Observers

[The God]

Dan,

I feel you have still not understood the problem with your jpg image equation in #88. Please note that in spacetime interval equation the j (or i) does not appear. You were not ducking, it's just that you did not get it.

Why dont you look at spacetime interval equation in some standard text and see if j appears the way you painstakingly typed.

 

[exchemist]

Actually this thread is about observers, and what observers are.

But no matter. As to quantum computers, here the notion of a synchronizing clock is somewhat different to the clock in a sequential computer.
For instance, if you suppose that an interference pattern is the result of a quantum computation, then where is the system clock?

If you have a which-path type of experiment (Wheeler's delayed choice), again where is the clock?

I've noticed there is a big difference between classical computing and quantum computing in regard to the way space (storage) and time (synchronized steps) look.
You have to more or less avoid thinking classically until there is some classical output (maybe). There is the rule of thumb that claiming you understand QM means you probably don't.

But if you can build a quantum computer, that must go some way towards a demonstration that you do understand it.
It's like you have to adjust what you mean by understanding.

Ah OK thanks.

This is clearly beyond me, so I'll bow out of the discussion.

 

[danshawen]

Dan,

I feel you have still not understood the problem with your jpg image equation in #88. Please note that in spacetime interval equation the j (or i) does not appear. You were not ducking, it's just that you did not get it.

Why dont you look at spacetime interval equation in some standard text and see if j appears the way you painstakingly typed.

You must be referring to the interval defined with all those (delta x)^2, (delta y)^2, (delta z)^2, and -(i*delta t)^2, is that it?

Then cool your jets. I wanted you to get a geometric picture of the situation, so set x = y = z = t = 0 for the origin. Plant a vertex of the resulting right triangle right at the origin of your coordinate system. And PRESTO! a Minkowski spacetime interval connecting two points separated by light travel time appears between the coordinates 0,0,0,0, and x,y,z,t2. (t1 = 0) mathemagically appears. The interval expression I gave in post #88 only needs a single physical dimension for each term under the radical because it is assumed the interval is referred to a geometrical origin 0,0,0,0 planted in space, as much as you all know I hate doing that in space without inertia.

Same arguments apply as I stated in post #88, only now you can geometrically see the assumption Minkowski made about time. To throw in the only other wrinkle; plant one entangled electron in atomic structure at 0,0,0,0, and adjust distance coordinates until its entangled twin is centered at x,y,z,t2.

Start a photon propagating from 0,0,0,0 and watch it ever so slowly approach x,y,z,t2. Flip the entangled electron at 0,0,0,0 and INSTANTLY, the electron at x,y,z,t2 changes state. The photon, on the other hand, is still between those points; in fact, it hasn't even moved in your mock Euclidean relativistic space yet. That's because Minkowski in his wisdom set time itself proportional to the speed of light. Works fine for v<c. Doesn't work at all for entanglement.

The entangled pair is part of the same waveform, you see? If we chose to do the same experiment with entangled photons, it would work only a little differently. You'd need to wait (light travel time) until the entangled photon actually reached x,y,z,t2, but after it passed that point, flipping the entangled state of its twin photon, trapped in a delay line or whatever, would also instantly flip the state of the photon at (or slightly beyond) the point x,y,z,t2.

Is that a little bit clearer than post #88?

A second central point came up in the discussion regarding QFT's edict that particles of matter or antimatter are ideal points. Finally I realized, they do this only because what follows in QFT simply does not wish to deal with the idea that it takes some as yet unspecified fundamental force to hold something like a quark or an electron together inside of atomic structure. Is it entanglement? Is it Higgs? Is it some combination? Who would know? No one who decided they don't really need to know; that's for certain. Someone with a very low regard for something as fundamental to physics as the conservation of energy, no doubt. By that reasoning alone, a single electron should simply fly apart due to the repulsive effect of its own negative charge, shouldn't it? What exactly keeps it from doing so? Does anyone really believe the Lagrangian for atomic structure is a complete compendium of the forces holding matter together without a knowledge of this force? That we even know anything at all about quarks is something of a miracle too.

No wonder such folk are loathe to hear about the discovery of something like the Higgs boson. How in the world they somehow managed to predict it even existed over 50 years ago is truly mathemagical.

Think more critically about the science and math you already think you know.

 

[arfa brane]

Entanglement, while we're trying to decide what it is, is about measurement after an interaction has occurred.

