Wednesday, March 02, 2016

Dear Dr. B: What is the difference between entanglement and superposition?

The only photo in existence
that shows me in high heels.

This is an excellent question which you didn’t ask. I’ll answer it anyway because confusing entangled states with superpositions is a very common mistake. And an unfortunate one: without knowing the difference between entanglement and superposition the most interesting phenomena of quantum mechanics remain impossible to understand – so listen closely, or you’ll forever remain stuck in the 19th century.

Let us start by decoding the word “superposition.” Physicists work with equations, the solutions of which describe the system they are interested in. That might be, for example, an electromagnetic wave going through a double slit. If you manage to solve the equations for that system, you can then calculate what you will observe on the screen.

A “superposition” is simply a sum of two solutions, possibly with constant factors in front of the terms. Now, some equations, like those of quantum mechanics, have the nice property that the sum of two solutions is also a solution, where each solution corresponds to a different setup of your experiment. But that superpositions of solutions are also solutions has nothing to do with quantum mechanics specifically. You can also, for example, superpose electromagnetic waves – solutions to the sourceless Maxwell equations – and the superposition is again a solution to Maxwell’s equations. So to begin with, when we are dealing with quantum states, we should more carefully speak of “quantum superpositions.”

Quantum superpositions are different from non-quantum superpositions in that they are valid solutions to the equations of quantum mechanics, but they are never being measured. That’s the whole mystery of the measurement process: the “collapse” of a superposition of solutions to a single solution.

Take for example a lonely photon that goes through a double slit. It is a superposition of two states that each describe a wave emerging from one of the slits. Yet, if you measure the photon on the screen, it’s always in one single point. The superposition of solutions in quantum mechanics tells you merely the probability for measuring the photon at one specific point which, for the double-slit, reproduces the interference pattern of the waves.

But I cheated...

Because what you think of as a quantum superposition depends on what you want to measure. A state might be a superposition for one measurement, but not for another. Indeed the whole expression “quantum superposition” is entirely meaningless without saying what is being superposed. A photon can be in a superposition of many different positions, and yet not be in a superposition of momenta. So is it or is it not a superposition? That’s entirely due to your choice of observable – even before you have observed anything.

All this is just to say that whether a particle is or isn’t in a superposition is ambiguous. You can always make its superposition go away by just wanting it to go away and changing the notation. Or, slightly more technical, you can always remove a superposition of basis states just by defining the superposition as a new basis state. It is for this reason somewhat unfortunate that superpositions – the cat being both dead and alive – often serve as examples for quantum-ness. You could equally well say the cat is in one state of dead-and-aliveness, not in a superposition of two states one of which is dead and one alive.

Now to entanglement.

Entanglement is a correlation between different parts of a system. The simplest case is a correlation between particles, but really you can entangle all kinds of things and properties of things. You find out whether a system has entanglement by dividing it up into two subsystems. Then you consider both systems separately. If the two subsystems were entangled, then looking at them separately will inevitably reduce the information. In physics speak, you “trace out” one subsystem and are left with a mixed state for the other subsystem.

The best known example is a pair of particles, each with either spin +1 or -1. You don’t know which particle has which spin, but you do know that the sum of both has to be zero. So if you have your particles in two separate boxes, you have a state that is either +1 in the left box and -1 in the right box, or -1 in the left box and +1 in the right box.

Now divide the system up in two subsystems that are the two boxes, and throw away one of them. What do you know about the remaining box? Well, all you know is that it’s either +1 or -1, and you have lost the information that was contained in the link between the two boxes, the one that said “If this is +1, then this must be -1, and the other way round.” That information is gone for good. If you crunch the numbers, you find that correlations between quantum states can be stronger than correlations between non-quantum states could ever be. It is the existence of these strong correlations that tests of Bell’s theorem have looked for – and confirmed.

Most importantly, whether a system has entanglement between two subsystems is a yes or no question. You cannot create entanglement by a choice of observable, and you can’t make it go away either. It is really entanglement – the spooky action at a distance – that is the embodiment of quantum-ness, and not the dead-and-aliveness of superpositions.

[For a more technical explanation, I can recommend these notes by Robert Helling, who used to blog but now has kids.]


  1. Wow, nice legs by the way! thanks for your posts...

  2. Time to quote Schrödinger(1935. "Another way of expressing the peculiar situation is: the best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts, even though they may be entirely separate and therefore virtually capable of being ‘best possibly known,’ i.e., of possessing, each of them, a representative of its own. The lack of knowledge is by no means due to the interaction being insufficiently known — at least not in the way that it could possibly be known more completely — it is due to the interaction itself.
    Attention has recently been called to the obvious but very disconcerting fact that even though we restrict the disentangling measurements to one system, the representative obtained for the other system is by no means independent of the particular choice of observations which we select for that purpose and which by the way are entirely arbitrary. It is rather discomforting that the theory should allow a system to be steered or piloted into one or the other type of state at the experimenter's mercy in spite of his having no access to it."

