Bohr's complementarity is completed with entanglement?

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In summary, the authors argue that the common treatment of wave-particle duality as a tradeoff between which-slit distinguishability, and the visibility of the resulting interference pattern is incomplete, and show how entanglement between the different degrees of freedom of a single optical field (a photon?) actually completes this relation, turning an inequality (like the uncertainty principle) into a simple equation. They also show that complementarity is important and broadly applicable in quantum science. Little disagreement exists today about complementarity's importance and broad applicability in quantum science, and from the paper it seems that Heisenberg held more to the onion with many layers idea of Feynman (who just had it as an idea
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Earlier this week, I found a very interesting paper pop up on the arXiv headed by Joseph Eberly, a notable figure in quantum optics and entanglement.

https://arxiv.org/abs/1803.04611

In it, they look at the common treatment of wave-particle duality as a tradeoff between which-slit distinguishability, and the visibility of the resulting interference pattern. Because one can certainly have the worst of both worlds (no distinguishability or visibility), they argue that this duality is incomplete, and show how entanglement between the different degrees of freedom of a single optical field (a photon?) actually completes this relation, turning an inequality (like the uncertainty principle) into a simple equation.

I don't know exactly what this has to say about the nature of reality and such, but thought it was cool enough to share:)
 
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  • #2
jfizzix said:
Earlier this week, I found a very interesting paper pop up on the arXiv headed by Joseph Eberly, a notable figure in quantum optics and entanglement.

https://arxiv.org/abs/1803.04611

In it, they look at the common treatment of wave-particle duality as a tradeoff between which-slit distinguishability, and the visibility of the resulting interference pattern. Because one can certainly have the worst of both worlds (no distinguishability or visibility), they argue that this duality is incomplete, and show how entanglement between the different degrees of freedom of a single optical field (a photon?) actually completes this relation, turning an inequality (like the uncertainty principle) into a simple equation.

I don't know exactly what this has to say about the nature of reality and such, but thought it was cool enough to share:)
I've had a quick read and it is new and satisfying. The equation they derive ##V^2+D^2+C^2\equiv 1## verifies that the maximally entangled 2-photon state (C=1) has zero interference visibility(V) and zero particle distinguishability(D). The principle seems to be that information is limited and you can't have all three in full.

There is also another great experiment that strongly supports the equation.
 
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  • #3
From the paper:
'Little disagreement exists today about complementarity's importance and broad applicability in quantum science'

Personally I don't even understand complementary - but of course that may be a failing on my part despite reading advanced texts like Ballentine where its not even mentioned. The essence (one of them anyway - it's not just one thing) of science is doubt and my failing may not reflect the true situation. But what can be said is a number of very knowledgeable people that post here, many with doctorates and professorships don't think it makes that much sense either. Of course they could be wrong. Anyway little disagreement would not seem to be the actual situation - even in Bohr's time both Einstein and Dirac disagreed - Dirac - nothing but the math ma'am, nothing but the math, - leave this sort of stuff out of it. Although when pushed on the matter believe it or not he agreed with Einstein - strange but true:
https://www.mitpressjournals.org/doi/abs/10.1162/posc.2008.16.1.103?journalCode=posc

And no - I can't download the full paper - maybe some reader can and post it but the following gives an idea of Dirac's view in a discussion with Heisenberg:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.485.9188&rep=rep1&type=pdf

He likely held more to the onion with many layers idea of Feynman (who just had it as an idea of what nature could be like - he wasn't really interested one way or the other- he just wanted to know more about the world) and I think Gell-Mann as well holds the same view.

Einstein - well his view towards QM varied but eventually he came to accept it as correct - but incomplete - his view was encapsulated in the ensemble interpretation which does not have complementarity - although agnostic to it may be a better description
https://en.wikipedia.org/wiki/Ensemble_interpretation

Here is my view. We have coins here in Australia - on one side you have a picture of the queen, on the other some animal. So sometimes, depending on which side is up its like a picture of the queen, or like a picture of an animal. So you could say the queen and the animal are complementary according to what I think Bohr is saying (as I said I don't really get it). But it's actually neither - its money - this complimentary business is just a verbose irrelevancy.