What does quantum computing tell us about entanglement, when it's something you can manipulate in abstract quantum circuits?
You use a combination of the CNOT gate and the Hadamard gate to entangle two states.

This 'logical' view of what it (entanglement) is, lays out an interaction in a left-to-right fashion, there is a notion of sequentiality.

 

[arfa brane]

This is what the state space of a single particle looks like: (the Bloch sphere)

​

Operators amount to rotations about the x, y, or z axes of the rest of the sphere. If you look at one of the single-qubit 'gates' like the Hadamard gate or the NOT gate, these are just rotations or compositions of rotations, after choosing an axis.

The kets |0> and |1> are vectors, the faint grey lines projecting from the state onto the xy plane are the complex amplitudes. Hence the magnitude of the state vector is given by Pythagoras' formula, you square the amplitudes and take the square root of the sum, amplitudes can therefore be negative, so the basis spans the entire sphere.
The two complex angles, well, there they are . . .

Unfortunately the Bloch sphere doesn't describe more than one qubit very well. In gate diagrams, there is an effective directed graph in which the qubits are just horizontal lines, vertically concurrent symbols (columns) represent operations:

The concurrence indicates a product space of two (in this case) 'Bloch units", the Hadamard gates transform these as distinct units (by rotating them equally), the CNOT gate is inverted.

 

[The God]

You must be referring to the interval defined with all those (delta x)^2, (delta y)^2, (delta z)^2, and -(i*delta t)^2, is that it?


Is that a little bit clearer than post#88..

Bold is mine inside the quote.

So you have i^2 or j^2 and that is -1, now pl revisit your #88 and tell us why that j appears inside square root?

 

[danshawen]

Bold is mine inside the quote.

So you have i^2 or j^2 and that is -1, now pl revisit your #88 and tell us why that j appears inside square root?

Not just 'j'; '(-1 * j)'. The time component of the interval is imaginary. That's all.

 

[The God]

Not just 'j'; '(-1 * j)'. The time component of the interval is imaginary. That's all.

 

[Q-reeus]

Not just 'j'; '(-1 * j)'. The time component of the interval is imaginary. That's all.

Uh uh - your formula for s in #88 won't give an imaginary time component. As you sort of abided by the -+++ convention there, an imaginary result using standard expression should only eventuate for a time-like interval, i.e. |(cdt)²| > |dx²+dy²+dz²|. And it should be purely imaginary. Your expression gives the square root of a complex number in general, except for the special case dt = 0 hence -j(cdt)² = 0.
What physical significance is to be attached to the square root of a complex number?

 

[arfa brane]

You might be wondering, as I am, where the complex coefficients are when |0> and |1> are on the z axis of the Bloch sphere.

What the coefficients are are complex numbers, say $$ \alpha, \beta $$ such that $$ {\alpha}^2 + {\beta}^2 = 1 $$.
And you need the inner product $$ \langle \psi | \psi \rangle = 1 $$

 

[arfa brane]

Of course.

The complex coefficients are in the Hilbert space, the Bloch sphere is a representation of two Hilbert-orthogonal states, |0> and |1>, in a measurement basis in which spin-up and spin-down are 180° apart. Coins are the same if you take heads and tails to unit vectors in $$ \mathbb R^2 $$.
A spinning coin is in a state where the outcomes are equiprobable on the z axis (thanks to gravity and flat surfaces), so you have an abstract probability vector halfway between heads and tails with equal real coefficients.

So that's one solution to $$ \alpha^2 + \beta^2 = 1 $$: $$ \alpha = \beta = \frac {1} {\sqrt 2} $$

So, that means the azimuthal angle is twice as large in the Bloch representation than in $$ \mathbb C^2 $$. To, hmm, restore the complex symmetry you map antipodal points on the sphere to orthogonal vectors in the Hilbert space.

 

[danshawen]

What physical significance is to be attached to the square root of a complex number?

Now, THAT, is exactly my point. Minkowski could not mathematically combine space coordinates with time coordinates directly, so he decided he would do so using an Argand vector space, just like his star student Hilbert continued to do with his "Hilbert spaces".

