbe equation. The continuity equation expresses the conservation of probability in function space, equation, is analogous to the classical Hamilton-Jokovic equation for a field, but it has this extra term, the quantum potential for the field. So by analogy with classical physics, we can then define total energy, we can define conjugate momentum, and then the vehicles that come from that, energy will be defined in this way, the contral of momentum like this, and from that we can then define, we can then find vector potential, electric field, magnetic field, energy density, momentum density, and this would be the B of those. I might point out one thing. If you take the functional derivative of the Hamilton-Jacobi equation, you end up with the analog of the classical wave The first part, you can see, is just the wave equation for the vector potential. We have the classical wave equation in this part. Now, for a radiation field in empty space, this will be equal to zero. So this term, this derivative of the quantum potential, is in some way a measure of how far away from classical, or the classical field that the quantum field will behave. And this is where our semi-classical wave field will always break down. potential, if this term is significant, the quantum effects will also be significant.

12:30 It's very difficult to work in function space, so it's much better and much simpler mathematically to just do what they're doing in field theory, is to take the normal mode, make expansions in terms of normal mode, so we have a vector potential in the functional momentum, and just make a Fourier series in terms of our normal world coordinates, q, k, mu, t and pi, k, mu, t. In our interpretation, the q, k, mu are regarded as functions of time. In our field theory, there will be operators, but the q and the pi, there's a combination of these that make the creation and annihilation operators. q is not a creation operator, but a combination of q and pi will be the creation operator, and there's a simple formula for that. Again, we can make the Schrodinger equation, the continuity equation in the Hamiltonian curve equation. Again, we get to the term. The wave function now becomes simply a function in Q k mu space. So we don't have a function anymore, and we have a much easier quantity to deal with. We simply have a function of Q k mu and t, and this gives the probability of a particular value of Q k. So in my series, If I work out my probabilities, this K1 has one value, K2 may have another value. Put this together, this defines one particular configuration. A different set of values will define a different field configuration. And so the picture is consistent with what we said earlier. Again, we can define total energy, et cetera. The part of home two approach is my equation of motion for the Q-K-U. From, again, by acknowledging classical Hamilton whose curve equation, we can define a country of a mentor, and we can insert this, equate this to the classical relation, one of the C-squared Q by DT, and this provides our equations of motion. If we can solve those, then we can express Q-K-U-T in terms of the initial values, my various vehicles, vector potential, vector potential, et cetera, in terms of the expungence of time and in terms of the initial values. I've just written down the field. I want to stay on this for two more. Just to give you an example of the field vehicles, you can derive all these forms of

15:00 essentially classical with the conjugal momentum given there. So to find our beigold, we will need to, for a particular state, to find the various derivatives of s with respect to the qk mu, then solve the equations of motion for the qk mu t, and in this way you can obtain your beigold as functions of time and the initial values. So you essentially have a completely classical picture of the But with the added element, that you have this non-mobile, this inherent non-locality within your field. Okay, so with that, with the background, with that, we don't speak the actual description of the reader-delayed choices in this model. I guess from the model itself you can probably guess very quickly what the basic description is going to be. To take my state, at each reflection I've taken a pi by two phase shift. So every time I have a reflection I introduce my pi over two phase shift. If I follow this through, I get a five phase change there. I end up with my state at the end. This is a contribution from this side and then a contribution from this side. And I have my final states after the beam splitter. And these are my states before the beam splitter. So my wave function before BS2 will be the first one. So this will be the appropriate description when the detectors are before beam splitter 2 in the which path configuration, and then, for my second, for my description path of b-splitter, I take the second with the wave function phi 2, phi c plus phi b, and that's such. Okay, if I analyze now the solutions before these two, I have my derivatives with a particular wave function of these and work through and I end up with these equations of motion which

17:30 then needs to be solved. Generally speaking, for the particular state that we're using, the ds by dqk for k doesn't equal plus or minus k alpha, or plus or minus k beta, will be zero. So all my Fourier coefficients, except for the two, one for each being, are zero. So I'm left with only two coupled differential equations to solve. It's not particularly difficult to solve them. You can solve these to get your, I call it QK mu alpha and beta, but they're still the normal one for them. And then I have my explicit solutions as functions of time, and the relationships that must exist between the initial values. Again, I put all these down, but it's essentially substituting the formulas, the Vs by the Q-k-mu, and for the explicit solutions of the Q-k-mu-t. And then you have explicit expressions as functions of time and the initial values of the various field vehicles. This is before the beam splitter. And I can do exactly the same thing for after the beam splitter. We can work out the derivatives of the state after the beam splitter and obtain our equations of motion. Again, we can solve these explicitly with these solutions. And again, these relationships between initial values are 10. And again, I can write my values. I'm not leading these up because I've just shown that they exist. I guess you're not going to look at them in detail. And one final relationship. We can relate the quantities before and after the beam splitter. It's just a matter of matching the boundary conditions and you end up with these relationships. So essentially, we have a full description of the Mark Center interperometer in terms of the field vehicles throughout the entering and leaving the interprometor. For the interference

