Bell’s theorem has been called “the most profound discovery of science”. However, there have been controversies on the deep implications of the theorem. This online workshop aims to highlight the existing debates and address the controversies. Read More
Workshop Date: Thursday, December 18, 2014 to Friday, January 16, 2015
Organizers: International Journal of Quantum Foundations
A new proof of nonlocality in standard quantum mechanics
December 19, 2014 at 3:40 am #1645
Consider a EPRB experiment in which there exists an ensemble of two spatially-separated and spin-1/2 particles (labelled 1 and 2) prepared in a spin singlet state. The spin of particle 1 is measured along two possible directions, a and a’, and the spin of particle 2 is always measured along the same direction $b$. Moreover, the two spin measurements on particles 1 and 2 are spacelike separated. Then, when the spin of particles 1 is measured along direction a or a’, the outcomes of the spin measurements on particles 2 have certain statistical distribution. According to the predictions of quantum mechanics, these two distributions for particles 2 are precisely the same.
My new proof of nonlocality is as follows. According to standard quantum mechanics, after each spin measurement on particle 1 (along direction a or a’), the wave function of particle 2 collapses from the initial entangled state to a definite spin state. The collapsed wave functions are different for different measuring settings a and a’. Then by introducing a possible nonlinear evolution for the wave function of particle 2 such as that suggested by Gisin (1990), the two statistical distributions of the measurement outcomes of particles 2 corresponding to the settings a and a’ may be different. This leads to superluminal signaling, in which there is definitely nonlocality (when assuming no superdeterminism). Since the introduced nonlinear evolution for particle 2 is local, there must exist nonlocality in the original EPRB experiment. Moreover, the nonlocality must assume the same form as the superluminal signaling resulting from possible nonlinear evolution.December 20, 2014 at 2:44 am #1656editorKeymaster
My proof of Bell’s theorem can be summarized in one sentence: since local evolution can change no-signaling to nonlocal signaling, the no-signaling also contains non-locality, the same non-locality in nonlocal signaling.
This new proof of the existence of non-locality seems too simple to be true. Critical comments are very welcome!December 22, 2014 at 3:11 am #1678Sheldon GoldsteinParticipant
Hi Shan. You are right. The proof is too simple. The mistake is this:
One should not take it for granted that the wave function is genuinely
physical, objective, ontic, in the proof of nonlocality. The proof
should depend only on the experimental facts of quantum theory. But for
your proof to work, you must assume that the wave function is ontic.
Otherwise you would not be able to conclude that the hypothetical local
nonlinear evolution of the wave function could lead to physically
different states depending on whether a or a’ had been chosen. Physical
differences out would require physical differences in. (And if a and a’
did directly lead to physically different situations, that would already
constitute nonlocality and you wouldn’t need the hypothetical nonlinear
Best, ShellyDecember 25, 2014 at 1:27 pm #1694
Many thanks for your very helpful comments on my new proof of Bell’s theorem. You pointed out a very important potential issue, and I should have clarified my argument concerning this point. Here are my answers.
First of all, as you may admit, when assuming that the wave function is ontic, my proof is valid. I think this is still an interesting result. But certainly, as you have emphasized, it is not a proof of Bell’s theorem.
Next, I think the psi ontology assumption may be avoided in my proof, since nonlinear evolution of the wave function may be not needed in my proof. Here I should have made my point clearer. It is also logically possible to introduce certain hypothetical local interaction with particle 2 or even with the device, which may change the outcome distributions to lead to nonlocal signaling. Certainly, if the wave function indeed represents the physical state of particle 2 (i.e. if the wave function is ontic), then such interaction may be equivalent to nonlinear evolution of the wave function.
Thirdly, I think even if my proof resorts to nonlinear evolution of the wave function, it does not necessarily assume psi ontology. The reason is that that different evolution of the wave function leads to physically different results or states of device does not necessarily require that the wave function is ontic. In standard quantum mechanics, different linear evolution of the wave function also leads to physically different results of measurement.
Lastly, I agree that physical differences out would require physical differences in. For example, if there is nonlocal signaling, then a and a’, which are physical different, can lead to physically different situations. But the wave function is a description of the in-between process involving the particles, and it is not necessarily physical or ontological only because the input and output are physical. Otherwise we will have a very simple proof of psi ontology.
Thanks again for your very helpful criticisms! Your further comments and criticisms are very welcome.
ShanDecember 25, 2014 at 1:30 pm #1697Sheldon GoldsteinParticipant
Hi Shan. The crucial point, as you say, is
> physical differences out would require physical differences in.
> But the wave function is a description of the in-between process involving the particles, and it is not necessarily physical or ontological only because the input and output are physical.
Remember that the only noncontroversial difference between the effects of choosing a vs a’ is in the distribution of B’s (collapsed) wave function. If that is not a physical diffference, the game is over and your argument can’t work. And if it is a physical difference the game is also over: you have nonlocality with no need to continue with further hypothetical interactions, nonlinear or otherwise.
Best, ShellyJanuary 13, 2015 at 2:58 am #1808
Thanks again for your further comments! I still think my proof of nonlocality does not depend on the meaning of the wave function. No matter the wave function is epistemic or ontic or something else, the wave function may be changed in a linear way by an usual external potential according to the Schrodinger equation, and it is also possible that the wave function may be changed in a nonlinear way by a special hypothetical external potential. The above proof only relies on the possible existence of the nonlinear evolution of the wave function (besides the collapse postulate and the Born rule). [If one thinks the proof depends on the meaning of the wave function, then whether or not nonlinear evolution may exist will depend on the meaning of the wave function, which seems to be implausible.]
Certainly, this new proof of nonlocality in standard quantum mechanics does not establish that our world is nonlocal. But the above result, if it is valid, may be still a surprise for some orthodox physicists. For the proof shows that no matter how to interpret the wave function, nonlocality always exists in standard quantum mechanics; one cannot resort to the possible lack of counter-factual definiteness or non-contextuality to avoid the nonlocality, though one might avoid it in another different quantum theory.
Best, ShanJanuary 13, 2015 at 12:32 pm #1810
I think my new strategy to prove nonlocality may be more helpful for understanding the nature of nonlocality, e.g. determining whether the nonlocality requires the existence of a preferred Lorentz frame.
It is noncontroversial that the existence of the above hypothetical superluminal signaling is incompatible with the theory of relativity, and it will lead to the existence of a preferred Lorentz frame. No matter which convention of synchrony is adopted, the preferred Lorentz frame can always be defined as the inertial frame in which the one-way speed of light is isotropic and the superluminal signaling is transferred instantaneously in space.
Now, the superluminal signaling is composed of two processes: a nonlocal process and a local subluminal process. Obviously, the local subluminal process does not lead to the existence of a preferred Lorentz frame. Thus the process leading to the existence of the preferred Lorentz frame must be the nonlocal process. In other words, quantum nonlocality must lead to the existence of a preferred Lorentz frame.
Note that this argument for preferred Lorentz frame is independent of whether my proof of nonlocality is valid, though they use the same strategy: nonlocality + locality -> superluminal signaling.
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