Classical Spheres, Division Algebras, and the Illusion of Quantum Non-locality:

Richard Gill

PS seems that the number of posts in the forum recently suddenly decreased from about 3050 to about 3000 so I suppose there has either been some major vandalism or some major active moderation.

My last word to JJC here: when he has posted the two computer files on internet and informed me that he has done so, he can claim 10 000 Euro from me; whether or not he gets them depends on whether or not they deliver the goods. If I disagree with JJC then we will call in the adjudicators.

The challenge is still open, not just to JJC, but to anyone. The rules (formulated jointly by RDG and JJC) are here:

Challenge rules on sciphysicsforums.com

Nick Mann

Interesting. Joy's toe-curling Gestalt-switch thread has disappeared, along with En Passant's of May 11 ("One might think that we can no longer produce works of great masters"). To guess the likely culprit or culprits (not En) you might apply the old legal adage: Cui bono?

But take heart because nothing's lost! Everything (or almost everything) posted on this thread during its lifetime is archived and searchable. Not that anyone's likely to bother but just in case. God I love technology.

Delete this and I'll just repost it.

Nick Mann

En's wasn't/isn't toe-curling.

Nick Mann

Should've said "Everything (or almost everything) posted to this forum topic during its lifetime ..."

Nick Mann

The final two posts on En's thread R.I.P.:

Nick Mann replied on May. 13, 2014 @ 23:04 GMT

Which reminds me. (From the sublime to the not-so-sublime.) You're adamant that Richard Gill owes you ten thousand euros. Are you planning to press your claim or allow him to welch? We could all use some closure here.

report post as inappropriate

Member Joy Christian replied on May. 13, 2014 @ 23:11 GMT

Watch this space.

report post as inappropriate

Richard Gill

Now, back on topic: take a look at:

arXiv:quant-ph/0108137

GEOMETRY OF ENTANGLED STATES , BLOCH SPHERES AND HOPF FIBRATIONS

REMY MOSSERI AND ROSSEN DANDOLOFF (2001)

Very nice paper. Published the same year, I believe, in J Phys A

Abstract. We discuss a generalization to 2 qubits of the standard Bloch sphere representation for a single qubit, in the framework of Hopf fibrations of high dimensional spheres by lower dimensional spheres. The single qubit Hilbert space is the 3-dimensional sphere S3. The S2 base space of a suitably oriented S3 Hopf fibration is nothing but the Bloch sphere, while the circular fibres represent the qubit overall phase degree of freedom. For the two qubits case, the Hilbert space is a 7-dimensional sphere S7, which also allows for a Hopf fibration, with S3 fibres and a S4 base. A main striking result is that suitably oriented S7 Hopf fibrations are entanglement sensitive. The relation with the standard Schmidt decomposition is also discussed

Greetings from a small island in the Baltic, excruciatingly slow 2G mobile internet

PS Still no submission of two files received yet from JJC or anyone else for that matter

Thomas Ray

Thanks for the link, Richard. It should help you understand the difference between Joy's framework of a complete measurement space of quantum correlations, and the assumption of quantum entanglement in the Hilbert space of separable states.

This paper was written a couple of years before Perelman's proof for the simple connectivity of S^3. The authors, using the definition of S^3 as a pair of complex numbers alpha, beta, such that |alpha|^2 |beta|^2 = 1 necessarily rely on the axiom of choice to get unitarity from the Hilbert space. I showed explicitly that eliminating the axiom of choice from the Hilbert space analytical framework eliminates decomposition, such that quantum mechanical unitarity is preserved in the continuous wave function. Therefore, no particles are entangled.

Richard Gill

Perelman did not prove the simple connectivity of S^3. Perelman proved Poincaré's conjecture that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

Mosseri and Dandoloff's mathematics seem to be correct, modest, and interesting. They did not shed light on quantum non-locality but did provide new tools for exploring the conventional framework of QM in the restrictive context of two entangled two-level systems. The connection between the Bloch ball and S^3 was already known. They extended this connection between quantum states and classical spheres from one two-level system to two. However no-one has been able to take this correspondence further.

Thomas Ray

"Perelman did not prove the simple connectivity of S^3. Perelman proved Poincaré's conjecture that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere."

Same thing, dude. If S^3 were not simply connected, the statement would be false. The space has trivial fundamental group -- what Perelman actually proved, equivalent to the Poincare conjecture, is Thurston's geometrization conjecture; i.e., finite geometric structures lie in the interior of surgically riven tori. If you understood Joy's framework, you would see that his complete space of measurement functions demands just such a structure, so that the function does not blow up at infinity.

"Mosseri and Dandoloff's mathematics seem to be correct, modest, and interesting."

Sure.

"They did not shed light on quantum non-locality but did provide new tools for exploring the conventional framework of QM in the restrictive context of two entangled two-level systems."

Except that if entanglement doesn't exist, the mathematics doesn't support a physical interpretation. One has to insert entanglement by hand to make this schema work.

"The connection between the Bloch ball and S^3 was already known. They extended this connection between quantum states and classical spheres from one two-level system to two. However no-one has been able to take this correspondence further."

Oh yes, they have. You just haven't caught up with it in the 13 years since this paper was published.

Heine Rasmussen

That S^3 (or any S^n with n>1 for that matter) is simply connected was proven by Poincaré himself, dude.

Thomas Ray

And once again, you are overlooking the relevant facts. Shall I be more specific, before you delete them again?

Thomas Ray

Perhaps Terry Tao explains it more clearly. I have posted this link before, to the article which also appears in Tim Gowers, ed., The Princeton Companion to Mathematics.