20:00 effects, it's clear, we have, I think it's that first diagram. For the interference effects, this is clear, we have a field interpretation, the field splits the first beam splitter, we go through, we end up detecting interference. So the question when you have a field approach is what happens in a which path detection. Here, what we can do just to see what happens is to actually model the detector by a hydrogen atom and then use the hydrogen atom to detect the position of the, of the position measured. I will say one thing. I'm assuming here that anatomy is a loquized chunk of something. So I'm assuming a lot about vermeil entities, that they are loquized and they're loquized chunks. I'll say a bit more about that next. And with this assumption, the particular criticism that Thomas made earlier, when he mentioned that people like the field approach, if you have a localized field at one time, you have a field that spreads out at later time. In other words, when that happens, how do Now that we can answer if we assume that atoms are low-farm. And the answer, what it's relevant here, we have here two beams that are split, yet we have to get one constant. How does that happen? Possibly the answer is obvious. Let me outline quickly what we can do. Using standard perturbation theory, we can actually deal with interaction of the field with an atom. So we take my Schrodinger equation, the radiation Hamiltonian, the atomic Hamiltonian, and the interaction term. And then we want to solve that. We want to solve that for the state before the beam splitter. So that's my initial field state, my initial atomic state. I will also assume the quantum is sufficient to ionize the atom, and then the interaction term. How long is standard preservation? Let me summarize this, perhaps focus on some of the conceptual elements. So just follow standard perturbation theory. I will write the wave function as a series of possible solutions.

22:30 And then by the standard form as a perturbation theory, I can work out the particular coefficients explicitly. I'll point to one significant term in all of this. Let me point to, in calculating the matrix element, this term arises where this is the initial state of the field, this is the final state of the field, and this is actually a creation operator. It's not a normal coordinate. Now we can see that it's a destruction operator actually. If it acts on one quantum state, just from the mathematics of field state, that's going to give you the ground state. By the authority of the field functions, this integral must give zero. In other words, the final state must be this one. This final solution, this is a part which represents the initial field function and the atomic wave function. So this represents a situation where nothing has happened. This represents the interaction. This part is outgoing electron plane waves with an associated probability. And this is the field left in its ground state. So if interaction occurs with the atom, we must have that an entire quantum is absorbed. So what we have is that the quantum is swept up non-locally from both beams and enters one and is absorbed by the atom on position. So in this way, a spread out field will be observed as a spot on a photographic plate. It's very uncommon. Where the criticism stands is, I don't think here we have a causal interpretation of fermionic fields along these lines so it's very interesting that this chap Colin has made a suggestion according to the different suggestions of Bell and it's interesting that it exists but for us the one criticism we face I think as yet a fermionic field cause interpretation has got me to read about Valentini has made very interesting suggestions where he field as fields of brassman numbers. It depends now whether you view a field of brassman numbers as an ontology or not, but I guess it would be possible. Let me ask the one question.

25:00 I have a general comment on this program he's going to deal with relations over there. Now, in the nomativistical theory, there's a very important model that you actually, in the category of different quality patterns, by the hymns, as much as the hymns, that you find the branches, perhaps, such that you can want you to be able to exhibit that you use a class so that it's a sort of re-development. That's fine in a moment, but you can't tell it's a part of what you want to do this. It's also fine to collect amenities in itself to give because the D.K. Holmes basis for massless desolation fields is roughly reduced to stay by instance. Now, we don't even expect something like that, either massive desolation fields or beyond the other than we've got, but it's not to respect this. There we'd expect D.K. Holmes basis was very sharp, and there it would be basis defined by fields. So that seems to be open by many particular ways of detection and that seems to create a fairly general problem for the university. Let me just summarize how I understood you quickly. You're worrying now about the solution of the measurement of the problem system. But in the bulk case, in the non-assisted ones, we have, whenever we do a measurement, we need to interact with our system and with the apparatus. The solution is going to be a series of terms. The eigenstakes of the operator of the measurement. is linked to a point of reading. So we have a whole series of terms. Now, the problem perhaps occurs because we see only one term of that series. And so from a normal perspective, we have a practice of the Bayesian function. Bayesian answers that because, not by getting rid of any of the solutions, but one packet contains the B of it, and the other packets remain negative. So they have a solution of the Bayesian problem. But a bosonic field case all solves the massive bosons as well, so actually, this one, Well, the cosmopolitan interpretation of the first fields we cover were the electromagnetic, or the Klein-Gordon arena complex. And it works exactly the same way for those fields. But the answer comes that the packets now occur in QK mu space. So it's not in all mu space, but we still have the same solution of the measurement problem. You have your configuration space for the apparatus, you have the QK mu space, you have this huge space in which these solutions of the appropriate