  3. Hmmm, in Zwiebach's notes, an entangled state is defined as one in which a superposition cannot be "defined away" by a clever choice of basis. Mixed states are not mentioned, (and the bases are tensor products). I suppose this amounts to the same thing as your explanation, but it's nice to see the connection to superposition.

    Congrats on the milestones!

  4. "a lonely photon that goes through a double slit" Externally labeling photons in any recoverable way obtains a pair of summed single slit patterns instead. Big lumps diffract, arXiv:1310.8343. arXiv:1506.06026 diffracts symmetric top molecular rotors. Challenge theory with experiment by internally labeling the particles.

    D_3-trishomocuban-4,7,11-trione is an extremely chiral (Petitjohn's CHI = 0.692889) rigid cage symmetric top molecular rotor. Its specific rotation dissolved in chloroform is -1786° [(1R,3R,5R,6R,8R,10R)-isomer]. Given one C_3 and three C_2 rotation axes, the 14 heavy atoms are four unique positions. If a molecular beam of only one enantiomer is diffracted, molecule by molecule, does it racemize as it traverses the slits in superposition (Hund's paradox, 10.1103/PhysRevLett.103.023202)?

  5. I'm curious if you've read Ruth Kastner's recent book "The Transactional Interpretation of Quantum Mechanics"? Seems to resolve all the quantum mysteries at the cost of an expanded ontology. Have you any comments on it?

  6. richard richard - that is TOTALLY inappropriate utterly sexist comment.

  7. Matthew,

    No, I don't know the book, sorry.

  8. Art,

    Yes... If you try to explain math with words, some things always get lost, sorry. An entangled state is generally a superposition of products of states. Whereas I was referring to superpositions of basis states, because these are the ones you normally encounter in popular science explanations. So what you say (or Zwiebach says respectively) is correct of course, and doesn't contradict what I said. The point is that entanglement can't be made to go away by changing the choice of basis states, whereas superpositions sometimes can be made to go away, and in particular if they are for basis states, always can be made to go away.

  9. "The only photo in existence that shows me in high heels."

    But in the photo, one can't actually see any heels (neither those of the shoes nor yours). A hidden variable, perhaps. :-)

  10. "richard richard - that is TOTALLY inappropriate utterly sexist comment."

    I agree that it is not really relevant, but it is sexist only if he doesn't make similar comments about the legs of male bloggers, a) assuming that they can be seen, b) assuming that they are nice, and c) for some definition of nice.

    On the other hand, it was her decision to post the photo.

    I think Chris Rock got it right in his Oscar monologue, when he took the mickey out of the suggestion that the press should ask actresses serious questions, like the ones they ask the actors, and not "what are you wearing". Paraphrased from memory:

    There is a reason for this: the men are all wearing the same thing! If George Clooney showed up in a lime-green tuxedo with a swan sticking out of his ass, you can be damn sure someone would ask "What the hell you wearing there, George?" :-)

  11. Here's something I don't understand and would love an explanation. Is the following correct? If Alice and Bob each get a piece of a total spin-0 entangled pair, and let's say Alice measures the spin in x-direction. Then if Bob measures the spin *after* Alice does in the x-direction, he'll get the same answer every time. However if Alice measures the spin in y-direction, and then Bob measures the spin (again after Alice, again in x-direction), he'll get a 50/50 answer on the horizontal spin.

    If the above is correct, then wouldn't the following work as a FTL communication method?

    Alice and Bob sync up their clocks, and then set up a continuous stream of entangled particles, and number them by time received. At pre-arranged times (say every second) Alice will take say 10 of them at a time, and measure the spin of all 10 in either x- or y-directions (which will be the 0 and 1 bits). At the same pre-arranged time + epsilon, Bob will measure the spin in x-direction, thus getting a (probabilistic) 0/1 measurement.

    The very first batch of particles we get will of course arrive at the speed of light, but after that first contact, we can keep communicating faster than light forever.

    Where did I go wrong?

  12. Unknown,

    I don't understand how they are supposedly communicating with this setting. What information does Bob get from his measurements and how? Best,


  13. Uhh, yeah, I think I see the problem - I assumed that the spin will magically always be the same when Alice measures in x-direction, but obviously it's not going to be, so from Bob's pov he won't be able to distinguish her measuring in x or y directions.

