High School Does the EPR experiment imply QM is incomplete?

[Mentz114]

I don't think this is correct; QM also predicts, for the entangled state in question, that, for each wing taken by itself, the probability of a photon passing its polarizer is 50%.

Right. And that is also the result predicted by $P(01|\alpha\beta)+P(00|\alpha\beta)=P(10|\alpha\beta)+P(11|\alpha\beta)=1/2$ and the other marginals work out the same.

I have it written out explicitly but it would be a long post.

 

[Boing3000]

I can read Javascript, but you can hardly expect everyone here to do so. When I asked for an explicit description I meant a description in words or math, something that everyone here could be expected to understand.

I have done that at post 13# and give more details at post #20. The problem is that you don't realize that the "algorithm" IS the EPR experiment. There is a one to one relationship between each step of the EPR setup within the algorithm. It is a straightforward implementation. There is no fuss, no weirdness added, no information added. The trouble is that some people here wants to add things (like speed/FLT) or "signal" (B(or A) can detect a change), when there is factually no such things anywhere to be found. For the last time I will repeat those steps. I expect again disappointment on you part, because you'll pretend then things are missing (like a function that take A and B as input). There is no such thing.

As a side note, I have taken a great deal of time to additionally explain that wanting to ADD more functionality in the algorithm, is not even possible, it would break its ability to reproduce QM correlation. So here is again those steps, with "uneeded" comment centered (but needed by DrChinese or you). Again those comments have nothing to do with explaining the algorithm result. But just explaining how they relate to incorrect interpretation.
On the right I will restate some general comment on how some step can be related to physical phenomena. (but it is just comments)
Things that are not written don't have to be added.

1. A pair of photon (A & B) is prepared in an entangled state.
   That state in the algoritm is $A{\rightarrow}V{\leftarrow}B$ representing a unique shared hidden variable V. V in this setup is the hidden polarization angle chosen randomly.
   
   
   It is trivial that at this point any function using A as input will always give the same answer with B as input, because all the information is in V, which is identical to both A and B. The key point is unique-shared that implement non-locality
   

   
   If V was not hidden, Alice(or Bod) could guess before hand, with more than 0.5 chance. She could arrange the polarizer angle to anything but V+-45°. But then this has nothing to do with entanglement. Just with previous knowledge, like in any simple polarization test
   ​
2. Each photon are separated (to arbitrary long distance) to Alice and Bod who are going to process to a measurement that will use inputs that only exist locally at Alice and Bod. Those input (two polarizer angle) are decided randomly and locally. Nor Alice nor Bod have any idea on how the other is choosing its angle. The result is "pass" or "don't pass". The result(and angle) are stored locally (but not reused) before later comparison.
   
   The algorithm don't care about space like separation. There is no position, nor distance anywhere.
   

   
   To stress this point further, the algorithm can execute Alice test before Bod or after, Bod or randomly before or after
   ​
3. Two interaction/measure happens at random angle $\alpha$ and $\beta$.
   In the algorithm a unique and identical function is used to represent this interaction. This function don't have any internal special knowledge except a random generator. This function is local because it uses two local input $(Site photon, Site polarization angle)$
   This function is atomic, meaning is cannot "run in parallel" in algorithmic jargon.
   This function return "pass" or "not pass" and can modify anything that is accessible to it $A{\rightarrow}V$ in the case of Alice, $B{\rightarrow}V$ in the case of Bob.
   
   
   It is trivial to understand how correlation is conserved at any angle, because the function can "drop" any value in V that can be reused later by any other input also referencing the same V. But again, there is no possibility for Alice to know if she get's to V before or after Bob. Nor does Bod. FAPP V is always a random hidden value with no "A" smell nor "B" smell. There is no link between A & B.
   

   
   The algorithm local measure function will sever the local link form the photon to its "old" hidden variable, and replace it by a new identical new one. The reason have nothing to do with the EPR results. It has to do with avoiding Alice to retest A shortly after, in the hope to guess if Bod (or Jon or whatnot) have modified it "in between". This implement "destructive measurement"/dis-entanglement
   ​
   The "test" is random() < Cos^2((Site photon->V)-Site polarization angle). This test is the classical proportion of light that goes trough a filter.
   if (true) -> set V to polarization angle and return "pass"
   if (false) -> set V to (polarization angle + 90°) and return "don't pass"
   
4. After running a (big) number of those experiments, all the results are compared again but with what the "actual" angle between Alice and Bod was in each case.
   The correlation is given by "number of pair that matched", versus "number of pair that don't"

 

[Mentz114]

I have done that at post 13# and give more details at post #20. The problem is that you don't realize that the "algorithm" IS the EPR experiment. There is a one to one relationship between each step of the EPR setup within the algorithm. It is a straightforward implementation. There is no fuss, no weirdness added, no information added.
[..]