Just my view - and it may be wrong - but I just wanted to say the situation is not quite how the paper presents it in its abstract in my view.

I will read the full paper when I get a bit of time.

Added Later:
Scanned the paper - the math looks OK but their interpretation goes right over my head. I will never get complementary and the paper just reinforces it. Maybe someone else can - but for me - its a lost cause. I am forever stuck in the group - as the paper says - 'The long record of frustrating and conflicting opinions about the meaning of complementarity is common knowledge.' Strange though it also says, as I quoted at the start, 'Little disagreement exists today about complementary's importance and broad applicability in quantum science' These authors are obviously not nincompoops but is it only me that sees a contradiction here.

I wonder what some of the professors that post here would say if they were given it to referee?

Thanks
Bill
 
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Well, I couldn't agree more with @bhoppa . Particularly I don't understand a quantum optician, who's really "entangled" with the foundations of quantum (field) theory as not many physicists in other areas of specialization. For over 90 years we now have modern quantum theory which is most clearly expressed in terms of Dirac's representation-free formulation. There is no "wave-particle duality" and no "complementarity" but just the probabilistic meaning of the notion of quantum state, and in fact that's verified, among all other high-precision observations in all areas of physics, by the many successful high-precision Bell tests done by quantum opticians. To teach "wave-particle duality" and "complementarity" and other gibberish of the philosophical circle around Bohr is an unnecessary burden for the students to learn modern quantum (field) theory, which is difficult enough, because one has to overcome our "common sense" which is due to the classical behavior of macroscopic objects of everyday life, which however is no contradiction to quantum (field) theory at all.
 
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  • #5
I cite an earlier paper of Eberly et al's in my https://arxiv.org/abs/1709.06711. Complementarity of measurements is entirely natural in classical signal processing because Fourier transforms relate different coordinate systems, position space and wave-number space, and, at least when thinking about noncommuting observables naively, that's essentially what Eberly et al. is channeling. One can see a recent video by 3blue1brown for an elementary account, . This has been known in general terms and scattered through the literature in various forms for 50 years at least, and one can even read it into a paper by Koopman in 1931.
I'm pretty sure the most important idea is that classical physics includes transformations, which can behave as observables in appropriate circumstances, as well as the obvious and always mentioned commutative algebra of observables: classical transformations almost always do not commute, just as operators in quantum theory don't. For anyone who knows classical mechanics well enough to understand the Poisson bracket, the Poisson bracket can be used to generate transformations. There's a moderately detailed explanation in Appendix A of the paper of mine that I mention above, as well as a heavier discussion in the main text of the relationship between random and quantum fields.
Eberly and others have been discussing noncomplementary measurements in classical optics for 15-20 years, and the Koopman-von Neumann approach to classical mechanics has existed in an undeveloped way since 1931, so one has to ask when someone will tell the story in a way that grabs the imagination of physicists. I don't know when that will be. The paper of mine that I link to tries again, but we have to assume that its try will again not be enough.
 
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  • #6
bhobba said:
Personally I don't even understand complementary - but of course that may be a failing on my part despite reading advanced texts like Ballentine...
Einstein ... his view was encapsulated in the ensemble interpretation which does not have complementarity...
Here is my view ... it's actually neither - its money - this complimentary business is just a verbose irrelevancy.
Bill
I take it, Bill, that when you say you don't understand complementarity, you mean you "just don't get" what Bohr and Einstein (or anyone else!) made of it? I'm pretty sure that you do understand what it actually means! Because it's meaningful to talk of a wavefunction as being a superposition of momentum states or position states but meaningless to talk about definite values of the two properties co-existing. So yes, you can label such properties as complementary. It's just a definition. But is there more to the actual concept - something I haven't come across?

Of course it's infinitely simpler to start with operators rather than semi-classical word-pictures. Which coming from a math-o-phobe must say something.