  14. A lot of this was really useful but I'm a bit confused on one or two counts.

    Are not there 2 inexplicit contexts in play? There's the theorist exploring the equations using knowledge, experience & ideas. In that context you can obviously redefine things in non-trivial ways because you also proxy for objective reality.

    But if there is well defined physical set-up like the two slit designed to exhibit some key attributes of superposition, then the scientist is much more constrained because the proxy for objective reality is now distributed between the scientist and the physical set up.

    In terms of the measurement core to the setup design of that experiment, the superposition is completely invariant. You could replicate the test 1000 times, and each independent replication repeat 1000 times, and human error & technology limitations aside, there wouldn't be a single case of any significant variation in the superposition.

    So paired to the actual setup there is exactly one superposition that is always there for the same set up, and for that reason it is legitimate to regard superposition as if it was a physical attribute.

    There are no known disagreements between theory and experiment in QM at the moment. We'd have to assume our experiments are reasonable captures of the features we are trying to understand. So surely, even though a superposition is indeed paired to a scenario, that does not imply that it is ambiguous.

    Surely the ambiguity is an artefact of the abstract/theoretic/imagination mode in which the physicist has to explore the equations using ideas applicable to actual manipulations of equations, while also proxying for objective reality.

    So in that scenario you can obviously change your goal, change the measurement and in so doing change the superposition. But that aspect does not reflect the real situation, because our experiments show us that the effect is impossible to modify without modification of the actual experiment, and only by the quantity corresponding to the size of the non-trivial component of the modification.

    Is that wrong?

    p.s. because of the "the sum of two solutions is a solution" feature, it follows that the range of distinct measurement scenarios that the well-designed and finalizing experiment constrains the scientist to can all be summed together to a single superposition. So there's no ambiguity in principle even at the abstract/theoretic level.

    Apologies if that is wrong but I hope I've provided enough information that anyone that that has the knowledge and is willing to do so, is able to help me out.

  15. "I agree that it is not really relevant, but it is sexist only if he doesn't make similar comments about the legs of male bloggers"

    I think people do and would do that, if the male blogger posted the one picture of him in frog-green shorts and a pair of dutch klompen or whatever.

    What the Richard or whoever wrote would probably have been inappropriate shouted across the room unless they were friends or that was the culture of the immediate environment that everyone bought into and enjoyed. But he did so knowing presumably that if Sabbine didn't appreciate the compliment it was in her gift to not publish it.

    I don't think the criticism was inappropriate either though, because men are currently put in a very difficult position with respect to paying compliments to women and girls because women and girls like compliments of this kind. The problem is a small minority of men that use compliments as a device to undermine and belittle. But it's horrid the way men are being tarred with that brush. It's wrong. But don't blame us, because we live in the same fear that you do, of the same extremists.

  16. Hello again, I'm back to blogging. Nice article, Bee.

  17. I posted the comment about my legs because I thought it was funny - you can't actually see my legs in the photo!

  18. Lucy M,

    Yes, what you say is wrong. What I was trying to tell you is that a statement like "the state is prepared in a superposition" is meaningless without specifying a superpostion of what. And no, this has nothing to do with confusing theory with objective reality. You can, for example, alternatively either measure the position or the momentum of some prepared state. You would then find, and that's been done thousands of times, that while the one can be very sharply measured (eg not in a superposition of momentum states, or at least not much), the other one can't (superposition of position states).

    Incidentally, this particular example has nothing to do with quantum mechanics specifically. It's a property of Fourier-transformations. A sharply focused wave-packed will contain a lot of different frequencies. That's all I am saying. Again, this is a well-demonstrated fact which is explained by, but not invented through, mathematics. Best,


  19. Bee,

    I guess common assumption is that the "normal state" of matter is that (almost) every particle is non-entangled to other particles. Is this assumption justified from physical facts?

    Is it possible to build (either a math model or a real setup) where several particles (e.g. atoms in crystal) are entangled with some/all of the other?

    BR, -Topi

  20. "I think people do and would do that, if the male blogger posted the one picture of him in frog-green shorts and a pair of dutch klompen or whatever."

    Time for the Quote of the Day:

    CALIFORNIA magazine, in an article on "The Man Who Invented Time Travel", even
    ran a photograph of me doing physics in the nude on Palomar Mountain. I was
    mortified---not by the photo, but by the totally outrageous claims that I had
    invented time machines and time travel.

    ---Kip Thorne

    Nice , Kip!

  21. "You could equally well say the cat is in one state of dead-and-aliveness,"

    I thought they do say that. The four states are 'cat is alive' , 'cat is dead', cat alive & dead', 'cat is neither alive nor dead'

    The same 4 for the two slits 'blob passed through slit 1),'...slit 2', 'both slit 1 and slit', and 'neither'

    Maybe that's just a way to cut a long story short for pedagogical purposes. But I really thought that was true because it is the complete logic.