I further researched some of John Duffield's (Farsight's) links to translations of Einstein-- The Berlin Years, only to find end notes riddled with corrections to the mathematical notation in the body of the text. So I'm not going to belabor the point that Minkowski, like many modern physicists, believed so much in the uniqueness of the invariant speed of light, that he also believe that it must be the basis of time itself, not pausing to think that he might have been making a serious mistake about that.

http://press.princeton.edu/titles/6161.html

In physical reality, the speed of light, c, is no more unique an invariant than the rest frame to which it is always referenced. Even the invariant velocity of light must be referenced to another velocity, even if it is a velocity of zero. You must always reference a motion to something else if you plan on using proportional math to compare them. The only reason any rest frame out of an infinite choice of inertial reference frames is unique is because of the centers of particles of bound energy moving with it. The centers of those particles is about as close a thing to "absolute space" as you are likely to find in this universe. An instant of absolute time exists because entanglement and a quantum spin of zero exists. A coordinate or position of Euclidean absolute space does not.

Just like he evidently was caught up in a whirlwind romance with quadratic equations, the idea of relating space and time in a novel mathematical way was simply too tempting to resist for someone like Minkowski, whether anything that he scrawled down made any real physical sense or not. To his credit, a lot of it did (make physical sense). But some of it did not. The speed of light as the basis of time did not. Minkowski (hyperbolic) spacetime rotations really did not either. Instead of using the rotated tail lights of a relativistic Cadillac, try the same experiment with a pair of entangled electrons in place of the tail lights. Not only do those simultaneous spin flips not rotate, it doesn't even make a difference how far apart or or close together they are, or even where they are in their orbitals or probability densities when they flip. There is no such thing as Minkowski rotation simply because he used the wrong model for simultanaeity. Light cones don't apply to entanglement either, and for the same reason.

I did a lot of thought experiments, some of them here, to verify the fact that Minkowski rotation really made no sense. Many of those were rather strongly opposed at sciforums on purely mathematical grounds, as if Minkowski could do no wrong. At least, I understand why now. His work was pretty good alright. It just wasn't as good as some here would have us believe.

And so we come now to the problem of arfa brane's difficulty with making those Bloch Spheres work as a model of entanglement for more than one qubit. I can already see, this is going to take some considerable effort to unravel, and it isn't going to be a pretty picture, either.

I'm going to need to go back to Feynman's original notes on quantum computing, because Hilbert's all seem to be a load of mathematical trash inherited from his favorite teacher Minkowski. Hilbert didn't a bit more understand the basis of entanglement than he knew anything about Feynman's "partons", Zweig's "aces", or Gell-Mann's "quarks" I doubt he even cared that he didn't know either. Hilbert was one of those who believed that Euclid's geometry held the keys to any mathematical or physical inquiry.

Hilbert's axiomization of mathematics:

https://plato.stanford.edu/entries/hilbert-program/

Fell short mainly because Einstein's friend Kurt Gödel derailed the entire effort before it was ever started. This was such a monumental failure of a great deal of effort basically for nothing, I can't understand why anyone bothers to read anything else Hilbert did his entire life. Einstein scooped him on General Relativity as well, but even Einstein found he could not deliver the theory without manipulations that contradicted some basic assumptions he needed to make for Special Relativity. I have already pointed out what some of those were, and why they were not inclusive of entanglement. So, Gödel's work ultimately meant that even his friend Einstein's life's work was not 100% complete or consistent either.

Part of the problem with the Bloch spheres seems to be that Special Relativity isn't necessarily going to work inside of an entangled bound particle of matter the way it does with unbound energy propagating a straight line trajectory, even though entanglement is possible for both modes of energy propagation (in a line, or spinning / or whatever else it does).

 

[Q-reeus]

A lot of thoughts there Dan, but I will just comment on your response to one issue. There is or rather are (two) formal mathematical solutions for your complex s of #88. Both being themselves complex quantities. Which implies some pretty bizarre physics. But hey maybe something wonderful might come out of such a situation - despite ny severe misgivings...

 

[danshawen]

A lot of thoughts there Dan, but I will just comment on your response to one issue. There is or rather are (two) formal mathematical solutions for your complex s of #88. Both being themselves complex quantities. Which implies some pretty bizarre physics. But hey maybe something wonderful might come out of such a situation - despite ny severe misgivings...

A second thought or two on the subject of relativity's demise in explaining quantum phenomenon is all that I'm asking here. To me, it has been a revelation. And I do want arfa brane's quantum computer to work.

Paradoxically, unique and alone among systems of mathematical reasoning, ONLY Kurt Gödel's reasoning about incompleteness and inconsistency seems to be both consistent and complete. Has no one other than Einstein noticed how rare this is?