Whether it's a useful concept is another matter. Even your coin analogy has a serious issue with the onticity of the unobserved face :) And saying that position and momentum are complementary may very well lead students to think that both properties do exist even though we can't measure both at once. Seems to me that you can probably leapfrog over complementarity and get straight to uncertainty relations.

Duality is a different idea altogether, I think, and one which has zero didactic or practical use whatsoever!
 
  • #7
Ok, if it's so simple, could you explain to us what "complementarity" means? I've also no clue. Understanding Bohr (as well as Heisenberg) is a challenge I've not mastered yet. It's the most enigmatic writing in 20th-century physics I encountered so far. To know what was really going on in QM at their time one has to read Born, Dirac, Schrödinger, Pauli, Sommerfeld, et al!
 
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vanhees71 said:
Ok, if it's so simple, could you explain to us what "complementarity" means? I've also no clue. Understanding Bohr (as well as Heisenberg) is a challenge I've not mastered yet. It's the most enigmatic writing in 20th-century physics I encountered so far. To know what was really going on in QM at their time one has to read Born, Dirac, Schrödinger, Pauli, Sommerfeld, et al!
That's why I was asking Bill what he meant.
 
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  • #9
We don't have to worry so much about what Bohr meant by complementarity. We can try to rationally reconstruct what Bohr might have meant or have been thinking, but that's interesting only up to a point, and Bohr presumably wouldn't recognize what we would be telling him was what he was thinking or meaning, because a lot of conceptual thinking has gone under the bridge since, until we brought him up to speed on all that stuff. I'm not writing off history, or not intending to, just sayin' that if one is trying to do physics then knowing the history but selectively forgetting some history is part of the modern story. We shouldn't forget Bohr totally, but it seems like there are pretty good reasons why we have selectively forgotten a large part of what he had to say about quantum mechanics.
It's not so hard, I think, maybe, to more-or-less understand that for two noncommuting operators, amongst other consequences, it is impossible, in general, to construct a joint probability density for the two probability densities we obtain for those two operators in a given state over the algebra that contains them, because there are mathematical theorems that say so. It's been clarified enough over the years that 3blue1brown can with confidence post the elementary video I linked to above. That measurement in a different basis (eigenstates of the position operator or eigenstates of the translation operator) is nontrivial is a not-so-weird classical thing as much as it's a quantum thing. That doesn't answer vanhees71's question, "explain to us what "complementarity" means? ... what was really going on in QM at their time", but it's what I've got.
 
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This is highly misleading! First of all nothing physical ever depends on a special choice of basis. No matter, whether I consider the wave function in position representation or momentum representation, a mixture thereof or any other representation, it always refers to the same basis-independent pure state ##\hat{\rho}=|\psi \rangle \langle \psi|##.

Furthermore, which observable you can measure accurately or not does not depend on the state of the system but only on your technical ability to measure. If you measure the position of a particle, its preparation in the state ##\hat{\rho}## implies that the probability density to find the particle at position ##x## is according to Born's rule
$$P(x)=\langle x|\hat{\rho} x \rangle=|\psi(x)|^2.$$
The same holds for momentum
$$\tilde{P}(p)=\langle p|\hat{\rho} p \rangle=|\tilde{\psi}(p)|^2.$$
That's for sure NOT what Bohr meant when inventing the (in my opinion superfluous) idea of "complementarity" since Bohr of course was well aware of the proper meaning of states and observables in quantum mechanics. He corrected Heisenberg on his wrong ideas with these notions in connection with his first paper on the uncertainty principle.
 
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  • #11
Let me turn it around, going back to an empiricist perspective. When we take raw data, we make choices that let us construct ensembles; for any given ensemble we can construct its moments, et cetera, which we can model to some approximation (i) by a probability space and a random variable, or (ii) by a stochastic state over an algebra (which is generated by just one operator, which has a spectrum that is the same as the sample space of the random variable). Now, if we construct multiple ensembles, there are some choices and some raw data for which we cannot construct a joint probability measure for some pair of ensembles. For the Kolmogorov axioms, this requires the introduction of some new concept, such as contextuality. For an algebra of operators, we don't have to introduce any new concepts, we can take it that the two operators are in general position, with the state and the operators to be determined by the raw data we have for the multiple ensembles we have already and possibly by making new choices of how we construct ensembles from the raw data.
One can say that "nothing physical ever depends on a special choice of basis", because indeed one wants the physics to be independent of how we choose ensembles, but the raw data is the ground floor of physics, and the choices that allow us to construct ensembles also determine, up to unitary equivalence, what operators and states we could use as a model, given those ensembles and that raw data.
Does that get me slightly closer to you?
 
  • #12
I'm a bit puzzled by people here saying that they don't understand complementarity.

Let's have a look at the definition of Scully et al. in [1]: "We say that two observables are 'complementary' if precise knowledge of one of them implies that all possible outcomes of measuring the other one are equally probable."

So everybody who is familiar with the uncertainty relations for canonically conjugate observables knows the most important thing about complementarity. I think that textbooks don't use the term anymore mostly because it is regarded as superfluous.

In the line of research which lead to the paper in the OP, it is however argued that complementarity should be regarded as more general than the uncertainty relations. Scully et al. talk about this in [1], and in [2], Englert derives an inequality which shows that there's a trade-off between the fringe-visibility in an interferometer and the amount of which-way information that is potentially stored in a detector. Since he does it without invoking a Heisenberg-Robertson-type uncertainty relation, he concludes that QM includes an additional amount of complementarity which isn't captured by the usual uncertainty relations.

---
[1] Scully et al., "Quantum optical tests of complementarity", Nature, 1991
[2] Englert, "Fringe Visibility and Which-Way Information: An Inequality", PRL, 1996
 
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  • #13
kith said:
Let's have a look at the definition of Scully et al. in [1]: "We say that two observables are 'complementary' if

It's a definition I suppose - but calling it complementary - I wouldn't - I would call it the uncertainty relations which is what it is. It's usually expressed as something like the following:
'Complementarity principle, in physics, is the tenet that a complete knowledge of phenomena on atomic scales requires a description of both wave and particle properties. The principle was announced in 1928 by the Danish physicist Niels Bohr. Depending on the experimental arrangement, the behaviour of such phenomena as light and electrons is sometimes wavelike and sometimes particle-like; i.e., such things have a wave-particle duality (q.v.). It is impossible to observe both the wave and particle aspects simultaneously. Together, however, they present a fuller description than either of the two taken alone. n effect, the complementarity principle implies that phenomena on the atomic and subatomic scale are not strictly like large-scale particles or waves (e.g., billiard balls and water waves). Such particle and wave characteristics in the same large-scale phenomenon are incompatible rather than complementary. Knowledge of a small-scale phenomenon, however, is essentially incomplete until both aspects are known.'

If the uncertainty relations is what is meant call it that - not complementarity which simply confuses.

BTW to me the above definition is not even understandable eg experiments have shown both particle and wavelike properties can exist at the same time. Have a look at the wave-functions of the hydrogen atom - they do not look like waves to me. Its like the coin analogy I gave before - sometimes its a picture of the queen, and sometimes a picture of an animal, really its neither - its money. If you want to redefine it as the uncertainty relations I couldn't care less - be my quest - at least that is precise an understandable - but I will call it the uncertainty relations - not complementarity.

If others want to extend the idea of the uncertainty relations - that's fine - but don't call it complimentary. Even if some physicists have defined it as that, and like I said I personally don't care one way or the other, it's not what is in common use which is more like the above.