  22. "I posted the comment about my legs because I thought it was funny - you can't actually see my legs in the photo!"

    Of course, the Queen of Spain has no legs!

  23. Lucy,

    You missed the point. The distinction I was drawing is between "The cat is in a superposition of two states, one of which is dead and one which is alive" (standard narrative) and "The cat is in one state of dead-and-aliveness which is not a superposition" (which is mathematically equally correct, but not what you get normally told - the reason is that there is no detector for the observable "dead-and-aliveness".)

  24. " It is really entanglement – the spooky action at a distance – that is the embodiment of quantum-ness, and not the dead-and-aliveness of superpositions."

    Are you saying superposition is just a construct? I thought it was fundamental because I thought it was the root of at least some 'quantum-ness', assuming that word means the same as "quantum strangeness" or whatever. Some examples below:

    - Reality does not exist until we measure : that seems clearly superposition related.

    - wave function collapse - the historical origination is the two slit experiment which is superposition related.

    - just the fact of the 4 possible superposition states (yes, no , yes&no, neither yes or no) are superposition related.

    I mean. Could I ask whether you can change the superposition of the two-slit experiment without changing the experiment?

  25. Hi sabine - sorry about this but it might be helping lots of other people who are learning and read your blog

    I might be but I didn't think I was missing the point. You contrasted the two states of 'is alive' and 'is dead' with a single state 'aliveness&deadness'. I was just saying that all three of those states are taught as being 3 of the 4 root states of {'dead','alive',both dead & alive (your one), ' neither dead nor alive' }

    It's just that I thought a superposition must have the complete logic....are the two states of 'alive' 'dead' a complete logic?

  26. Do you wear red lipstick, Lucy? That is the historic mark of a felatrice. Eye shadow mimics coital engagement. High heels bunch the wearer's calves and buttocks suggesting lordosis, all signs of sexual receptivity. They also configure the foot's top as distal end of the leg, making it appear longer and slimmer. The Northern European standard of beauty is parity-inverted as Diversity Barbie.
    A professional scientist's credentials are performance not anatomy. OTOH, success generally requires a certain metaphoric attitude regardless of its carrier.

  27. Hi Lucy,

    I didn't think you posted your comment because you knew you were missing the point...

    Your story about the "four root states" is nonsense. You need two root states in this case, which is what I refer to as basis states. What I am telling you is that it's up to you to chose them. You can chose "dead" and "alive" and then the cat is in a superposition of dead and alive. Or you can take the superposition of "dead" and "alive" and declare it as one new basis state "dead or alive", which is no longer a superposition. Hence, what is and what is not a superposition is a matter of choice, not fundamental.

    There are btw infinitely many superpositions of "dead" and "alive" depending on how you divide up between dead and alive. Could be 50/50 or 20/80 or 99/1 etc.

  28. Hi Sabine,

    thanks for the post, this is a very fundamental point.

    I guess you know about Cramer's transactional interpretation of QM without reading Ruth Kastner's book. I think some of your readers would appreciate a post about it because (as far as I know) it is the only interpretation that makes real sense; in particular for the mechanism of the "spooky action at a distance".

    Thanks again

  29. "You need two root states in this case, which is what I refer to as basis states. What I am telling you is that it's up to you to chose them. You can chose "dead" and "alive" "

    What about the logic of the set up? Why you explain in terms of non-trivially choosing different states (i.e. that don't mean 'dead' or 'alive' or equivalent.

    What are the possible outcomes given that setup?

  30. uncle al - it sounds like you pay.

  31. Lucy,

    I have no idea what your question for the "logic of the set up" is supposed to mean. To begin with, let me make sure we both know that you cannot actually bring cats into a state of 'dead-and-aliveness'. This is just an example for any type of system with a 2-dimensional Hilbertspace. There's no "logic" in this setup, there simply is some setup, which is the experiment. For that you have an equation, you solve it, that's how it works. All of which, btw, I explained in the above blogpost. The possible outcomes depend not on the preparation of the state, but on the measurement. That, too, I explained in my blogpost. Best,


  32. It is really entanglement – the spooky action at a distance – that is the embodiment of quantum-ness

    Allow me to point out that - in my opinion as an interested layman - the use of the term "spooky action at a distance" in relation to quantum entanglement is one of the great tragedies in physics, and has led generations of students and legions of amateurs badly astray. There is no action at all involved in entanglement, other than the initial event where it is created; it is merely a correlation between the outcomes of measurements performed at different parts of a system, as you also pointed out in your article. This cannot be stated often enough - entanglement has nothing to do with action, nothing to do with the exchange of anything. In a sense, it is just a trivial statistical observation, and not at all surprising or mysterious; one just needs to learn to look at it in the right way.