1. The algorithm don't care about space like separation. There is no position, nor distance anywhere.
   

   
   To stress this point further, the algorithm can execute Alice test before Bod or after, Bod or randomly before or after​

It sounds as if you have caught some of the essentials - the randomness of the 'before/after'.
My own simulation based on the equations I gave generates data sets that are indistinguishable from actual 2-channel EPR experiments.
Like yours nothing it is added except the shared state. I don't know why you think there's anything special about it. It is standard programming.

Could you let me have some simulated data for analysis ?

 

[DrChinese]

Have you ever consider that there is another possibility ? That you simply never tried to understand what I say instead of trolling me and threatening me ?
.

You have things backwards. You are flirting with forum rules by putting forth your own (often incorrect) ideas. Meanwhile, PeterDonis is being patient and giving you latitude.

BTW, did you ever fix your simulation?

 

[DrChinese]

The correlation is given by "number of pair that matched", versus "number of pair that don't"

Assuming you are making proper correlation be Matches - Non-matches: the results will vary from 1 to -1 as theta changes. The correlation is 0 at 45 degrees, 1 at 0 degrees, -1 at 90 degrees. At 60 degrees, it should yield -.5 (.25-.75).

 

[kurt101]

The sum of the probability amplitudes for that outcome. You appear to be misunderstanding what that means.
Suppose there are three possible outcomes; call them A, B, and C. For outcome A, suppose there are two ways that can happen, with amplitudes $a_1$ and $a_2$. For outcome B, suppose there are three ways it can happen, with amplitudes $b_1$, $b_2$, and $b_3$. And for outcome C, suppose there is only one way it can happen, with amplitude $c$.
Then the probabilities are: for A, $\left( a_1 + a_2 \right)^2$; for B, $\left( b_1 + b_2 + b_3 \right)^2$; and for C, $c^2$. There are no "cross terms" between outcomes.
You should re-think things in the light of the above.

First of all I agree with the just of your post that there are no "cross terms" between outcomes. I made a superposition of the word "outcome" in my previous post on this thread. First to mean outcome in the way you use it and second to mean as the path it took (in a way my brain thinks of the path as an outcome). That being said I think I did use it correctly in the thread you closed. Sorry for the confusion, but my underlying point is still valid (at least in my mind). If I am still mixed up, I appreciate your effort in correcting me!
So let me rephrase with your example. For outcome B there are 3 ways that it can happen. This gives a probability of b₁b₁ + b₁b₂ + b₁b₃ + b₂b₁ + b₂b₂ + b₂b₃ + b₃b₁ + b₃b₂ + b₃₃.
You are multiplying all of the ways the path b₁ can interfere with plus all of the ways path b₂ can interfere with plus all of the ways path b₃ can interfere with.
Again my point is that only 2 paths can interfere at a time to give you a definite state.

And maybe I don't really understand the definition of superposition, but either way, I don't see any evidence that there is really a mixture of states other than in then in a probabilistic sense. And the math suggests that there is only ever a mixture of 2 paths for any given instance. This seems similar, if not equivalent to the entanglement collapse algorithm that myself, Boing3000, Mentz114 and maybe others have posted on this forum; where there is only ever a mixture of 2. Maybe that should not be surprising, given that the collapse algorithm does give the same result as QM.

 

[PeterDonis]

For outcome B there are 3 ways that it can happen. This gives a probability of b1b1 + b1b2 + b1b3 + b2b1 + b2b2 + b2b3 + b3b1 + b3b2 + b33.