I think possibly the key to Gödel's success is how narrow his focus was on the problem at hand, something it no doubt shares with Special Relativity. It also helps that he was reasoning ABOUT mathematics, a tool with crisply demarcated limitations already imposed by its very nature.

GENERALization it seems, must always fail at some point related to consistency. It is just not possible to symbolically define rules that apply in all situations. You simply can't know all there is to know about everything there is. And so you must specialize if only because a flea or a blind man can't grasp the totality of something as large as an elephant at once.

SPECIALization, on the other hand always fails in the category of completeness. It is not possible to ignore everything outside one's narrow expertise with respect to one's own discipline without the possibility that some aspect of the specialty problem area has been neglected. No matter how narrow your focus, you can't possibly know everything about anything, regardless of any artificial limitations placed on its scope. Quantum physics would be the best example so far.

The symbolic nature of mathematics also works against a finite mind attempting to grasp the infinite, but it is the only tool blind fleas with finite minds have to work with.

AN OBSERVER MUST ALWAYS CHOOSE A DIRECTION IN WHICH TO OBSERVE. This is the essence of what an observation is, even if all that is observing is the cold, dead focusing lens of a scientific instrument. The narrower the field of view is, the greater directional selectivity, and the greater the tunnel vision effect. We are all as blind fleas.

 

[Write4U]

Danshawn said,
AN OBSERVER MUST ALWAYS CHOOSE A DIRECTION IN WHICH TO OBSERVE. This is the essence of what an observation is, even if all that is observing is the cold, dead focusing lens of a scientific instrument. The narrower the field of view is, the greater directional selectivity, and the greater the tunnel vision effect. We are all as blind fleas.

As Antonsen says; "different perspectives lead to greater understanding" of what is being observed.

 

[arfa brane]

Wolfram Mathworld has this to say about the Poincaré sphere (aka the Bloch sphere).

The Poincaré (Bloch) sphere provides a geometric representation of a pure qubit (quantum bit) state space as points on the surface of the unit sphere

. Any point of the surface represents some pure qubit.

The mixed qubit states can be represented by points inside of the unit sphere, with the maximally mixed state laying at the center. The red line from the center to the surface of the sphere corresponds to the pure state and has unit length. For mixed qubit state the length of line must be less than 1. (references here to a diagram)

The most general single qubit state has the spinor form

, where

and

are spherical polar coordinates with

and

.

So the complex 2-vector has half the zenith angle $$ \theta $$ (sorry, I initially said it was the azimuth and it isn't) as the Bloch sphere rep. This seems to be a consequence of having to map the measurement operators (Pauli matrices) to 3-dimensional space, which is of course where measurement operators operate.

What do entanglement gates do?

 

[Write4U]

Arfa said,
What do entanglement gates do?

The million dollar question.

I am usually able to fundamentally imagine the concepts of GR and QM, but entanglement seems almost mystical. It has no apparent limitations, other than its mirror entanglement.
Could entanglement happen in a different plenum of our universal dimensions?

 

[arfa brane]

From Wikipedia:

Quantum mechanics is mathematically formulated in Hilbert space or projective Hilbert space.

The space of pure states of a quantum system is given by the one-dimensional subspaces of the corresponding Hilbert space (or the "points" of the projective Hilbert space). For a two-dimensional Hilbert space, this is simply the complex projective line ℂℙ1. This is the Bloch sphere.

So the Bloch sphere is a re-labelled Riemann sphere? And since it's also the complex projective line, maybe there's a way to view entanglement in that context?
Except, you have two spheres, or projective lines. Their product is a tensor space, sort of a vector with twice as many dimensions.

Also note that because this Hilbert space 'drops' into the same one where coins have real probabilities--measurements are real-valued--you can do some of the gating stuff with real vectors, the tensor products are then quite trivial: 00 is the tensor product of two 0-states (I've omitted the Dirac notation), and so on. Tensoring two real bits (not qubits) gives a real 2-bit register.

Like the example you see of an XOR gate being equivalent to a CNOT gate, modulo some thing or other.

 

[The God]

A lot of thoughts there Dan, but I will just comment on your response to one issue. There is or rather are (two) formal mathematical solutions for your complex s of #88. Both being themselves complex quantities. Which implies some pretty bizarre physics. But hey maybe something wonderful might come out of such a situation - despite ny severe misgivings...

Nope, that is typo. No new physics.

 

[Q-reeus]

Nope, that is typo. No new physics.

No typo. Maybe misunderstood my drift. Agreed no new physics.