Holosplit

 Recently I had to update Mathematica on my laptop and after having solved the challenges of the license manager that keeps looking different every time I have to use it, I learned that Mathematica 14 can now officially work with finite fields.

This reminded me that for a while I wanted to revive an old project that had vanished together with the hard drive of some old computer: Holosplit. So, over the last two days and with the help of said version of Mathematica I did a complete rewrite which you can now find on Github.

It consists of two C programs "holosplit" and "holojoin". To the first you give a positive integer \(N\) and a file and it spits out a new file ("fragment") that is roughly \(1/N\) of the size. Every time you do that you obtain a new random fragment.

The later you give any collection of \(N\) of these fragments and it reproduces the original file. So you can for example distribute a file over 10 people such that when any 3 of them work together, they can recover the original. 

How does it work? I uses the finite field \(F\) of \(2^3=256\) elements (in the Github repository, there is also a header file that implements arithmetic in \(F\) and matrix operations like product and inverse over it). Each time, it is invoked, it picks a random vector \(v\in F^N\) and writes it to the output. Then it reads \(N\) bytes from the file at a time which it also interprets as a vector \(d\in F^N\). It then outputs the byte that corresponds to the scalar product \(v\cdot d\).

To reassemble the file, holojoin takes the \(N\) files with its random vectors \(v_1,\ldots,v_N\) and interprets those as the rows of a \(N\times N\) matrix \(A\). With probability

$$\frac{\prod_{k=1}^N \left(256^N-k\right)}{(256)^{N^2}}$$

which exponentially in \(N\) approaches 1 this matrix is invertible (homework: why?). So we can read one byte from each file, assemble those into yet another vector \(e\in F^N\) and recover

$$d=A^{-1}e.$$

Besides the mathematics, it also poses philosophical/legal questions: Consider for example the original file is copyrighted, for example an mp3 or a video. The fragments are clearly derived works. But individually, they do not contain the original work, without sufficiently many other fragments they are useless (although not in a cryptographic sense). So by publishing one fragment, I do not provide access to the original work. What if others publish other fragments? Then my fragment could be the last remaining one that was missing. If there are more, any individual fragment is redundant so publishing it strictly speaking does not provide new information. 

The Bohm-GHZ paper is out

 I had this neat calculation in my drawer and on the occasion of quantum mechanic's 100th birthday in 2025, I decided I submit a talk about it to the March meeting of the DPG, the German physical society, in Göttingen. And to have to show something, I put it out on the arxiv today. The idea is as follows:

The GHZ experiment is a beautiful version of Bell's inequality that demonstrates you get to wrong conclusions when you assume that a property of a quantum system has to have some (unknown) value even when you don't measure it. I would say it shows quantum theory is not realistic, in the sense that unmeasured properties do not have secret values (different for example from classical statistical mechanics where you could imagine to actually measure the exact position of molecule number 2342 in your container of gas). For details, see the paper or this beautiful explanation by Coleman. I should mention here that there is another way out by assuming some non-local forces that conspire to make the result come out right never the less.

On the other hand there is Bohmian mechanics. This is well known to be a non-local theory (as the time evolution of its particles depend on the positions of all other particles in the system or even universe) but what I found more interesting is also realistic: There, it is claimed that all that matters are particles positions (including the positions of pointers on your measurement devices that you might interpret as showing something different than positions for example velocities or field strengths or whatever) and those have all (possibly unknown) values at all times even if you don't measure them.

So how can the two be brought together? There might be an obstacle in the fact that GHZ is usually presented to be a correlation of spins and in the Bohmian literature spins are not really positions, you will always have to make use of some Stern-Gerlach experiments to translate those into actual positions. But we can circumvent this the other way: We don't really need spins, we just need observables of the commutation relation of Pauli matrices. You might think that those cannot be realised with position measurements as they always commute but this is only true as you do the position measurements at equal times. If you wait between them, you can in fact have almost Pauli type operators.

So we can set up a GHZ experiment in terms of three particles in three boxes and for each particle you measure whether it is in the left or the right half of the box but for each particle you decide if you do it at time 0 or at a later moment. You can look at the correlation of the three measurements as a function of time (of course, as you measure different particles, the actual measurements you do still commute independent of time) and what you find is the blue line in

GHZ correlations vs. Bohmian correlations

   

You can also (numerically) solve the Bohmian equation of motion and compute the expectation of the correlation of positions of the three particles at different times which gives the orange line, clearly something else. No surprise, the realistic theory cannot predict the outcome of an experiment that demonstrates that quantum theory is not realistic. And the non-local character of the evolution equation does not help either.

To save the Bohmian theory, one can in fact argue that I have computed the wrong thing: After measuring the position of one particle at time 0 or by letting it interact with a measuring device, the future time evolution of all particles is affected and one should compute that correlation with the corrected (effectively collapsed) wave function. That, however, I cannot do and I claim is impossible since it would depend on the details of how the first particle's position is actually measured (whereas the orthodox prediction above is independent of those details as those interactions commute with the later observations). In any case, at least my interpretation is that if you don't want to predict the correlation wrong the best you can do is to say you cannot do the calculation as it depends on unknown details (but the result of course shouldn't).

In any case, the standard argument why Bohmian mechanics is indistinguishable from more conventional treatments is that all that matters are position correlations and since those are given by psi-squared they are the same for all approaches. But I show this is not the case for these multi-time correlations.

Post script: What happens when you try to discuss physics with a philosopher: