getting exam results tomorrow.

Getting A level exam results on thursday.

Getting final A2 results tomorrow.

Getting my Masters Certificate in Particle Physics in January.

Correlations without Correlata?

While Bell's theorems aim at the derivation of a statistical inequality that violates quantum predictions, the physicsts Greenberger, Zeilinger and Horne provided in 1989 a new theorem "without inequalities", therefore yielding a "stronger" result than Bell's, that again heated up the ongoing debate about the philosophical implications of quantum theory. We will focus on a simplified version of the proof given by Mermin [1].

Imagine a gedanken setup (figure above) that consists of a particle source that creates triples of spin 1/2 particles. After the emission, they are sent into areas of spacetime that are far apart from each other - so far that the velocity of light would not suffice to signal information between them while experimenters do nearly simultaneous spin measurements on each of the particles (spacelike separation). Each measurement yields binary results, one can only get +1 or -1, and the measurements are modeled in the quantum mechanical formalism using Pauli spin matrices σ. Their upper case index {1, 2, 3} denotes the number of the particle, whereas the lower case index {i,j,k} denotes the spatial direction in which the spin is measured. One can choose i,j,k=x,y,z independently for each of the particles. In particular the following four "observables" are of interest, meaning that we either measure all combinations of one x-setting and two y-settings or the triple x-setting. Those combinations can be simultaneously mesured because all four operators do indeed commute (though the single particle Pauli matrices do not commute of course).

At this point it is useful to become aware of the fact that the dimension of the whole system's Hilbert space is 2^3=8, since each particle's Hilbert space is 2-dimensional and the dimension of the whole Hilbert space is the product of the constituent's dimensions due to the tensor product. Bell's setup involves only two particles and therefore the Hilbert space dimension is only 2*2=4. Having noted this, the following maximally entangled wavefunction describes the joint state of those three particles:

Where the kets denote the basis of the σ_z Pauli matrix. Some computation yields the following eigenvalues:

Note that the triple x-measurment (last line) does have a negative eigenvalue - this will be the source of the derived contradiction. So far so good, we're done with the part that relies on the purely quantum mechanical formalism. The interesting thing is, that with this particular setup one can, after the measurement of two particles, determine with certainty what the result of the third must be. E.g. we measure the x-direction on particle 1, the y-direction on particle 2 and the y-direction on particle 3 (first line of the equations above). Then the product of all outcomes must be +1. In the case that the outcomes of particle 2 and 3 are -1, we can directly determine that the first particle's outcome must be +1. More generally, if both y-components yield the same result, the x-outcome must be +1. If both y-measurements yield a different result, the x-outcome must be -1. This is in accordance with the quantum mechanical predictions above. The possibility of determining values with certainty makes one suspect that it might be reasonable to assume that there are some predetermined values prior to the measurement. Let's elaborate this train of thought: We denote the predetermined measurement value with the variable m. Its upper case index stands for the particle's number and the lower case index for the spin component. Then, according to our findings, the products of any combination of two y-measurements and one x-measurements yield +1.

If these three equations hold, also the product of these equations must be:

Since all of these are scalars, we should be able to simply rearrange them. Any y-value appears twice, which is why they contribute each a +1 [simply because (-1)^2=(1)^2=1].

But then only the x-values remain and their product must be +1. But this is a clean contradiction to the quantum mechanical prediction! The eigenvalue to the operator corresponding to a triple x-measurement is -1. Since the mathematics is clear, the assumptions leading to the formulation of predetermined values must be flawed somehow - and there's by far no consensus which assumption is responsible for this mess. Either some notion of "locality" must fail, or there are no "elements of reality" - the possibilites are various. Just to mention one conclusion that could be drawn from this: "Correlations have physical reality; that which they correlate does not." [3] So one might give up the idea that we have to deal with "real" correlata - but this is not mandatory. One can save the idea of "reality" with certain restrictions, it's a matter of the own philosophical worldview, so no one can say for sure that the GHZ proof or Bell-type proofs in general are "no go theorems" for hidden variable accounts. Having noted this, I recommend to really understand what's going on in the GHZ proof above, before jumping to conclusions by adopting a certain interpretation.

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Further reading: [1] Mermin: "What's wrong with these elements of reality?" doi: 10.1063/1.2810588 [2] Greenberger, Zeilinger, Horne: "Bell's theorem without inequalities" doi: 10.1119/1.16243 [3] Mermin: "What is Quantum Mechanics trying to tell us?" arXiv:quant-ph/9801057v2

This is a proof for why we do not make an assumption on Locality!

If you allow each of your entangled GHZ Kets to be shared by Alice, Bob, and Charlie, lightyears away from each other, each of whom can measure their ket in whichever basis they please, say either bases X or Y, then on the locality assumption, you expect Eigenvalues -1.

