NIST Physicists Coax Six Atoms Into Quantum 'Cat' State

Science Daily | National Institute of Standards and Technology ^ | 2005-12-03

[grey_whiskers]

Try this:

1 = 1

1/-1 = -1/1

(cross multiply)

-1 * -1 = 1 * 1

(-1) ** 2 = (1)**2

(take square roots)

-1 = 1

Tee hee!

[VadeRetro]

(take square roots)

Which points up that an operation performed on both sides of an equation had better return a single, unique value.

[Quark2005]

The technique sounds impressive when written about, but I'm pretty sure it doesn't allow communication faster than light.

You are absolutely correct - this is a common misconception about entanglement that I've even seen a few physicists fall into.

Quote from a 1997 physics news article Signal Travels Farther and Faster Than Light

"Whatever the nature of the connection between entangled particles may be, nearly all physicists agree that it cannot be used to transmit messages faster than the speed of light. All it can do is assure that a random choice by one entangled particle is instantly echoed by its distant partner. This is not the same thing as transmitting information, the experts say, and therefore it does not violate relativity theory."

[onedoug]

Hmm. Is that supposed to refer to wife beating in the Bible? How?

[King Prout]

Try this:
1 = 1
...
...
...
-1 = 1
Tee hee!

for making my brain hurt after a long work day:

*SMACK!!!*

[grey_whiskers]

*SMACK!!!*

Now my HEAD is spinning in both directions :-)

But I deserved it...

[griffin]

HA! Let's see the muslimes do this!

[The Red Zone]

It is a classic formatting problem, which I'm surprised to see is mirrored in the original source article.

[The Red Zone]

Cat Burglars beware.

[The Red Zone]

So should another name be used for the states of the entangled particles. Dog states, anyone?

[snarks_when_bored]

(...nothing up this sleeve, nothing up that sleeve...)

But, for the record, the positive square root of the square of a quantity is equal to the absolute value of the quantity; it is not equal to the quantity with its original sign. That is,

√[ x² ] = |x| .

Hence, since

√[ (-1)² ] = |-1| = 1

and

√[ (1)² ] = |1| = 1 ,

there's no contradiction.

Shame on you, g_w, for giving KP a headache!

[grey_whiskers]

Sorry to disagree with you for once, snarks, but the last words in my post were (and I quote from memory):

Tee hee.

It was a jest.

(I saw it on in a T-shirt catalog once, next to the "Reunite Gondwanaland" and "If you're not part of the solution, you're part of the precipitate".)

Cheers!

...and BTW, great job on the radical signs in HTML in your post.

[snarks_when_bored]

It was a jest.

Mine was, too. I guess I'm getting heavy-handed (not to mention heavy-headed) in my dotage!

[TChris]

The NIST work, while a long way from full entanglement of a real cat’s roughly 1026 atoms...

Wow, that must be one small cat!

[TChris]

This situation is sometimes called quantum indeterminacy or the observer's paradox: the observation or measurement itself affects an outcome, so that it can never be known what the outcome would have been if it were not observed.

I still have a problem with the generalization that an obervation cannot be made without affecting the outcome. I would accept that in some cases, even most cases, that an observation would affect the outcome, but absolutes seem to be dangerous territory.

[SubMareener]

Ah! The time-dependent one-dimensional Schrödinger equation! Be still my beating heart! The gentle curves of your partial derivatives are so much more romantic than the harsh Bras and Kets of the Dirac formulation! ;-)

[Paul Ross]

Bump. Neat.

[William Tell]

Factoring polynomials to determine roots is one technique that can be used. One is not actually "dividing" by the factor which is known, but rather one is simply ignoring that factor and observing the other two.

For example, we wish to know "a" and "b" from the following equation:

x^3 +1 = (x+1)(x-a)(x-b)=0

The values of "a" and "b" will be the other two cube roots of "-1".

I'm not aware of a "procedure" for factoring polynomials which have complex roots.

But there is a way to find these roots using the known geometric properties of roots in the complex number plane.