Thanks
Bill
 
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  • #14
Well, yes, it's a definition, and it's very clear, contrary to Bohr's writings on complementarity. He makes out of this notion some very general but also very vague philosophy. Particularly the article by Englert looks very interesting, because it aims at quantifying the either vague notion of "wave-particle duality", which in my opinion doesn't exist in the original hand-waving sense of "old quantum mechanics". For me it's overcome by "modern QT", which quantifies everything in a clear, however probabilistic, way, and no apparent paradoxes a la "wave-particle duality" are present in "modern QT" anymore. I'll study Englert's paper later in some detail.
 
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  • #15
vanhees71 said:
Well, yes, it's a definition, and it's very clear, contrary to Bohr's writings on complementarity. He makes out of this notion some very general but also very vague philosophy. Particularly the article by Englert looks very interesting, because it aims at quantifying the either vague notion of "wave-particle duality", which in my opinion doesn't exist in the original hand-waving sense of "old quantum mechanics". For me it's overcome by "modern QT", which quantifies everything in a clear, however probabilistic, way, and no apparent paradoxes a la "wave-particle duality" are present in "modern QT" anymore. I'll study Englert's paper later in some detail.
The paradoxes also go away if you put particle interactions into a black box and just define an interaction in terms of possible outcomes - all of which occur in superposition. Obviously a lot of modern QT is there inside the black box, but quantum paradoxes themselves are, as far as I can tell, always due to interpreting a superposition as a proper mixture.
Would you agree with that? Even a little bit? :)
 
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  • #16
bhobba said:
It's usually expressed as something like the following: 'Complementarity principle, in physics, is the tenet that a complete knowledge of phenomena on atomic scales requires a description of both wave and particle properties. [...]'
Bohr introduced the notion of complementarity in 1928, a time where neither the publication of the general uncertainty relation (Robertson, 1929), nor the publication of Dirac's book (1930), nor von Neumann's introduction of Hilbert spaces (1932) had happened yet. So one could argue that Bohr simply lacked the tools to explain the concept nicely to us.

On the other hand, I think you are right that for Bohr, complementarity was more than for Scully et al. For him, the concept was deeply related to the interpretation of QM.

bhobba said:
If others want to extend the idea of the uncertainty relations - that's fine - but don't call it complimentary. Even if some physicists have defined it as that, and like I said I personally don't care one way or the other, it's not what is in common use which is more like the above.
Well, terminology evolves and may vary from subfield to subfield. The quantum optics people have chosen to use the clear definition of complementarity which I cited above. I appreciate this because it builds neatly upon my previous understanding.

Generally, I like looking at the writings of the old masters through the lens of modern terminology because it sharpens my understanding of both (I especially recommend Malcolm Longair's "Quantum Concepts in Physics" for this).
 
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kith said:
On the other hand, I think you are right that for Bohr, complementarity was more than for Scully et al. For him, the concept was deeply related to the interpretation of QM.
Lande is quite ascerbic about this. There's too much to quote but you may be able to find a copy of New Foundations of Quantum Mechanics or just read the Google review. It's dated and polemical but IMO very much worth looking at.
 
  • #18
vanhees71 said:
Well, yes, it's a definition, and it's very clear, contrary to Bohr's writings on complementarity. He makes out of this notion some very general but also very vague philosophy. Particularly the article by Englert looks very interesting, because it aims at quantifying the either vague notion of "wave-particle duality", which in my opinion doesn't exist in the original hand-waving sense of "old quantum mechanics". For me it's overcome by "modern QT", which quantifies everything in a clear, however probabilistic, way, and no apparent paradoxes a la "wave-particle duality" are present in "modern QT" anymore. I'll study Englert's paper later in some detail.
Oh yeah, the other thing is, the OP did imply the paper treats complementarity and duality as much the same thing. So I guess they are making their theorem a statement about reality. But is it all just weaving a dream?
 
  • #19
Derek P said:
Lande is quite ascerbic about this. There's too much to quote but you may be able to find a copy of New Foundations of Quantum Mechanics or just read the Google review. It's dated and polemical but IMO very much worth looking at.
Interesting, whenever I think I know them all, another interpretation of QM pops up. I'm not sure whether I will get access to any of his writings soon, so maybe just two quick questions.