No, it doesn't. As I said in my previous post, it give a probability of $\left( b_1 + b_2 + b_3 \right)^2$. But since these are complex numbers, and the probability is a real number, you actually need to take the squared modulus, i.e., the probability is actually $\vert \left( b_1 + b_2 + b_3 \right) \vert^2$. Or, to write it another way: $\left( b_1 + b_2 + b_3 \right) \left( b_1 + b_2 + b_3\right)^*$, where the asterisk denotes the complex conjugate. None of these things are the same as what you wrote; none of them are the same as just multiplying out the two factors of $\left( b_1 + b_2 + b_3 \right)$. The only correct way to describe the process in words is that you add together all of the amplitudes for the different ways a particular outcome can happen, and then take the squared modulus of the result.

my point is that only 2 paths can interfere at a time to give you a definite state.

And this is not correct. You are mistaken about how the probability is computed from the amplitude. There is nothing in that process that corresponds to "only 2 paths can interfere at a time". See above.

The rest of your post just compounds this error.

 

[PeterDonis]

I have done that at post 13# and give more details at post #20.

If those posts made sense we would not still be having this discussion.

There is a one to one relationship between each step of the EPR setup within the algorithm.

Do you mean the EPR setup with local hidden variables, as described in Bell's paper? (Apparently you do--see below.) If so, that setup must satisfy the Bell inequalities, so it cannot reproduce the QM predictions. I thought you were claiming that your algorithm reproduced the correct QM predictions.

That state in the algoritm is $A{\rightarrow}V{\leftarrow}B$ representing a unique shared hidden variable V. V in this setup is the hidden polarization angle chosen randomly.

There is no "hidden polarization angle" in the correct QM model. This V is a local hidden variable, in Bell's terminology, and it means your algorithm (if it is correct in all other respects) will produce results that satisfy the Bell inequalities, not results that match the correct QM predictions.

I don't see the point of trying to make sense of the rest of your description since the above shows that you already have the most important point wrong.

 

[PeterDonis]

My own simulation based on the equations I gave generates data sets that are indistinguishable from actual 2-channel EPR experiments.

Are the code and generated data sets available somewhere?

 

[Mentz114]

Are the code and generated data sets available somewhere?

I can do one or more of the following
a) let you have datasets as csv text files
b) Let you have the simulation if you have a windows machine so you can make datasets
c) the code is Delphi and the source is available

The simulation does exactly what is described by the probabilities I worked out.

 

[PeterDonis]

I can do one or more of the following
a) let you have datasets as csv text files
b) Let you have the simulation if you have a windows machine so you can make datasets
c) the code is Delphi and the source is available

For me, a) and c) would be fine; I don't have a Windows machine but I can read Delphi source code. If you'd rather send them privately, PM me.

 

[Boing3000]

Like yours nothing it is added except the shared state. I don't know why you think there's anything special about it. It is standard programming.

I don't know why you think there's anything special about it. I have said over and over that is absolutely basic. You've just quoted a post when I said " It is a straightforward implementation. There is no fuss, no weirdness added, no information added."

Could you let me have some simulated data for analysis ?

The code is free to use, there is nothing special about it.

Actually the only thing special, is that is does not have anything special. The sharing of variable is the default behavior in computing. Doing otherwise requires extra work. Everything is classical.

I am sure yours will be fine to, especially if you use QM formula (or complex number) or if some function use input from A & B (which seem to be the case).
Mine do not.

 

[Mentz114]

I don't know why you think there's anything special about it. I have said over and over that is absolutely basic. You've just quoted a post when I said " It is a straightforward implementation. There is no fuss, no weirdness added, no information added."The code is free to use, there is nothing special about it.

Actually the only thing special, is that is does not have anything special. The sharing of variable is the default behavior in computing. Doing otherwise requires extra work. Everything is classical.

I am sure yours will be fine to, especially if you use QM formula (or complex number) or if some function use input from A & B (which seem to be the case).
Mine do not.

OK. The essential thing is that the entangled particles share one wave function. So quantum theory implies a 'shared register' which does not depend on time or distance separation and so is non-local.

 

[DrChinese]

OK. The essential thing is that the entangled particles share one wave function. So quantum theory implies a 'shared register' which does not depend on time or distance separation and so is non-local.

However, as I keep mentioning: the angle of the first measurement (say on A) is an INPUT to the equation that yields a value for the other (B). The measurement context is critical.

 

[DrChinese]

The code is free to use, there is nothing special about it.

What's special is that you have not repaired it yet, and still refer to it as something for discussion. Alice=0, Bob=120 yields around .75 on your model, the correct is -.50 correlation (25% match rate).