Explaining how Perelman used the Ricci flow to prove Thurston's geometrization conjecture, Tao makes clear the role of simple connectedness. When we hypothesize about the shape of the universe, the measurement function it accommodates always depends on its manifold connectedness. A multiply connected space such as the torus works for quantum mechanics, though it cannot be a space of complete measurement functions, which dooms quantum theory to mathematical incompleteness. Not so for the classical schema on a simply connected manifold:

"In higher dimensions, the Ricci flow can develop singularities before perfect symmetry is attained, but it turns out that it is possible to perform surgeries on the singularities that develop this way, so that the manifold becomes smooth again and one can restart the Ricci flow process. (The surgery may however change the topology of the manifold, for instance it might convert a connected manifold into two disconnected pieces.) In three dimensions, it has recently been shown by Perelman (though with not all details fully fleshed out yet) that Ricci flow, when augmented by surgery to remove the singularities, does indeed convert an arbitrary manifold (obeying some mild assumptions) into a finite union of some very symmetric (and explicitly describable) pieces; the precise statement of this conclusion was known as the geometrisation conjecture of Thurston. One consequence of this conjecture (which is now a rigorous theorem proved by Perelman) is the Poincare conjecture: any three-dimensional manifold which is simply connected (which means that any closed loop on the manifold can be contracted smoothly to a point, without ever leaving the manifold) can in fact be smoothly deformed into a 3-sphere (which is to four-dimensional Euclidean space as the usual two-dimensional sphere is to three-dimensional Euclidean space)."

Heine Rasmussen

There is nothing wrong with Terence Tao's text, for sure. So does this mean that you now understand that Poincaré's conjecture was not that "S^3 is simply connected", but rather something different?

Thomas Ray

Heine -- LOL! Do you understand that simple connectedness is requisite to geometric uniformization? And more relevant to the topic -- that an intrinsic geometric flow is independent of a coordinate system? I think I'm done here.

pjackson

Richard,

Please stop removing my posts showing your claim to be false. Your 'supporting logic' did not falsify my proposition A = B C. Accusations must be justified.

Brendan is reviewing all Orwellian removals and I expect a number to re-appear shortly.

Peter

Richard Gill

Peter, I think your logic is wrong. You think mine is wrong. Those two facts have been established twenty posts ago. Neither of us is going to change their minds. No one else is interested. Leave off the personal insults aimed at me. Every time you go off topic in order to make insulting personal remarks, I'll report your post as inappropriate.

Reminder: the topic is "Classical spheres, division algebras, and the illusion of quantum locality".

Richard Gill

I do not "remove posts showing that my [RDG's] logic is false". I *report* posts containing what I feel is inappropriate content, as I understand the meaning of the phrase "inappropriate content". I studied the FQXi forum rules in order to find out what FQXi means by this.

A post claiming that I *remove* posts which allegedly show I am wrong is a very good example of a post with inappropriate content.

Thomas Ray

Get the whole story right. Particularly that of finite fundamental group. Dude.

Richard Gill

Forum rules: Per the scientific and educational purposes of the forums, users should adhere to a code of conduct where they discuss in a civil, helpful, and constructive way. Comments should not contain language or content that is:

Rude or disrespectful;

Combative or overly aggressive;

Unpleasant or offensive by common workplace standards;

Targeted at individuals, rather than at science content.

[deleted]

Mosseri and Dandoloff (2001; quant-ph/0108137) represent single qubit pure states by points on S^3 through noting that two complex numbers alpha and beta with |alpha|^2 |beta|^2 = 1 determine a point on S^3. The problem with this picture is that (a) there are no mixed states in the picture, (b) the overall phase of a Hilbert space state vector is superfluous. Thus in their picture, pure states are represented by curves on S^3, not points; and there are no mixed states.

In fact, pure qubit states can be usefully represented by *points* on S^2: the Bloch sphere. Mixed states become geometrically points inside the sphere, ie in what one might call the Bloch ball. They have coordinates x, y, and z satisfying x^2 y^2 z^2

Richard Gill

... = r^2 < or = 1. Now define t = sqrt(1 - r^2). We now obtain two points (x, y, z, +/- t) on S^3 representing a qubit mixed state. It turns out that the geometry of S^3 is intimately related to quantum fidelity, a fundamental concept in quantum state reconstruction and other quantum informational tasks.

No idea if there are generalizations to higher spheres (higher spins, multipartite systems).

Thomas Ray

"Thus in their picture, pure states are represented by curves on S^3, not points; and there are no mixed states."

This is true in my picture, too. (3.3) When continuous curves are exchanged for discrete points, we get the Poincare Conjecture in physical terms.

"In fact, pure qubit states can be usefully represented by *points* on S^2: the Bloch sphere. Mixed states become geometrically points inside the sphere, ie in what one might call the Bloch ball. They have coordinates x, y, and z satisfying x^2 y^2 z^2"

And this local geometry lacks the extra degree of freedom required to satisfy the global topology.

Joy Christian

In the last paragraph of this paper I make the following observation:

"I hypothesize that the space we live in respects the symmetries and topologies of a parallelized 3-sphere, which is one of the infinitely many fibers of a parallelized 7-sphere. The EPR correlations are thus correlations among the points of a parallelized 3-sphere, whereas quantum correlations in general are correlations among the point of a parallelized 7-sphere."

This hypothesis is fully spelt out in this paper as well as this paper (see, especially, the main theorem and its proof on the pages 12 to 16 of the last paper).

Joy Christian

I messed up the first link. Here is the correct link.

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