  33. Markus,

    I agree with this... I recall being myself very confused about what exactly is supposedly "acting". Be that as it may, it is arguably a term that *has* spread very widely, so I wanted to make clear it's the very same thing that I am talking about.

  34. Hi Sabine - We were using the example of Schrödinger's cat so what I meant by the logic of the setup (i'm sorry if it was wrong terminology), is the Schrödinger's cat puzzle as stated:

    "a cat imagined as being enclosed in a box with a radioactive source and a poison that will be released when the source (unpredictably) emits radiation, the cat being considered (according to quantum mechanics) to be simultaneously both dead and alive until the box is opened and the cat observed."

    What I'm querying is this idea that there are no constraints or few constraints on how you define the superposition, *when* there is a stated problem or experiment.

    What I'm asking is, once you've redefined your position, or defined the superposition away (as you example of the single state). Whatever you do, in a context of Schrödinger's catyou still need to return to the the problem as stated, and have something sensible to say.

    So the cat takes fatal poison if a decay occurs, and remains alive if it doesn't. So what words are you going to use for your superposition, and do they mean 'dead' and 'alive'. Or what will you say if you have defined your superposition away?

    We're still in the context of Schrödinger's cat.


  35. Lucy,

    As I wrote in my blogpost "what you think of as a quantum superposition depends on what you want to measure" and I already told you above "there is no detector for the observable "dead-and-aliveness"". In your example you are picking preferred states (dead and alive) by the measurement you want to do. Saying that a state is "in a superposition of" something is not ambiguous and perfectly fine. Saying that the state "is in a superposition" (period) is ambiguous. Hence, it is perfectly fine to say "the cat is in a superposition of dead and alive". Saying that "the cat is in a superposition" is entirely meaningless.

    (I am not posting your other comments because you are just repeating misconceptions.)

  36. finally as I think you have indicated you don't want to discuss this further. Will it be ok if I bring a noted QM expert to your blog to discuss your claims? That'd quite an attraction for people I should think.

  37. @ Lucy M said...
    uncle al - it sounds like you pay.

    They paid me as a professional sperm donor: Tay-Sachs negative and a 5σ IQ. Piecework is OK...but overtime was brutal. Do your chair parade in Google lest you drool inconsistencies. Your output will bet less messy. Add citations,
    Phys. Rev. Lett. 116 090405 (2016), arXiv:1509.02749, doi:10.1103/PhysRevLett.116.090405
    "All but one of them include a photon path that, after being initially entangled, doesn’t interact with any other beam path before it reaches the detector."
    "I still find it quite difficult to understand intuitively what exactly is going on"

  38. It seems to me that the crucial and key difference between classical and quantum mechanics is simply the notion of a discrete state, i.e {dead, alive}.

    I say this because l think that entanglement depends upon the ability to distingish between the two.

    I also think that you can think of it this way;

    {dead =R}, {alive = B}, {dead or alive = M} and {either dead nor alive = Black} these all being valid state. And the computing possibilities are profound.

  39. Lucy,

    The problem with your repetitive comments is the following. You seem to believe that there is something to be discussed or interpreted here. There is not. I am merely telling you facts about quantum mechanics. You can look these up in any textbook, but I understand that there are a lot of people who don't like reading textbooks, so I am offering here a brief summary of the most important facts. I have now explained to you several times why it is ambiguous to say that a state "is in a superposition" without explaining a superposition of what, but have come to the unfortunately conclusion that you don't actually want to understand this, you just want to repeat your misconceptions. You are welcome to "bring" a "noted QM expert" if that makes you happy.

  40. Claver,

    No, it is not. Unfortunately, this is another misconception which is very common. It is correct that some observables become discretized in quantum mechanics, but others do not, and besides this, you can have discretization also in classical systems - it's a matter of boundary conditions. A typical example are standing waves in cavities that can take on only discrete values of wavelengths. On the other hand, there are many operators in quantum mechanics whose spectra are continuous. The most general case is a mixture of discrete and continuous values. The defining feature of quantization is the existence of conjugate variables which cannot simultaneously be measured precisely.

    (Incidentally, I have a post upcoming on Forbes on misconceptions about quantum mechanics some time next week or so. I'll post a link here.)

  41. Dear Sabine - you've thrown quite a few pejoratives around, but why could you not have answered my original question at its strong point, instead of assuming some boilerplate laypersona misconception, without reading carefully at all just in case that isn't that.
    It's immensely disrespectful and arrogant to do that. And then follow it with all the put downs.
    You act like being challenged by a layperson is beneath your dignity. But very few physicists deserve the sort of deference you want. You're yet to make your mark in physics. You blog to laypeople, but treat their efforts with contempt. You did not read my question carefully. You have not understood it.