The locality assumption states that the choice among the two possible measurement bases X or Y made by Alice cannot affect the measurement result obtained by another observer Bob or Charlie when they are space-like separated, or outside each other’s light cone.

The locality assumption is demanding that it is the choice of basis of one observer which should not influence the measurement outcome of another observer.

In most of the literature i’ve come across, the GHZ state takes the form:

where zero is to mean the up state and 1 the down state.

Then my eigenvalues are yours sign-flipped!

So the quantum mechanics measurement will give you +1, and the locality Assumption is a false one when we accept the quantum measurement to be true.

Since the locality measurement is false, Alice’s choice of basis does affect Bob’s / Charlie’s outcome. This is the so called spooky action at a distance - One observer instantaneously affects the measurement of the others. Adding to that, there is no dynamics involved, its purely a statistical action.

But it does not violate relativistic causality - no information can be sent this way over space-like separated distances between Alice or Bob or Charlie.

Quantum nonlocality is how two or more parties can share entangled states across a quantum channel in Quantum computing! 

getting exam results tomorrow.

Getting A level exam results on thursday.

Getting final A2 results tomorrow.

The growth of knowledge* *Degrees not actually required. Just time, effort, and dedication.

As the disk sector is being completed, the centroid spirals into the center.

Fewer than a dozen lines of Wolfram Language code created this hypnotic view of a Roman Surface: https://wolfr.am/dYwe52BS

Beneath the water of Mangroves

 - Wikipedia

The code that took America's Apollo 11  to the moon in the 1960's has been published

When programmers at the MIT Instrumentation Laboratory set out to develop the flight software for the Apollo 11 space program in the mid-1960s, the necessary technology did not exist. They had to invent it.

They came up with a new way to store computer programs, called “rope memory,” and created a special version of the assembly programming language. Assembly itself is obscure to many of today’s programmers—it’s very difficult to read, intended to be easily understood by computers, not humans. For the Apollo Guidance Computer (AGC), MIT programmers wrote thousands of lines of that esoteric code.

Here’s a very 1960s data visualization of just how much code they wrote—this is Margaret Hamilton, director of software engineering for the project, standing next to a stack of paper containing the software:

The AGC code has been available to the public for quite a while–it was first uploaded by tech researcher Ron Burkey in 2003, after he’d transcribed it from scanned images of the original hardcopies MIT had put online. That is, he manually typed out each line, one by one.

“It was scanned by an airplane pilot named Gary Neff in Colorado,” Burkey said in an email. “MIT got hold of the scans and put them online in the form of page images, which unfortunately had been mutilated in the process to the point of being unreadable in places.” Burkey reconstructed the unreadable parts, he said, using his engineering skills to fill in the blanks.

  “Quite a bit later, I managed to get some replacement scans from Gary Neff for the unreadable parts and fortunately found out that the parts I filled in were 100% correct!” he said.

As enormous and successful as Burkey’s project has been, however, the code itself remained somewhat obscure to many of today’s software developers. That was until last Thursday (July 7), when former NASA intern Chris Garry uploaded the software in its entirety to GitHub, the code-sharing site where millions of programmers hang out these days.

Within hours, coders began dissecting the software, particularly looking at the code comments the AGC’s original programmers had written. In programming, comments are plain-English descriptions of what task is being performed at a given point. But as the always-sharp joke detectives in Reddit’s r/ProgrammerHumor section found, many of the comments in the AGC code go beyond boring explanations of the software itself. They’re full of light-hearted jokes and messages, and very 1960s references.

One of the source code files, for example, is called BURN_BABY_BURN--MASTER_IGNITION_ROUTINE, and the opening comments explain why:

About 900 lines into that subroutine, a reader can see the playfulness of the original programming team come through, in the first and last comments in this block of code:

In the file called LUNAR_LANDING_GUIDANCE_EQUATIONS.s, it appears that two lines of code were  meant to be temporary ended up being permanent, against the hopes of one programmer:

In the same file, there’s also code that appears to instruct an astronaut to “crank the silly thing around.”

“That code is all about positioning the antenna for the LR (landing radar),” Burkey explained. “I presume that it’s displaying a code to warn the astronaut to reposition it.”

And in the PINBALL_GAME_BUTTONS_AND_LIGHTS.s file, which is described as “the keyboard and display system program … exchanged between the AGC and the computer operator,” there’s a peculiar Shakespeare quote:

This is likely a reference to the AGC programming language itself, as one Reddit user . The language used predetermined “nouns” and “verbs” to execute operations. The verb pointed out 37, for example, means “Run program,” while the noun 33 means “Time to ignition.”

Now that the code is on GitHub, programmers can actually suggest changes and file issues. And, of course, they have

You can find the official Apollo 11 AGC source code on GitHub here

[source]