27:30 will be packets in the space so the whole series of times when we pick out one packet so the solution of the field collapse I've described is not part of the collapse mechanism from the state so I actually will solve it but I'm going to leave a family for the situation Thank you. First of all, I would like to thank to ,, who gave me a chance to have this . And the title of 12, I would say, is a highly degenerative measurement for a mortality test. And I put the subtitle as a finite . I just mean here the locality cast is nothing but a violation of balancing policy. My name is Wan Min's own second year PhD student at the University of Belpris and I'm working in a quantum optics and quantum information group with Professor Swain and Dr. Kim. The theme of talk was given in order, like I will discuss about the idea of the measurement and in terms of observable, I will see some structure of observable and will inquire about number of parameters for the unitary operator. And then I will define a certain correlation observable which describes the amount of correlation from the measurement with that observable. And if the

30:00 If the correlation is a certain property in any type the type system can have, then we can think about the correlation observable, which can give outcome as an amount of correlation. And I will get some condition for that observable, and I will relate how that correlation observable can be used for deriving false equality. Then, I will apply those for the continuous variable state. If we like to have just the outcome measurement from a continuous variable state, which naturally means each outcome is infinitely degenerated. So I will give a certain permission operator which can give just a definite outcome for the continuous measurement of a continuous variable state, and I will show a violation of values and equality for a specific continuous variable state here, and I will give a permission. We know certain de-outcome measurements can be expressed in a d-dimensional diverspace as a hermitian operator, which we call observable. And such a hermitian operator can be represented by d-by-d matrix. And with the 100-discity condition, we know there are real parameter is d squared for describing d outcome measurements. So, which means if we have a d squared of real parameter, then we can fully characterize our measurement configuration and setting. So, things like that, we can describe a certain observable, which gives a d outcome, as a point in a d squared vector space. Such a representation is possible using what is called a generalized block vector representation. Here just the A and A, J are the real parameter for the observable, and here I is the identity matrix and SJ is the group generator, or in general case is SUD group generator, and for

32:30 For two outcome measurements, this is a polystremetric sigma X, sigma A, and sigma Z. In a more spherical sense, the observable can be represented in a given basis as a spectrum of eigenvalue for each outcome we assign. So this is the eigenvalue for each outcome, and this is given certain measurement setting as a basis, given basis, and this basis can be transformed by the unitary operator, and this unitary operator can also canonically parameterize using a group generator like this. Here P is just a real value which specifies a certain unitary transformation for a transformation of basis. And this unitary operator will give a transformation of generalized block vector we just I just just show you in the previous slide and that SUD you need to operate it can be realized, realizable experimentally as an array of beam splitter and phase shift which is called of the biased multiple beam splitter, and this was suggested by Chukovsky and his worker in 1997. So among these different expressions of observable, can we make a connection between of those? Of course. So if we make a relation between the eigenvalues and generalized block vector, then like A0 So it's nothing but a summation of all possible eigenvalues and the norm of a generalized block vector is a square of eigenvalues, a summation of a square of eigenvalues, but we only have a d number of eigenvalues while we have a d squared number of real values for generalized which means we need a d-square minus d number real parameter

35:00 to describe a given basis of a measurement. And so we can ask, like, is that really true that like a d-square minus d real parameter can fully describe a unitary operator? The answer is yes, yes, and we can visualize this situation for the case of Stangela measurement because if we think about Stangela measurement which is just a two outcome measurement, then we assign the number for each outcome as IMW, spin half, and minus spin half, And we need to specify the direction of Stangellac magnet, which needed just two number of real parameters to be described. So that corresponds to the unitary operator, and the rest of the parameter describes the eigenvalues. And that can be proved previously using a subgroup generators, but I'm not going to go through the group here. So I just show you the structure of measurement which gives a D different outcome. and we here are interested in our correlations of a certain system, certain two systems, then can we describe a correlational syllable in terms of measurement of each part? If like that, the situation should be like this of measurement of observable for side A and measurement for the side B should describe certain correlations and that should be taken as a correlation observable. But this is not the case because this situation just means measuring each side while it does not compare each measurement outcome.

37:30 So this is the general form of a correlation observable. So if we consider classical communication for the purpose of comparing each outcome, we should assume it has a form like this. here muij, what we call here correlation table, or correlation waiting factor. And this will give the amount of correlation from the local measurement and classical communication for the purpose of comparing each other. Then, is this correlation of SOHO can describe real correlations, quantitative correlations of subsystems? The answer is we need more condition. We need more condition to take that as a correlation observable. One very important constraint we should give there is no local property that correlation of probability should have because if we measure one part while we don't measure the other part, then that measurement shouldn't give any information about the correlation part, which means we trace out one part, then that should be always give a number zero. And the other condition we should think is if we measure each part and we have a spin-up and spin-up, then we can think as a perfectly correlated. Then, if we agree that it's perfectly correlated, then we should take spin down, spin down, also should be taken as a maximum correlation. And then we give a constraint in the name of translational symmetry. And that should satisfy these conditions. And last condition, we think, is a maximal discrimination, which means if we set 0 and 0 as maximally correlated, and 0 and D minus 1 outcome is maximally anti-correlated, the number between those should give amount of correlation, number between 1 and minus 1,