 

[Mentz114]

However, as I keep mentioning: the angle of the first measurement (say on A) is an INPUT to the equation that yields a value for the other (B). The measurement context is critical.

That is true and implied by the equations, I think. But knowing which came first seems to break something in my probability model.

 

[Boing3000]

Assuming you are making proper correlation be Matches - Non-matches: the results will vary from 1 to -1 as theta changes. The correlation is 0 at 45 degrees, 1 at 0 degrees, -1 at 90 degrees. At 60 degrees, it should yield -.5 (.25-.75).

OK, that's again different from what you were asking at post #31. Just for you I have added both. Correlation A is what you require now. correlation B is what you required at post #31

BTW, did you ever fix your simulation?

That's so kindly asked for a simulation that don't need any fixing. Peter knows it, he can read java-script, and the program is trivial.

The simulation is NOT build around any "special" input value, your type of correlation is meaningless globally. Globally what is interesting is that QM (3/4) beats Classical probability (2/3), you have made an entire site about it David.

 

[Boing3000]

If those posts made sense we would not still be having this discussion.

A discussion happens only when there are two people that are willing to listen.

Do you mean the EPR setup with local hidden variables, as described in Bell's paper? (Apparently you do--see below.)

There are 3 unambiguous word forming a sentence in post #48 "It does both.". No matter how mny time I have write "non-local", I am countered with a "local", which can mean "you don' read my post" or "you are trolling me"

But I am quite sure it is non-nonsensical to have an algorithm that can reproduce both classical (the "wrong" one) and QM (the "good" one), even though it is a click away.

I thought you were claiming that your algorithm reproduced the correct QM predictions.

I thought it too. But apparently that simple claim is very complicated to understand.

There is no "hidden polarization angle" in the correct QM model.

Bell's proof is not part of the QM formalism. It is a logical proof that can apply to QM statistics. Nobody is arguing the QM model.
There is a "hidden variable" in Bell's proof. In the simulation that variable is an angle.

This V is a local hidden variable, in Bell's terminology, and it means your algorithm (if it is correct in all other respects) will produce results that satisfy the Bell inequalities, not results that match the correct QM predictions.

And yet, you have seen the code, and it works both as a classical simulator and a QM simulator.

I don't see the point of trying to make sense of the rest of your description since the above shows that you already have the most important point wrong.

It is thus clear that you don't bother to get what I say correctly, nor what can be said generally about the EPR and Bell's proof, nor how it can be very simply explained in term of simple logic object, nor what non-locality mean in a non-philosophical/spooky kind of way.

It is kind of ironic, given that I got the motivation after reading DrChinese site.

 

[kurt101]

No, it doesn't. As I said in my previous post, it give a probability of $\left( b_1 + b_2 + b_3 \right)^2$. But since these are complex numbers, and the probability is a real number, you actually need to take the squared modulus, i.e., the probability is actually $\vert \left( b_1 + b_2 + b_3 \right) \vert^2$. Or, to write it another way: $\left( b_1 + b_2 + b_3 \right) \left( b_1 + b_2 + b_3\right)^*$, where the asterisk denotes the complex conjugate. None of these things are the same as what you wrote; none of them are the same as just multiplying out the two factors of $\left( b_1 + b_2 + b_3 \right)$. The only correct way to describe the process in words is that you add together all of the amplitudes for the different ways a particular outcome can happen, and then take the squared modulus of the result.

I left out the complex conjugate because it did not seem important to the point I was making. So if you include complex numbers and the complex conjugate operation then you have:

For outcome B there are 3 ways that it can happen. This gives a probability of b₁*b₁ + b₁*b₂ + b₁*b₃ + b₂*b₁ + b₂*b₂ + b₂*b₃ + b₃*b₁ + b₃*b₂ + b₃*₃.

You are multiplying all of the ways the path b₁ can interfere with plus all of the ways path b₂ can interfere with plus all of the ways path b₃ can interfere with. The only difference between the complex and the non-complex case seems to be that in the complex case the other path being interfered has an opposite rotation to it.

And with the scenario involving complex numbers you are still left with terms that have 2 paths. So assuming this is the correct math, why wouldn't you take this to mean that for any given instance only 2 paths can interfere at a time to give you a definite state.