  42. Dear Lucy,

    I have answered your questions, repeatedly, with as much patience as I can master. Clearly your inability to understand my replies is my fault. I find it interesting that you think I am being 'disrespectful' by explaining the basics of quantum mechanics, whereas you are being respectful by questioning that I know undergrad physics. You have taken up enough of my time with this nonsense - don't bother submitting further comments. Good bye,


  43. Hi Bee,

    Not sure where this fits in the hierarchy of “truthiness,” but it is informative on the historical context of the terminology and takes a broader look at the question of interpretation.

    The Transactional Interpretation of Quantum Mechanics by John G. Cramer

    Be nice to hear thoughts on the basic idea of the paper.


  44. I'm not sure if the story of the legs was a metaphor for "Because what you think of as a quantum superposition depends on what you want to measure." but happy International Women's Day.

  45. >If you crunch the numbers, you find that correlations between quantum states can be stronger than correlations between non-quantum states could ever be.

    1. Do you mean Bell's theorem? In this case it is correlations between measurement results, not states, right?
    2. In this case your statement is also wrong: e.g. Newtonian gravity is a non-quantum theory, which can in principle create even stronger correlations than quantum mechanics can via entanglement, because Newtonian gravity is also a non-local theory.

    The violations of Bell inequalities do not challenge "non-quantum theories" but rather theories that are both "relativistic and deterministic" (be they quantum or not). I know you want to present non-technical explanations, but the challenge is that they are simple, yet correct ;-).

  46. Maurice,

    Yes, I was referring to Bell's theorem. What you say is correct, it makes other assumptions which I didn't explicitly state (which I have however extensively elaborated on elsewhere). My point was just to say that you for certain won't ever understand Bell's theorem if you don't know what entanglement is...

  47. Coincidentally I also picked the sentence "If you crunch the numbers, you find that correlations between quantum states can be stronger than correlations between non-quantum states could ever be." to comment on, which I see was also done by the last commenter before I post this.

    I just wanted to say that I thought that was a good way to sum up Bell's Theorem - a good choice of words. So J have said it.

  48. Hi Sabine,

    Are u able to admit a mistake? The error was not that you did not explicitely list all necessary assumptions for the derivation of Bell's theorem (of course u can be brief...) but that you indirectly implied that an assumption of "non-quantumness" is necessary, while it is not. Therefore an attentive non-expert reader will take away the wrong message: "the experimental violation of Bell inequalities confirms the quantumness of entanglement." I realize that it is a just a mistake in your writing and not in your understanding of the subject. But for your readers that does not make a difference...

  49. Maurice,

    I thought I understood what you were saying, but now I think I don't. Bell's theorem demonstrates that correlations between measurement outcomes in quantum mechanics are stronger they would have been for a local hidden variables theory. That's what I tried to briefly capture with the sentence in my blogpost. There are further assumptions that enter the derivation, such as the lack of correlations between the prepared state and the detector settings, and various other loopholes. I don't know what you mean when you say I "implied" that "non-quantumness" is necessary - necessary for what? Neither do I understand what in my phrasing you are complaining about.

  50. > Bell's theorem demonstrates that correlations between measurement outcomes in quantum mechanics are stronger they would have been for a local hidden variables theory. That's what I tried to briefly capture with the sentence in my blogpost.

    But you didn't catch it because you did not write "...between local (L) and pre-deterministic (D) states [i.e. local HV theory] could ever be." but "...between local and non-quantum (N-Q) states could ever be." Besides non-existence of loopholes L & D are the necessary assumptions for the proof of Bell's theorem. So an attentive reader will have to conclude that from what you wrote that N-Q is a necessary assumption for the proof of Bell's theorem. From the experimental confirmations of Bell's theorem we learn that one of the assumptions necessary for its proof must be wrong. So the attentive reader must conclude that N-Q cannot hold, which is simply wrong.
    > I don't know what you mean when you say I "implied" that "non-quantumness" is necessary - necessary for what?
    For the proof of Bell's theorem.
    Expressed in other words: in the blog-post you wrote effectively:
    "Bell's theorem demonstrates that correlations between measurement outcomes in quantum mechanics are stronger they would have been for a non-quantum theory." which is incorrect.
    You're seem to be obsessed with being always right. Each time I point out a mistake, long justifications and leaving the dubious stuff in place instead of a simple "thanks, corrected!".