40:00 and that should be equally spaced between 1 and minus 1 as like a measurement of angular momentum in a z direction should countries from j to minus j is spaced equally. So this condition, with these three conditions, we can uniquely decide, determine that eigenvalue mu jk is like this, so we can finally have what can be called a correlation observable. And that correlation observable gives an amount of correlation in terms of a correlation function with a configuration of each part. And this gives the amount between minus and 1. But we can ask for here, if this correlation can discriminate the correlation, what we can say, quantum correlation from classical correlations, we can ask. Like this, the answer is quite negative because for certain products, they also can give maximal correlation. two cases also bound from minus N1. So for the purpose of discriminating the quantum correlation, we need more than one measurement set in each part and what we need finally is called balance inequality. So as a linear combination of these correlation functions with the two measurements set each side we can have a certain function whose bound is proved as two for the local realizing model and and that is uh that is the same balcony which was found Collins and his work last year so we can happily use this balcony for uh as a quantum collision and one of the main power of specific version of Bell's inequality is the result of the violation of this Bell's inequality

42:30 is well agree with the numerical search of amount of noise which can break quantum correlation of the system, and this means, for the case of higher violation of this Bell's inequality, then it means the system is more robust against the Y noise. So, this Bell's inequality can be used comparing how much of the system has a quantum correlation. So, this is good. However, in their study, they used a certain, like, specific type of a unitary operator, which is called QFD. QFD is not quantum field theory. And so they use only this unitary operator, QFD, which is parameterized for only one single parameter, and it can be questionable that this unitary operator is optimal measurement for the outcome measurement, the answer is yes and no, or in this case, answer is yes for the case of maximally entangled state we found, because with the single parameter mutual equation, which we can call quantum free transformation, can give any value between 1 and minus 1 of correlation functions, while for the case of partially entangled state, we thought of optimized value of a violation of a CHFH version of balance inequality for the case of this state, which is generally partially entangled state, and we found the it's a smaller violation of Bell's inequality than SU2 unitary operation, which means QFD is not optimized unitary operation for the partial integral state, and we should consider SUD unitary operator for that Bell's inequality if we want optimal value of violations. Okay, so with this thousand equality and measurement, we would like to apply these to the continuous

45:00 variable system in many reasons. One of them is we would like to know how much of a contemplation that continuous variable state can have and the other one is in this specific study we want to compare the amount of violation of Bell's inequality with a different number of outcome for a single system. We need the objective single system to comparing a different number of outcome. So we choose as a continuous variable state for that and for that we need a certain observable which can give a finite a finite outcome from a continuous density matrix, a continuous state. And this observable only has a diagonal component, and each component means observable for the outcome measurement. And alternative expression of this observable can be like this, which shows us it's nothing but a projection of a D-modular basis into the same state of a continuous variable density matrix. So using this observable also means also means if we apply this measurement on a continuous variable state, it gives the same results as like applying just the non-degenerated measurement on a certain d-dimensional dense matrix. And this implies we can, instead of using this measurement, we can make a mapping of a continuous variable state into the And then we can do the same thing we did in a D-dimensional state. And so, is this a possible legitimate kind of a mapping? Then we can say, formatively, because this mapping can be proved as a linear trace-preserving complete positive map because it has a cross-representation like this.

47:30 Okay, let's now see a specific example of a formalism I showed you. Then we can take a certain example of untangled continuous variable state, which has a good character. It's a tumor-squeezed baking state. It can be represented as a number basis like this. Here R is a squeezing parameter, and this state can be generated using a non-degenerated optical parametric amplifier. And this state is example, goes EPL state at the limit of R goes infinity. And with the mapping I just show you, the tumor squeeze state can be mapped like this. And this n is nothing but a normalized factor, so at the limit of Argo's infinity, these states are projected into a maximally entangled d-dimensional state. So, if we plot of a violation of Bell's inequality using the observable I just introduced, and then optimize in an SUD space, using an SUD configuration in each side, Then we have this graph, so what we have as a result here, and each graph means this is two outcome measurements, and this is three outcome, and four outcome, five outcome, and this is tangent type of scale squeezing parameter, and what we have at the infinite squeezing, it has a value like this, but that is not a maximum violation, and at the finite squeezing limit, it gives a maximum violation, and a higher outcome gives a higher violation of balance inequality for the high squeezing limit, while lower squeezing limit was inversed. So this result is summarized like this, and one more thing is, no matter what the number

