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True Instant Messaging Using Quantum Entanglement

According to some textbooks, the light-speed communication barrier is impossible to break even though entangled particles can seemingl...

Friday, July 27, 2018

True Instant Messaging Using Quantum Entanglement

According to some textbooks, the light-speed communication barrier is impossible to break even though entangled particles can seemingly send information back and forth instantaneously. We will explore why this light-speed limit is thought to be true, then we will debunk it. (No doubt Einstein will roll over in his grave.)

If you are familiar with the Star Trek series, you may have wondered how Captain Kirk can instantaneously communicate with Starfleet Command which is light-years away. As it turns out, the twenty-third century has an Alice and Bob. Alice is a Starfleet admiral. Bob is the captain of the USS Schrodinger Cat. Using quantum-entanglement communication, Alice will attempt to warn Bob that the Bohr (who are far worse than the Borg) are attacking a nearby Federation colony: Einstein IV. To accomplish this, Alice must send Bob the following quantum state:

The entangled system that is essential to their communication is mathematically described below:

Equation 2 above shows the first 0 and first 1 belong to Alice (A) and the second 0 and second 1 belong to Bob (B). Alice's first step is to calculate the tensor product of equations 1 and 2:

Alice then applies a CNOT gate to equation 3:

Alice uses a Hadamard gate (H) on equation 4. Several steps ensue. Feel free to skip to the final result: equation 9.

At equation 9 we end up with four entangled states, each with a probability of 0.25. The table below summarizes which states belong to Alice (A) and which states belong to Bob (B):

Alice does a measurement, so only one of the four entangled states will be received by Bob. If Alice measures |00>, Bob will receive the quantum state Alice intended to send, but if Alice measures any of the other states, Bob will need to apply an X-gate and/or a Z-gate to get the right state. How does Bob know which gates to apply? Alice gives him a call and tells him! Once Bob receives Alice's call (no faster than light speed), Bob performs one of the following operations (9 to 21):

Anyway, that's the argument against true instant messaging. The uncertainty of quantum entanglement defeats its purpose. To know which state is the right state, Bob must wait for Alice's signal which could take light years to reach him. The Bohr will surely destroy Einstein IV! On the other hand, what if Alice is an airhead? Or, what if the communication system is full of tribbles? Let's try thinking outside the box and see what can be done. To simplify matters let's convert those messy entangled states to simple variables:

Now let's add a second entangled system. That gives two entangled systems. We label each with i or j subscripts:

Alice measures both Ai and Aj and gets A2 and A3 respectively. Bob sees B2 and B3 respectively. (See row 1 at table below.) When Bob was a cadet he was trained to ignore this first initial state. Alice continues to measure Ai until she gets the desired state A1 (see row 3). How does she let Bob know that B1 is the right state? Does she give him a call? No! She measures Aj until its state changes while leaving Ai alone. At row 4, Bob sees that B3 has changed to B2. Any change is a signal that Bi is now the correct state. So Bob knows B1 is the state Alice intended.

At close range, this form of communication is highly impractical; however, since Alice and Bob are light years apart, the up to 16 measurements needed to make getting the proper state a 99% certainty is well worth it! That being said, perhaps we can improve this new system. How about making the probabilities work for us instead of against us?

Below are the new rules Alice needs to get the state she wants 75% of the time instead of 25%:

The best way to understand the above rules is a straightforward example. Alice measures both Ai an Aj. No matter what state she gets, both she and Bob understand that this random state shall be labeled 1. (See row one at the table below.) This 1 bit is not part of the message--it simply sets things up. Alice wants to send Bob a 0 bit, so she measures Ai until it changes from A1. She does nothing to Aj which is A2. At row 3, Bob notices that B1 has changed to B4 and B2 is unchanged. He (or a CPU) interprets this as 0. At row 4 he sees that B4 has not changed and B2 has changed, so he records another 0 bit. At row 5 Alice does not measure Ai, only Aj and gets A3. Bob notices the change from B1 to B3. Another 0 bit. At row 6, Alice measures Ai, not Aj. Bob sees the change in Bi but not Bj, so he records a 1 bit ... and so on.

When all is said and done, Bob receives the following message from Alice: 100011. The first bit is ignored, so the message is really 00011.

Alice can improve the odds of getting the desired bit by using superpositions with more than four possible states. Imagine two pairs of dice (36 states). Roll both pairs. Label whatever you get 1. To make the same bit, leave the first pair alone and roll the second pair until you get a different state. To make the opposite bit, leave the second pair alone and roll the first pair until its state changes. The chance of getting the bit you want on the first try is 35/36 or 97%--practically a sure thing!

Tuesday, May 29, 2018

Re-normalizing Feynman Diagram Amplitudes in a Non-arbitrary Way

Quantum electrodynamics (QED) is perhaps the most precise and successful theory in all of physics. There is, as I've mentioned in previous posts, a peculiar characteristic within the theory's math: infinities keep cropping up. In this post we deal with the infinities that appear in the math when calculating Feynman-diagram amplitudes.

If you read the previous post, you recall Paul Dirac having a problem with re-normalization. He said, " I must say that I am very dissatisfied with the situation, because this so-called 'good theory' does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way."