Also to be clear, by instance I mean a single measurement.

 

[PeterDonis]

There are 3 unambiguous word forming a sentence in post #48 "It does both.".

Does both wrong, apparently. See below.

Bell's proof is not part of the QM formalism. It is a logical proof that can apply to QM statistics

No, it is a logical proof that shows that no local hidden variable model can produce the same statistics as QM. QM statistics violate Bell's inequalities.

it works both as a classical simulator and a QM simulator.

Apparently you haven't read the posts by myself and @DrChinese telling you that your "working" simulator is giving obviously wrong answers.

You are now thread banned, since you continue to make incorrect assertions and you refuse to listen to people trying to correct you.

 

[PeterDonis]

with the scenario involving complex numbers you are still left with terms that have 2 paths.

Because you are taking the squared modulus of a sum, which means you can never have terms with more than two factors.

why wouldn't you take this to mean that for any given instance only 2 paths can interfere at a time to give you a definite state.

Because that's not what taking the squared modulus of the sum means. "Interference of different ways something can happen" is what the sum of amplitudes itself means. "Interference" means adding amplitudes for all of the different ways an outcome can happen, to get a total amplitude for that outcome.

Taking the squared modulus of an amplitude--whether it's an amplitude obtained by adding a bunch of other ones, or just one amplitude corresponding to only one way that a particular outcome can happen--means "calculating the probability of an outcome". That's all it means.

 

[PeterDonis]

why wouldn't you take this to mean that for any given instance only 2 paths can interfere at a time to give you a definite state.

Another indication of why this must be wrong: the terms in the squared modulus of the sum include, for example, $b_1^* b_1$. By your logic this would mean "path" $b_1$ is interfering with itself. But that's not what "interference" means; a single way an outcome can happen can't "interfere" with itself.

Again, the resolution of all this is that "interference" means taking the sum of the amplitudes, not taking the squared modulus of the sum.

 

[DrChinese]

That's so kindly asked for a simulation that don't need any fixing. Peter knows it, he can read java-script, and the program is trivial.

As a software professional, I assumed you would fix the program immediately. I was surprised when you didn't. But it works nicely now. I think our point of departure is around your statement: "There is a "hidden variable" in Bell's proof. In the simulation that variable is an angle."

a) Bell assumes there is a function that can replicate the QM prediction arbitrarily closely. It does not specifically require a hidden variable, but it could - no particular objection to that in an algorithm either.

b) You call the hidden variable an angle. There is no angle that can serve to give us the QM statistics unless it is one of the measurement angles A or B. Which is exactly what you do. If A is measured first: A's measurement angle is copied to B's setting. That's what you do in your code, and that's how it should be done.

// this is kind of the crux of the matter,
// even for a single photon, to keep 100% probability to still have the same polarization at the same angle later on, we have to change its polarization

if (isPolarized) {
photon.polarization.angle = detectorAngle; // not so incidentally, if polarization is **shared** by some other photon, it will then be 100% correlated too
}
else {
photon.polarization.angle = detectorAngle + (Math.PI/2); // 90 degree will force 100% non-polarized
}

c) So all is good. Your code is essentially transmitting the value of the measurement angle on A so that B is polarized identically to A (that being prior to the measurement of B). A good non-local influence from first measured to second measured. Which as you say in your comments, doesn't matter which comes first.

Overall, I like the way you laid out your code and the straightforward interface.

 

[PeterDonis]

Bell assumes there is a function that can replicate the QM prediction arbitrarily closely.

As a clarification: Bell assumes that there is such a function, but then proceeds to prove that, if there is such a function, it cannot have the "locality" property he defines (because no function that has that property can replicate the QM predictions).

 

[PeterDonis]

You call the hidden variable an angle. There is no angle that can serve to give us the QM statistics unless it is one of the measurement angles A or B. Which is exactly what you do. If A is measured first: A's measurement angle is copied to B's setting. That's what you do in your code, and that's how it should be done.

And, as you note, such an algorithm explicitly encodes a non-local influence from A to B (assuming A is measured first). But Bell's "hidden variable" was explicitly defined as something whose value was already determined at an event that is in the past light cone of both measurement events.

 

[Lish Lash]

Bohmian Mechanics has faster-than-light signals, but for us to never see them or use them requires our ignorance of the particles' positions to mask the signals precisely.