50:00 of outcome we considered, it always gives a violation of balance inequality. So with all this, I can make a conclusion. So in a study of the outcome measurement, what we found is the number of parameters which fully characterizes the unitary operator is d-sphere minus d and what we did for the next section was I defined a certain correlation observable and with a certain condition I found its eigenvalues and we found that it can be used for driving Collins version of equality and and what we did for the highly degenerated measurement is we use all these schemes for the continuous variable state and and for the the outcome measurement on the continuous variable state we can found certain map which is a completely positive map and we so we see how much of 1000 equality violated with tumor speed state using the outcome measurement. Thank you. Well, I mentioned 3, the maximum violation does not correspond to the maximally untangled state. No, no, in this case, no. Can you explain it with your formalism? Because it's still a mystery why this data isn't... I know Asin and their work already showed partial entangles can give a higher violation of Bell's inequality and there was a mystery and I have a conjecture on that but couldn't conform is there are possibilities that partial entangles can be decoyed, can be less decoyed than maximal entangled state So, and this belt inequality is agreed with the amount of noise thing, so we conjectured that partially in angle state is more robust against the white noise, but that is nothing

52:30 but a just conjecture, I can't, I just, I just couldn't say more than that. How much does the maximum violation grow for D towards infinity? That is a very huge simulation because we have too much parameter here. We tried until the 10 outcome measurements, and what we saw was that it was almost approaching to an asymptotic number, like almost 3.1, stuff like that. I don't remember the exact number, but it was asymptotically approaching to a certain value. But that was just numerical search. I got a bit confused earlier in the discussion when you were defining your correlation observable and you said you had to introduce constraints of considering local classical communication But in that sort of familiar case of the observable you consider only doing just two cubits which are entangled, it's just some spin direction this over here and some spin direction over here. And you said, well, that would just be measuring one on either side. There's no correlation there. That's precisely what we're doing, just measuring one here and one here, and there's no tracing between the two. Right. That's because I just tell you about like a measurement on a single qubit. But if we consider our ensemble of ensemble, then we come to like a, like if we have like a spin-up and spin-up, spin-down, results like that, or let's say two measurements, then that doesn't have any correlation at all, which means I didn't consider here, I didn't say about anything about the unitary break up or that observable. So that is just a comparison. I just explain it for the purpose of explaining the observable.

55:00 Could you say why you don't consider the standard correlation coefficient, the theory of random variables? standard correlation coefficient. When you're defining a correlation zero, why don't you consider it just a standard correlation coefficient? Defining the theory of random variables. Why didn't I consider like just omega a and omega but that's the product of omega A and omega B? No, no. Omega B? What's that? Why do you not consider, I mean, Bell's measure, correlation measure is just a special case. The correlation coefficient of random variable. Why didn't you just take that general correlation coefficient expression to create the correlation? Because I assume that a certain convention can be measured by measuring each subsystem. And I start from them, and like, after measuring it, we need a classical communication for comparing each outcome, so I think I didn't understand the answer to my question. because that isn't representing the operator you're measuring, that classical communication isn't representing the conditional operator you're measuring on both sides. Um, it's, um, like, okay, I, I just, I just, like, weighting factor which, which can represent the amount of collisions. then I didn't mention about the rotation of measurements configuration. If we have like the orthogonal measurements then even though there is spin up and spin down, then it shouldn't give any correlation, which means that one is spin up and the other one is spin like randomly distributed which means when I explain that that correlation observable I just assume intrinsically measuring in the same direction So I just explained in that case, if you rotate each measurement setting, then amount of correlation should be changed. So, I don't know.

57:30 No, I'm sorry. If you have two random variables, you can define the correlation coefficient as just the variance. Right. Divided by the product of the standard deviations. Yeah, covariance. I think it would be covariance. It's between your 10 dimensions, for instance, you have 100 bears, 10 at each side. It's very complicated because he wants to optimize the variation of bear inequalities. This is a way to reduce the problem. Is that means you, instead of I considering all possible mu ij, why did I consider a certain specific . Sorry, I can't. I think maybe we should . Thank you again. Thank you very much. So, finally, this session might be seated. A criteria included in the panel. I should mention that this is a revised and short-referred talk given two months ago in a workshop on Holism and Physical Theories, organized by Holism. And I'd like to clarify what we can make of Holism in quantum mechanics. And it was a short motivation before I could get out on it. So, the question of whether one of you can advise sort of for the homilisms that are positive. Why? What is it that makes it a holistic theory? And other theories, possibly not. And I'd like to propose an operational criterion to decide whether other theories are holistic. And with that I mean to decide from the formalism operations it allows for and things you can do and from some property assignment rules whether or not the theory does interpret it is holistic.