This seems to be a realistic particle interpretation. So does he talk about superluminal influences? And what's the difference between Landé and de Broglie-Bohm?
 
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  • #20
Derek P said:
Lande is quite ascerbic about this. There's too much to quote but you may be able to find a copy of New Foundations of Quantum Mechanics or just read the Google review. It's dated and polemical but IMO very much worth looking at.

It's dated 1965, which essentially makes it pre-Bell.
 
  • #21
kith said:
This seems to be a realistic particle interpretation.
Hey, come on, I'm not advocating any interpretation here! I was commenting on this:
kith said:
On the other hand, I think you are right that for Bohr, complementarity was more than for Scully et al.
Seems Lande would agree. But yes, it's real particles.
kith said:
So does he talk about superluminal influences? And what's the difference between Landé and de Broglie-Bohm?
No pilot wave because the particle interacts with a grating "as a whole". Somehow it exchanges momentum according to the spatial spectrum of the grating.
DrChinese said:
It's dated 1965, which essentially makes it pre-Bell.
But not pre-Everett!

I have no idea how Lande deals with entanglement. My guess would be that as single particle interactions are local, real and probabilistic, the interpretation will eventually find itself needing macroscopic superposition. Whether Lande anticipated this, I can't remember: it was 48 years ago that I read the book. But it was a good antidote to wave-particle duality, that's all I know.
 
  • #22
Derek P said:
The paradoxes also go away if you put particle interactions into a black box and just define an interaction in terms of possible outcomes - all of which occur in superposition. Obviously a lot of modern QT is there inside the black box, but quantum paradoxes themselves are, as far as I can tell, always due to interpreting a superposition as a proper mixture.
Would you agree with that? Even a little bit? :)
In some sense that's what's done in relativistic QFT: The only interpretation of particles is in terms of asymptotic free states, and the observable predictions are in S-matrix elements. Of course, the interactions are not in a black box but described by the action principle, i.e., by the Hamiltonian of the quantum fields.
 
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  • #23
vanhees71 said:
In some sense that's what's done in relativistic QFT: The only interpretation of particles is in terms of asymptotic free states, and the observable predictions are in S-matrix elements. Of course, the interactions are not in a black box but described by the action principle, i.e., by the Hamiltonian of the quantum fields.
You can, and some would say you should, do much the same thing even in first quantisation: keep everything as a wavefunction until an actual observation occurs. That way you don't have to worry about what's "really" going on as the wavefuction evolves. We know what comes out at the end and that's all that matters. So no paradoxes. See no evil, hear no evil, speak no evil! :biggrin:
 

1. What is Bohr's complementarity?

Bohr's complementarity is a principle in quantum mechanics that states that certain physical properties of a particle cannot be observed or measured simultaneously. This means that a particle can exhibit both wave-like and particle-like behavior, but never at the same time.

2. How is Bohr's complementarity related to entanglement?

Bohr's complementarity is completed with entanglement, which is a phenomenon where two or more particles become connected in such a way that the state of one particle cannot be described independently of the other. This means that the properties of the entangled particles are no longer complementary, but instead, they are strongly correlated.

3. What is the significance of Bohr's complementarity and entanglement?

Bohr's complementarity and entanglement are both fundamental principles in quantum mechanics and have major implications in our understanding of the nature of reality. They challenge our classical understanding of cause and effect and suggest that particles may be interconnected in ways that we cannot fully comprehend.

4. Can entanglement be used for practical applications?

Yes, entanglement has been used in various practical applications, such as quantum cryptography, quantum teleportation, and quantum computing. These applications take advantage of the strong correlations between entangled particles to perform tasks that are not possible with classical systems.

5. How does Bohr's complementarity and entanglement affect our understanding of the universe?

Bohr's complementarity and entanglement have revolutionized our understanding of the universe, particularly in the field of quantum mechanics. They have challenged our traditional view of the universe as a purely deterministic system and have opened up new possibilities for understanding the fundamental nature of reality.

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