  51. Maurice,

    For the purpose of this sentence that you find so offensive, non-quantum means local hidden variables. As much as I welcome suggestions for how to improve my writing, I do not think that "between local (L) and pre-deterministic (D) states [i.e. local HV theory] could ever be" would have helped any reader understand anything, unless they already know what this means. As I wrote already in my first reply to you "what you say is correct," and I herewith want to thank you for your suggestion to improve this sentence, but I think the original version is more useful for the average reader.

  52. Awesome write up Bee.

    Just one question for you (and apologies if I missed this anywhere). Richard Feynman once said:

    "a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery."

    He was referring to the double slit experiment (and the associated wave-particle duality that is exhibited). I feel like this jives with the superposition component that you analyze first. But was he wrong to not take entanglement as the basis for quantum weirdness?

    Any help is appreciated as always!

  53. Pete,

    What Feynman was probably referring to was the measurement process, not the existence of superpositions per se. You can explain the interference pattern on a double slit by them being due to waves. What this does not explain is the probabilistic nature of the wave-function (for single particles). This is a postulate - something that has (of yet) no deeper explanation (and may never have), so you can rightfully refer to it as a "mystery". I doubt that Feynman was referring to the existence of superpositions themselves as "mystery". As I explained in my blogpost, superpositions exist also in other theories, and in addition their definition is ambiguous. Neither would I call entanglement "mysterious." My point was merely to say that entanglement is a less ambiguous property.

    I would actually agree that the measurement is the weirdest/most mysterious aspect of quantum mechanics. Best,


  54. I presented the "correction" of your sentence not as a suggestion to include it in a corrected blog piece. Rather - if u wouldn't be immune to advice - my suggestion would have been to remove the whole reference to Bell's theorem because it strictly does not teach us anything about QM itself (the subject of your blog post) but about hypothetical deeper theories. Einstein, Podolsky & Rosen pointed out that QM cannot be both local and pre-deterministic, and Bell pointed out that - provided Bell equations are violated w.o. loopholes - nature cannot be both local and pre-deterministic.

  55. Maurice,

    I am considering your advice, otherwise I wouldn't engage with you to begin with. But it is hard for me to consider a suggestion that "would have been" something you didn't say to begin with. Now that you you clarified you want me to remove the sentence, let me clarify that I will not remove it. I don't consider it misleading, in addition I have explained exactly what the technical meaning is above, and frankly I think you only pick around on this sentence - and on me - because you want to demonstrate that you know something about Bell's theorem. We've all heard it, thank you.

  56. "What Feynman was probably referring to was the measurement process, not the existence of superpositions per se. "

    No - Feynman, like most people, does indeed identify interference - this "radical change to the method of combining [superposing] probabilities of independent alternatives"* - as the central mystery/difference in QM.

    * QM and Path Integrals (p2+).

  57. Paul,

    Maybe. I am actually not very interested in psychoanalyzing Feynman... The sentence you quote I would have interpreted as saying that the central mystery/difference is the peculiar combination of *probabilities* (via the absolute square), not the existence of superpositions per se. Interference and superpositions just are not a distinctly quantum effects. These are phenomena you have in other (non-quantized) theories as well. What is a distinct quantum effect is that these superpositions give rise to probability distributions.

  58. Sure - and of course there's no need to 'psychoanalyze' Feynman or guess that he meant "the measurement process" when we can just read what he wrote. And, hopefully, no-one ever means that superpositions (and/or interference) per se are distinctly quantum - that would be silly.

    "What is a distinct quantum effect is that these superpositions give rise to probability distributions."

    I think that rather depends on how you view quantum theory - as some kind of mechanics (mysteriously) giving rise to probabilities or as some kind of probability theory applied to mechanics:

  59. You look about 7 feet tall in this picture; were you approaching the speed of light?

  60. Bee,

    With regard your explanation of entanglement and superposition, is there a broader interpretation here? Is there a larger meaning to be found in these phenomena or is this simply just the way it is, no more questions, end of story?

    You refer readers to Robert Helling’s notes:

    “Often (in fact any time we do not consider the whole universe as one big system) we are interested only in subsystems of bigger systems…

    This is formalized by taking the Hilbert space of the “whole world” to be a tensor product H=H1⊗ H2 of a Hilbert space, H1 for the smaller “system” we are actually interested in and another Hilbert space H2 of the rest of the world (often called “the environment”).”

    In this philosophical, big-picture context one has the image of systems nested one within another like Russian babushka dolls with the “whole world” being regarded as the largest babushka.

    Is it possible that quantum phenomenon could be interpreted as indicating that the entire physical universe (with its physics and physicists) is contained within a higher-order babushka whose geometry cannot be directly assessed, but only inferred through the shape of its contents?