1:00:00 And I would like to contrast my approach to what I would call the standard approach of holism in terms of supervenience by showing that the supervenience approach is too limited Okay, and a short outline, so I'd like to mention what I call the standard approach to the whole list, and address my view of how to view the whole list. Given criterion to start with a lot of theories on this in terms of operational means, and apply it to orthodox quantum mechanics, that is to do the criteria in quantum mechanics and to show that it is indeed holistic, but without the need of intelligence. And, I'd like to conclude. Okay, what I call a standard approach to holism is supervenient standards on So a physical theory is holistic, even only if some of the properties of the whole must be revealed on either or both the properties of the parts and on the mutual interactions that can occur according to the theory. And what's been taken as a parametric example of holistic theory would be called orthodox quantum mechanics. It has a certain whole, For example, the bipartite bottom stage which has a singlet and a triplet state, it could be in a singlet or a triplet state, and both states have completely different spin properties, completely anti-correlated or completely correlated. But in both cases, the individual systems have the same reduced state and no spin property at all. So, to a difference in global properties does not correspond to a difference in local properties or subsystem. So there's no supervenience and best of the theory is holistic. This is just an example of a supervenience approach to show that quantum mechanics is holistic. However, I'd like to approach this example from a different point of view and show that It's not a full story because it's too idealized.

1:02:30 It only looks at certain states and all of what we can do with the states and find out properties. So I look at what normalistic physical processes, according to the theory, can actually be performed instead of solely considering state descriptions, which I call operational status. If that is possible to determine, using only local means and classical communication, whether or not one is dealing with either the single or the triplet state. And that's quite easy. You just measure on each subsystem the spin in the z-direction. Compare the results using classical communication. And if you have the same parity and the state was a triplet, the parity is not the same when the state was a singlet. So using only local means of classical communication the two belt states can be distinguished after all. So the two different local properties can be obtained by using local meaning as a classical indication. So I would say there's no indication of holism in this case, which is contrary to the paradigmatic example, the supervenient approach. So how should we then address holism? I would say that, first of all, we need some sort of property assignment rules, and some theory gives natural property assignments, other than not, then we need a certain interpretation. For example, quantum mechanics, you can have Bohmian interpretation, where you have a property assignment on a phase-space for the vehicles that are in classical physics, or you can look at orthodox quantum mechanics, where you have the eigenstate eigenvalue to assign properties to normal systems. Further, besides the property assignment rule, I think we should focus on properties which are assigned to composite systems and to its parts. I call these the global and local properties, respectively. And then what sort of theories could be candidates for holding? Well, these theories that contain local properties which cannot be determined from the local properties subsystems and either from the mutual interactions and in determining I'll consider physical that is normalistic constraints but how to determine these property cycles so my method is an operational spouse I think we should

1:05:00 say the physical realizability of measuring global properties that occur And we should take it as a constraint that we only use as local operations of classical communication, abbreviated by LACC for the relations community. And the guiding idea is that only LACC, local operations of classical communication, provides only the local properties for taking into account mutual interactions. And using these kind of operations, the question is, can we still find out global properties? If not, we find out theories are distinct. If we can't find all global properties, theories are not so distinct. So, let's share my criteria and I will formalize it a little further. So I would say a physical theory with a property assignment rule is holistic if and only if some determination of the global property assignments cannot be implemented or performed by local operations and classical communication. And it works for any physical theory with a property assignment rule and a specification of what local operations and classical communication is. And how does it relate to supervenience approach? Well, if the theory is not holistic in supervenience approach, there's neither in this regard, not the other way around. I've abbreviated this as such, and we can see that this whole, because if properties of the whole supervene the properties of the parts, And just measuring those properties allows determination of the properties of the whole. But if global properties of the whole do not supervise properties of the parts, it could very well be the case that global properties may be obtainable using local properties of classification. This was what was going on in the example. That certain local properties did not supervise properties or at least the states of the subsystem, find out the global properties by using local operations and classical communication. I'd like to apply this criterion to orthodox policy and I just take this property assignment

1:07:30 rule, by the eigenvalue eigenstate rule, that is, a physical property, a physical property property that a consumer quantity has a fixed value, if and only if its state is an item state of the corresponding operator to that observable. And the value would then be the item value, the property. And I consider only ideal for modern measurements. So, formulated in terms of quantum mechanics or of those quantum mechanics, the criterion is that a part of quantum theory is holistic if and only if some of its global property assignments cannot be termed by local quantum operations and classical communication. Well, given the property assignment rule, what is mobile operations and task of communications? Well, let me first specify how to view a general quantum process. It just takes a certain density state on a system to a different density operator. And those could be different in the spaces. So it just transforms a certain density operator to another density operator. And this mapping should be a completely positive trace non-increasing map. So it should be an operator acting linearly on emission matrices such that if it is tensed to a bigger system, it still takes density operators to density operators. This is at least a completely positive requirement that if you do nothing on the system which happens to be James Hamilton, you still transform states into states, and also known by form operations. So these general processes, what is the constraint of local operations plus communication in But, well, the class of L2C operations consists of the two following two elementary operations. Do something on system A and nothing on B, or do something on B and nothing on A. And the class of communication comes in and you can condition on which result one of the two obtained.