Let's see if we can re-normalize Feynman-diagram amplitudes in a non-arbitrary way. First, we define the variables:

Next, let's do a typical textbook calculation and reveal how the infinity arises. Below is the Feynman diagram we will be working with. A and A' are particle and anti-particle, respectively:

The diagram progresses from bottom to top. There are two vertices. The particle (A) and anti-particle (A'), with momenta p1 and p2, meet at the first vertex and annihilate each other. A boson (B) is released. It has an internal momentum q. At the top vertex it creates a new particle (A) and anti-particle (A') with momenta of p3 and p4.

To find the amplitude M, we need a dimensionless coupling constant (-ig) for each vertex. This coupling constant contains the fine structure constant (see equation 1) There are two vertices, so we square the coupling constant (see equation 2):

To conserve momentum we use the Dirac delta function (see 3 and 4). Momenta p1 and p2 are external momenta heading in, and q is the internal momentum heading out (see 3). At 4, q is incoming momentum and p3, p4 are outgoing momenta.

For boson B's internal line we need a propagator, a factor that represents the transfer of propagation of momentum from one particle to another:

We integrate over q using the following normalized measure:

We put all the pieces together to get equation 7. We begin solving the integral at equation 8:

We can solve the integral more easily if we set q equal to p3 and p4. Using some algebraic manipulation, we arrive at equation 11:

Note that at equation 11 we have a red portion and a blue portion. To get the solution at equation 12, we simply throw away the blue portion! We can just imagine Dirac rolling over in his grave. Further, equation 12 is supposed to be the probability of the event illustrated in the Feynman diagram. But probabilities are dimensionless numbers. This probability has dimensions of 1/momentum squared!

Experiments may show that equation 12 is correct within a tiny margin of error, but can the math that leads to it be more sloppy and arbitrary? Sure it can. But let's try to make it less sloppy and arbitrary. We can start by changing the normalized measure:

Next, we can recognize that momentum is conserved, so the Dirac delta functions will equal 1:

As a result, a lot of the stuff we arbitrarily threw away is now properly cancelled. We end up with equation 19:

If we evaluate the integral, we get an infinity (see 20). The good news is we can convert that infinity to the expression at 21. If we introduce a gamma probability amplitude factor, the infinity becomes a finite number at 21b.

We make a substitution at equation 22:

If we throw away the blue section at equation 22, it makes logical sense when you treat that section as all the probable outcomes that could have happened but didn't happen when the observation was made. The observer saw the expression outlined in red--the eigenvalue. That eigenvalue is paired with what is supposed to be its probability amplitude. Notice if we multiply this amplitude by the gammas in the summation, we get the probability amplitudes for all the eigenvalues that add up to infinity. As a result, the right side of equation 22 is no longer infinite. If we take the sum of squared probability amplitudes multiplied by their respective eigenvalues we get the expectation value.

The expectation value is not what we want, however. We want the actual observed value outlined in red, so we ignore "what could have happened but wasn't observed" outlined in blue. This approach is logical instead of arbitrary.

Now, let's see what we can do to fix the dimension problem. At 23 we pull out a momentum unit and set it to one. This leads us to a new solution at 24:

At 24 we end up with an eigenvalue multiplied by a probability amplitude--and the dimensions come out right. The eigenvalue fits nicely into Einstein's energy equation:

So we have a solution for four-dimensional spacetime. For three-dimensional space, we get equation 27:

At 27, the eigenvalue is just q, the internal momentum of the Feynman diagram. The probability of q is the same as the Feynman-diagram event. We obtain the probability by squaring the phi amplitude:

In conclusion, if you encounter an infinity in QED math, it is OK to discard it. It's not really arbitrary to do so, because you are only interested in what you observed. You are not interested in an infinite number of probable events you didn't observe in your experiment.

Saturday, May 26, 2018

Finding the Flaw that Necessitates Renormalization

Here's what Paul Dirac had to say about renormalization:

"Most physicists are very satisfied with the situation. They say: 'Quantum electrodynamics is a good theory and we do not have to worry about it any more.' I must say that I am very dissatisfied with the situation, because this so-called 'good theory' does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small – not neglecting it just because it is infinitely great and you do not want it!"

So let's see if we can find the flaw that causes infinity to appear in equations and necessitates the ad hoc method of neglecting it in an arbitrary way. First, let's define the variables:

Consider the integral below. It adds up the Coulomb potential energy between two particles. The result is infinity.

If the location of each particle is uncertain and/or there is a superposition of states, we might assume, that at each location there is some energy, and, if we add up each of those energies from zero to infinite r (the distance between the particles) we end up with infinite energy!

Let's assume, arguendo, there is infinite energy. We could get that result if we take the average energy and multiply it by infinity:

Of course, when we measure or observe the two particles, we find the energy is not infinite. So why did the math give us infinity? Well, notice there were no probabilities involved when we solved the integral.