This is a misrepresentation of Bohmian Mechanics. To claim that the Bohmian pilot wave posseses a "faster than light" velocity is to imply that it propagates through the medium of 4D spacetime. If that were the case, it would become a local phenomenon subject to relativity, in contradiction to the non-local nature of Bohmian Mechanics. The pilot wave is instead manifest in complex-valued configuration space, the non-local domain where the quantum wave function is defined. This is consistent with the pilot wave's non-relativistic simultaneous guidance of all particles with which it is entangled.

The issue is whether a non-local influence is occurring. Clearly, that is a viable possibility per Bell. And, for example, Bohmian Mechanics postulates a manner in which that can occur. However, there is mutual influence between A and B in that model.

This is not how Bohmian Mechanics describes correlation of entangled particles. In Bohmian Mechanics, there is no "mutual influence" between entangled particles - ALL guidance propagates unidirectionally (non-locally) from the pilot wave to the entangled particles. The particles exert no "influence" (i.e. transmission of information) back to either the pilot wave or each other. The pilot wave simply evolves in configuration space in accordance with the quantum wave function.

 

[DarMM]

This is a misrepresentation of Bohmian Mechanics. To claim that the Bohmian pilot wave posseses a "faster than light" velocity is to imply that it propagates through the medium of 4D spacetime. If that were the case, it would become a local phenomenon subject to relativity, in contradiction to the non-local nature of Bohmian Mechanics. The pilot wave is instead manifest in complex-valued configuration space, the non-local domain where the quantum wave function is defined. This is consistent with the pilot wave's non-relativistic simultaneous guidance of all particles with which it is entangled.

Out of quantum equilibrium there is superluminal signalling right? Alteration of Bob's statistics by Alice's experiments (usually called signals/signalling) even when Bob is outside Alice's lightcone. I'm not talking about a wave moving in 4D space, just signalling faster than light could achieve.

 

[Lish Lash]

Out of quantum equilibrium there is superluminal signalling right? Alteration of Bob's statistics by Alice's experiments (usually called signals/signalling) even when Bob is outside Alice's lightcone. I'm not talking about a wave moving in 4D space, just signalling faster than light could achieve.

Once again, you are postulating the physical 4D spacetime manifestation of a "superluminal signal" that propagates at a faster-than-light velocity to convey information between mutually entangled particles. No such signal exists in Bohmian Mechanics. The Bohmian pilot wave manifests in complex-valued configuration space, a domain where the relativistic concept of "faster-than-light" is meaningless. This is the essence of the pilot wave's non-locality, which impels it to act simultaneously (rather than locally) in guidance of all particles with which it is entangled.

 

[DarMM]

Once again, you are postulating the physical 4D spacetime manifestation of a "superluminal signal" that propagates at a faster-than-light velocity to convey information between mutually entangled particles. No such signal exists in Bohmian Mechanics. The Bohmian pilot wave manifests in complex-valued configuration space, a domain where the relativistic concept of "faster-than-light" is meaningless. This is the essence of the pilot wave's non-locality, which impels it to act simultaneously (rather than locally) in guidance of all particles with which it is entangled.

Yes, I get the basic ontology of Bohmian Mechanics, but this does constitute superluminal signalling as per the standard definition of those terms. I'm not assuming the superluminal signal is physically propagating like a wave or something in 4D spacetime. Signalling in the sense of altering Bob's statistics in a way that can convey information (when out of quantum equilibrium) and doing so before light would be capable of doing so.

 

[Lish Lash]

I'm not assuming the superluminal signal is physically propagating like a wave or something in 4D spacetime. Signalling in the sense of altering Bob's statistics in a way that can convey information...

I have the impression you're envisioning this "superliminal signal" as something akin to the way a shadow projected by a laser situated on Earth can appear to dart across the surface of the moon at a faster-than-light velocity. Here's an entertaining examination of this phenomenon:



As explained in the link, the phenomenon is real, but no information is transmitted at FTL speeds from (4D spacetime) "point A to point B". That is likewise the case with mutually entangled particles in Bohmian Mechanics. Regardless of how it may appear to occur in our 4D spacetime measurements, no information is conveyed between entangled particles (at any 4D spacetime velocity whatsoever). All such correlations are produced by the non-local guidance of the pilot wave, acting simultaneously on all particles with which it is entangled.