1:10:00 For example, say A performs a measurement, and she can communicate her results to B, after which B performs a measurement, conditioned on the result A received. So that would be this method. So A performs her operation, after which B performs his operation, conditioned on what A did. This could be a whole sequence of such communication, balance, and measurements, or evolutions. So, now we have criterion in the orthodox formula, and we know what a property sign can be. I'm considering what orthodox quantum mechanics is. And what low-operative industrial communication is. So now the question, is quantum mechanics ballistic, right? According to my criterion. Well, it is. And I take an interesting example. not needing entanglement, you can take an example of needing entanglement, that does need entanglement. But here's an example of not needing entanglement. So, suppose you have physical quality, which the first point of operator R, that has a set of nine eigenstates, where nine plays one to nine. And we look at the following property sign of the rule. If the system is in an eigenstate, sign I, it has the property that the quantity has to value R. And here comes the trick. Suppose this observer works on a bi-partite system where each system has three dimensions and you have the following set of orthonormal ion states on that operator. Obviously this notation is clear. For example this state is the state 0 on the first system and the superposition state 0 plus 1 on the second system. And I did not mention the normalization. The thing to notice is these are all product states. So there's no entanglement involved. And perhaps, superficially, you would think, well, since these are all product states, you can measure the observable r of which these are ion states. Well, let's see what that implies. So we want to determine if the composite system as a property, the value of the observable, is one of these nine numbers.

1:12:30 And we would like to use only local operations and classical communication for like A and B. So we have to determine which item state A and B have or project on during the measurement. Which one of these product states that we project on already have. I'm going to model this a little bit, so say we project on i to say psi i, then two-dimensional outcome i, you can associate a certain bottom operation where the trace is just renormalization, where you consider these projection operations. The way I write this in this form, because you want to have classical records for a, for party A and party B, which they can communicate and you can catch conditional, conditional items. So even though you project on a multi-part accident, I use the classical records of the algorithm for A and B. And these can be considered to be local properties of the subsystems of A and D. And using any of such projections and putting them in this form, it amounts to determining the global property assignment given by the observable R. Well, it turns out this cannot be done using local operations across communication. That was first done by Benedict Co-workers, which coined this phenomenon that monocality without entanglement. I thought it was a confusing term. But anyway, this is the trick you have. Observer which has only product states. Nevertheless, its item states cannot be distinguished, or in my case, its property assignments cannot be gained using only local operations and classical communication. I can sketch the proof, we have these nine states, and say the A and B perform projective measurements in any of their operation or communication rounds, the distinguishability of the state is spoiled, and it happens in any local place, it's A and B projected or measuring. And in other words, in some of the states, the A and B alone is non-authorical, and I

1:15:00 Bennett and Colbert are made to look at, to view this. So these nine states, let's see if this works. So these nine states can be depicted in a title like this, where the Psy-1 has a special form, but for example side two and side three are are both in this time to measuring amounts to making a cut so you can see that any cuts any of the horizontal vertical cuts one of the four tiles and spoiling the distinction finish with it. It's way to depict what's going on. Okay. Well, I'd like to conclude now that I sketched an operational criterion for holism that determines whether any physical theory with a property assignment rule is holistic. And I showed that the supervision approach of limited use. It collects operational criteria that any theory allows for. A theory tells you what you can do or not do. And I would like to use this quote that we can do more for example in this case or less than one of states or any physical state that maybe seemed to tell us. The theory does not only specify what states are allowable but also what we can do with the states. And orthodox polymechanics is holistic, according to my criterion, but it's not a tango which makes it holistic. And I'd say this to be an example of the idea that we might get fundamental new insights from investigating what we can and cannot do, or not can. I call it a terminological analogy, because what we can and cannot do machines that have higher efficiencies than current machines, which we cannot, we find the second law of thermodynamics. So the idea is that by looking at what we can or cannot do, we might get, hopefully, maybe, analogous results, laws that are possible or not possible, from which we can drive part of the theory.

1:17:30 Okay. Good to do. Well, thank you very much. It's a beautiful talk. Personally, I feel now I understand what this 99 paper is about. Okay. But I have a joke comment to you, and that is I think that people in the philosophy community will like still using the traditional supervenience definition for what holism should be. But I think that you can grab jargon from quantum information people and talk about bound holism like they talk about bound entanglement. It seems to me that the way to sell your view which is really attractive is to say that the traditional approach being metaphysical misses a distinction about whether we can access and the standard example of entangled states is trivially accessible because we can't find out that the state is entangled. Your example of triplet versus singlet, what's important about Bennett, et al, is that they show that there is, in effect, a bound holism. That's how I would sell it myself, however. It's just a matter of words, really, but I'd like to say that what Bennett shows and what you've brought out is that the metaphysics is crude, because it misses a distinction, and what it misses is the existence of bound holism. That's just a naive question following up Jeremy's remark. I mean, if we have the internal mind and focus holism on entanglement, that's somehow more natural for us that we would consider an entangled state holistically. And you've shown, well, we can get something with nine states or so, or we don't need entanglement. But in a way, isn't that a rather specific class of cases than that you can find such holistic situations that are not so physically obvious like entanglement. And I mean, if at the end was your analogy