    To the provincial reader of physics there are some lapses in the script that raise questions:

    Why in the universe’s grand expanse of time and space does each far-flung mote and moment progress redundantly according to the “laws of physics”?

    Why within the redundancy of physical law do new things arise?

    Why are wave-like and oscillatory dynamics so predominant in the universe with scales varying across many orders of magnitude?

    Why hydrodynamic quantum analogs?

    Why the multitude of uncertainty relations?

    Why wave/particle descriptions and probabilistic equations?

    Why states?

    Why energy?

    Why information?

    For what it is worth there is a simple experiment that suggests the existence of a higher-order babushka. A measurement of the angle created between a mason’s plumb bob and his spirit level will be found to be exceedingly close to ninety degrees. Perhaps it is simply a matter of properly interpreting the results.

    And as always, thank you.

  61. Bee,

    Old post, sorry for bringing it up again. But I have apparently been uncinciously pondering your word thoughout the year...

    You state: "Most importantly, whether a system has entanglement between two subsystems is a yes or no question."

    Which I interpreted "whether a system has entanglement is a yes or no question"

    Now I understand that the difference is major.

    Having a two qbit system, with four base states |00>, |01>, |10>, |11>.
    Which can have a entangled setup: 2^.5(|01>+|10>).

    But changing the basis to: |0>, |1> stating the leftmost qubit, and |e>, |n> stating whether the rightmost qubit is equal (e) or not equal (n) to the leftmost.

    With that basis (omitting the coefficients), |01>+|10> equals to |0n>+|1n> which turns to be seprable, ie not in entangled state.

    Did I got it correctly. Is the question of entanglement (with yes/no answer), a valid question only after we have restricted the basis by selecting how we divide the system to subsystems?

    If this is the case, I do not understand how the entanglement would be fundamentally different from superposition.

    Br, -Topi

  62. Topi,

    That's right, the addition 'between subsystems' is relevant. You can have superpositions of a system that doesn't have subsystems. Ie |1> + |0> (with proper weights) is a superposition, but not an entangled state. Best,


  63. Bee,

    I didn't mean single particle system (most basic superposition).

    What I tried to ask: is it really so that a system with more than one subsystem can be at the same time at non-entangled (separable) and in entangled state (non-separable) (depending on the selected basis), if-and-only-if the observer's freedom to select basis is not restricted by the selection of subsystems (in my example the first setup was selected to have subsystems spin+spin, while the second was selected to have subsystems spin+equality)?

    If so, should the "between subsystems" be more exactly stated as "between fixed subsystems"?

    BR, -Topi

  64. Topi,

    I'm not sure I understand your question. A system can be entangled between subsystems A and B but not between subsystems C and D. You can't make the entanglement go away with a choice of basis. A system can be in a superposition (of basis states) in some basis and not in some other basis. Best,


  65. Bee,

    Let's stick on two-qubit subsystem.

    On original basis (basis 1):

    psi = c0|00> + c1|01> + c2|10> + c3|11>.

    The rule was: "selecting a new basis the entanglement doesn't go away".

    By setting, c0 = 1/sqrt(2), c1 = 0, c2 = 0, c3 = 1/sqrt(2), we have an example state:

    psi = |00> + |11>, which is an entangled sate.

    The rule does not restrict me in selecting the new basis (apart from assumed orthonormality)

    I select the new basis vectors as:
    |Dn> = |01>
    |De> = |00>
    |Un> = |10>
    |Ue> = |11>

    where |D> vs |U> is measured with operator OL, and |n> vs |e> is measured with operator OR.

    OL =
    |0, 0, 0, 0|
    |0, 0, 0, 0|
    |0, 0, 1, 0|
    |0, 0, 0, 1|

    OR =
    |1, 0, 0, 0|
    |0, 0, 0, 0|
    |0, 0, 0, 0|
    |0, 0, 0, 1|

    yielding basis change operator:

    T =
    |0 1 0 0|
    |1 0 0 0|
    |0 0 1 0|
    |0 0 0 1|

    and now the example entangled state in basis 2 is

    psi_2 = T psi = T(|00> + |11>) = |De> + |Ue> = (|D> + |U>) * |e>
    (* = tensor product)

    which is the definition of separability, which I assume equals to non-entangled state.

    ie. in Basis 1, psi is entangled, and in basis 2, psi is not entangled.

    Where did I fail?

    BR, -Topi

  66. Topi,

    Your states are products of basis vectors. The basis vectors would be eg |1>_A, |0>_A for system A and |1>_B, |0>_B for B and your states would be something like |1>_A \otimes |1>_B and so on. There isn't any unitary transformation acting on A or B respectively that will make the entanglement between A and B go away. Best,



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