Every complex number can be represented in the form "a+bi", where a is the real component and b is the imaginary component.

Mapping such a number in the complex number plane results in a single point.

There is an alternative method for designating complex numbers which use the magnitude and direction of the vector which is represented by that same point.

The direction along the positive real axis is at angle zero. The direction along the positive imaginary axis is at angle 90 degrees. Any complex number can be represented by the angle and magnitude of the vector to that point from the origin.

The real number "1" is represented as (1,0), which means we have the vector which is one unit long pointed along the zero degrees line to the right of the origin.

The real number "-1" is represented as (1,180), which means we have the vector one unit long which is pointed leftward along the negative real axis.

The imaginary number "i" is represented as (1,90) which means we have the vector one unit long which is pointed upward along the postive imaginary axis.

The imaginary number "-i" is represented as (1,270) which means we have the vector one unit long which is pointed downward along the negative imaginary axis.

Note that there is NOT a unique representation for each complex number. For any complex number (r,theta) there are an infinite number of representations in the form, (r,theta + n*360) where n can take on any integer value.

The neat part is that finding "roots" of numbers using this "magnitude-angle" representation is relatively simple. One simply finds the real root of the magnitude and divides the angle by the root being sought.

To find one of the "second" roots of a number ( the square root) simply find the square root of the magnitude and divide the angle by 2.

The remaining roots have the same magnitude but are rotated through angles which cause all of the roots to be equally spaced throughout 360 degrees.

A first example: find the two square roots of "1". The number "1" has a magnitude-angle representation which is (1,0). The positive square root of the magnitude is "1", so this is the magnitude of both square roots we are looking for. To find the angle for the first root, we divide the angle of the original number by 2, giving "0". So the first square root of (1,0) is (1,0).

To find the other square root, we recognize that two equally spaced roots will be 180 degrees apart, so that the angle for the second square root is 180. The magnitude of the second square root is the same magnitude as the other root, so the second square root is (1,180).

The first square root of "1" is therefore the vector which is "1" unit long and has angle "0", so it is identical to the original number.

The second square root of "1" is the vector which is "1" unit long and has angle "180", so it is the number which is one unit to the left of the origin of the complex number plane, or "-1".

Now lets find the cube roots of "-1".

We start by representing "-1" in magnitude-angle form, or (1,180). Then we find the first of the three cube roots by taking the cube root of the real-number magnitude, which is "1". We find the angle of the first root by dividing the original angle by "3", which gives us "60". So the first cube root of "1" is (1,60). That is, it has magnitude "1" and is in the direction sixty degrees above the positive real number axis".

The other two cube roots have the same magnitude as the first cube root, but are equally spaced around 360 degrees. This means we add 120 degrees to the angle of the first root to get the angle of the second cube root. We add 240 degrees to the angle of the first root to get the angle of the third cube root. The two roots have magnitude-angle representations of (1,180) and (1,300).

The final step is to convert the cube roots to "a+bi" form to get the complex roots being sought.

The first cube root was (1,60). That is, magnitude "1" at angle "60". This means that the real-coodinate is "1/2" and the imaginary coordinate is "sqrt(3)/2". The complex number is then "1/2 +i*sqrt(3)/2".

The second cube root was (1,180). This is simply the real number "-1", as expected.

The third cube root was (1,300). That is, magnitude "1" at angle "300". This means that the real-coodinate is "1/2" and the imaginary coordinate is "-sqrt(3)/2". The complex number is then "1/2 - i*sqrt(3)/2".

Now we can plug these values back into the original equation as follows:

x^3 +1 = 0 = (x+1)*(x - 1/2 - i*sqrt(3)/2 )*(x - 1/2 + i*sqrt(3)/2 )

I'll let you multiply out the expression on the right to demonstrate that its value is "0" (Let's hope I made no errors.)

Note that the method above can be used to find arbitrary roots of any complex number. Multiplications, divisions, and other arithmetic operations can be similarly performed.

[FastCoyote]

"Doesn't work that way. You can't send information faster than light."

I believe the jury is still out on that one.