1:20:00 with the second law, then this would be just, I mean, yeah, there are these things out there as the second law can be violated easily in this room, given the whole city of Leeds around us. but how specific is this? Well, it must be specific because they produce a paper on it. No, not in that sense specific but are there many such state groups you can find? I think this is an example of a large class it's just a feature that you have a certain observable which has product states, a composed observable and a component system subsistence states are not authoritative anymore. So that's sort of the general form of trying to find more of such But I agree with you that at first, Thailand seems a natural candidate, and we're at least in this community. We think it's all around us, and we're used to that. I would still say that, maybe even if it is only a small subclass, The reason it exists tells us or not to both identify all this entertainment. Sure, but if it were a measure of zero, what would it be? How big is measure zero? Well, again, your analogy. Okay, yes. Think about humanity. I have two points. One, the second one relates to Jeremy, so perhaps 2.8. The first one was just, distinguishing a single and triplet state by local operations of classical communication, which I can't remember how one does that, but Ken, is it possible to do it without, in a single shot measurement, without sharing prior entanglement? Because if you have to share entanglement to begin with, then you're cheating, so you're already using the state process and all that stuff. In the case I don't do this, you can do it in a single shot. I'm only distinguishing two bell stats, two of the four maximum entanglement stats on the . But it will distinguish all four. We need more resources, we need to do more. But I, it has been claimed that it can be done using only local operations class communication

1:22:30 I think called pre-distributed randomness, shared randomness. Yeah, but is there a container of randomness or just? No, no, just a bit of classical, this string, which is pre-shared, but it's random. I tried to sort of model what's going on, and I have my doubts, if you can do it. It's a good job, I'm a distribution of all four bell states. It's still an open problem, for me at least, if you can do it or not do it, it's a lot of operations. But it was a specific example that the surveillance approach calls it on this, because of this example, the theory would be on this. And I would say, well, it's too limited, it's a good approach. You can't do more. You can't find out if you're buying a problem. But that's not considering, for example, in the Bell Operator, then it's considering just a multi-general operator, isn't it? Yes, it is. So you could say, well, is genuine realism to consider this other operator? The Bell Operator, which rejects onto the whole entangled basis, then you can't, those properties can't be determined. So, there's no line of saying that no title leads some reason, or at least some of it. Someone needs to phrase it for them. When you pointed out, I think that the property in vegetation will be used as interpretation. Now, if you have an interpretation like Saeber, which I guess the property in vegetation would just be something like the entire state is the property of the state. So, in the proper state, it should be such and such a metric . Now, from that perspective, there seems to be a problem with the definition of holism abuse if we're considering single measurements, because, for instance, even on a single system, which isn't in any sort of process at all, no measurement will distinguish between this and this. You mean through normal forms? Yeah, sure. I take a never-type proper infatuation rule. like say it's either this stage or that stage. Now, no one can, no only thing like that but it's normal definition yet we still can't distinguish the two by any single definition. Now we could of course do the distinction between zillions of copies but I guess if we had zillions of copies we could distinguish the states and the values and so on.

1:25:00 Well, first I agree on you that, you know, it's interpretation-dependent, which rule you could choose. And about the average, probably Simon's rule, it could fairly be the case that probably Simon's rule does not lead to a means of design, but I don't know enough about the average. Criteria for holism, whether there is criteria for holism, so it can also depend on the property imagination rule? Well, obviously not. I think if the property salmon rule is clear, then you can. You can ask yourself, well, is this a global property? And then can I find it out by using only local properties to pass to the detention on sepsis? But since they can't distinguish now the colonial states have all the whole single system, then the sentient rule says now the colonial states make out anti-projections then. Doesn't that make that state realistic, by definition? Maybe, yes. But I don't know nothing about the efforts there. I just want to go ask, what exactly does Bernie et al. prove that they, is it about one-shot distinction or is it with many, many copies allowed? Many, many copies and also using sort of weak measurements, I use a POVM. Even with that you can't? No, I can't. Local, I guess, just like, in this week. But since I limited myself to von Neumann measurements, or ideal von Neumann measurements, but that's why it was so long as well, because the proof for the non von Neumann case is very long and long enough. So, thank you. Thank you.

1:27:30 Thank you. I just wanted to try to help a big, as you get a kind of a house manager, and what's the house manager? When he gets dropped or he gets dropped, because there are a lot of things that are in fact you're running to get up with. And why is this? I'm not going to say that I'm not going to say that I'm not going to get you right. I'm not going to get you right. I'm not going to get you right. I think you're right. Thank you. Thank you. I think you've really made philosophy out of it, it's been a big deal, especially in

1:30:00 science. When you look, what is going to happen, I suppose, I think, what is going to happen to you? I don't know. I don't know. Thank you.