Suppose we assume that, since there is an infinite number of states the particles could be in (due to the distance apart (r) being anywhere from zero to infinity), there must be an infinite number of probabilities. Those probabilities must also add up to one. The average probability is therefore 1/infinity:

If we multiply the average energy by infinity, we get infinity, but if we multiply that by 1/infinity, we get the average energy or expectation value:

This is the same result we would get if we summed each probability and each energy eigenvalue:

When we observe and measure the energy, we get the different eigenvalues. The average energy we will observe is the expectation value. So, it makes perfect sense to multiply the absurd infinity by the average probability. After all, we want our math to agree with nature.

Now, let's consider an example from QED (quantum electrodynamics). We want to calculate the total vacuum energy or ground-state energy. One typical way of doing this is to integrate over k-space. We begin with equation 8 below and work our way to equations 14 and 15 (note: variables including but not limited to Planck's constant are set to one):

At 14 we see the ground-state is infinity. Ridiculous! At 15 we renormalize by subtracting the infinity from the total energy (H). This is exactly the kind of thing Dirac complained of, so let's take what we've learned above and apply it to this situation. We know we can get infinity by multiplying the average observed ground-state energy by infinity:

Even though we are dealing with a field instead of individual particles, let's quantize the field by imagining it is made up of individual particles--each with it's own energy state and finite eigenvalue, and, more importantly, each finite energy has a probability associated with it. Also, the totality of these particles, at any point in time, have an overall state with a probability associated with it. We can imagine an infinite number of possible particle states and overall states with finite energies adding up to infinity, so there must be an infinite number of probabilities that add up to one. The average probability is, once again, 1/infinity:

We get the average ground-state energy if we multiply the infinity by the average probability:

Note that equations 20 and 22 are in agreement. The solution is not infinity, but the expectation value or average ground-state energy? Not quite. The solution is definitely not infinity. Additionally, we are not interested in knowing the average energy. We want to know the total energy, say, in a given volume V.

So the next step is to divide the average energy by a unit volume (Vu):

Now we have an energy density. According to WMAP, the vacuum energy density is approximately what we have at equation 24. At equation 25 we multiply the density by the volume we are interested in to get the total "finite" ground-state energy.

Equation 26 shows the energy above the ground state is no longer the total Hamiltonian (H) minus infinity, but the total energy minus a finite vacuum energy.

Wednesday, May 16, 2018

Warp Drive Mathematics and Physics

"Scotty!" barked Captain Kirk, "we need more power!"

"I don' know, Cap'n!" replied Scotty, "we're on impulse engines alone!"

This classic exchange comes from the Star Trek series. It takes place in the 23'rd century, a time when there is warp-drive technology. In this post we work out the mathematics and describe the physics behind warp drive.

What exactly is warp drive? According to the series, it is powered by dilithium crystals. Warp drive pulls the starship's destination closer and pushes the ship's starting coordinates further back. Essentially, the spacetime shrinks in front of the starship and stretches out behind it. This implies shorter spacetime wavelengths in front and longer spacetime wavelengths in back. It is possible to derive an equation that models this. Let's begin with the classic Hamiltonian:

Why the Hamiltonian? It is the sum of kinetic and potential energy. We can think of kinetic energy as energy needed to move a particle through space. Potential energy is, of course, stored energy, or, time energy, since a particle moves through time when it is at rest.

Energy conservation suggests that when there is more kinetic energy (more movement through space), there is less potential energy (less movement through time), and vice versa. At equations 4 and 5 below, we show the equivalency of time and potential energy; and, space and kinetic energy:

We can also create a Minkowski diagram:

From the Minkowski diagram we can derive the Lorentz factor (see equation 11 below):

If we start with the Planck mass squared, we can derive and define the spacetime wavelength (lambda) as well as proper time (lambda/c). (See equations 16 and 17):

Using a scale factor (alpha) we can build a second energy equation equal to the one we derived from the Minkowski diagram.

At equation 20 we set the kinetic energy equal to the gravitational energy. Gravitational energy is the warped spacetime that allows the starship to stay at rest, yet, seemingly move through space. It actually moves with space rather than through it. This enables the starship to reach destinations at super-light speeds.

At 21 and 22 we equate the classical Hamiltonian with the energy's quantum representation. The alpha scale factor makes this possible. Also, notice energy would not be conserved without it. When gravitational energy increases, the wavelength (lambda) decreases. This conserves energy on the right side of equation 22, but the left side can become infinite. Dividing the left side by alpha fixes this problem.

Using a bit of algebra we derive equation 29 below:

Equation 29 is the warp-drive equation. We know that massive galaxies move away from us faster than light if they are far enough away. Equation 29's first term contains Hubble's constant and bar-lambda. This is a velocity with long wavelengths or vast distance. The second term contains a velocity with short wavelengths or distance. The greater the difference, the faster the starship moves with space. It's like dark energy pushing from behind and gravity pulling in front. We can use an integral to sum every point in space along the path between the longest wavelength to the shortest:

At 30 and 31 we show how energy is conserved in spite of the fact that gravitational energy seems to have no upper limit. Shorter wavelengths (lambda) offset the longer wavelengths (bar-lambda):

Below we restate equation 26 at 32. From there we show how Einstein's field equations can be derived.

The fact we can derive the field equations confirms that the warp-drive equation is a solution.