I’ve been working on a book on this subject (very close to being finished but still may make a few more changes): https://www.foleosoft.com/docs/stochastic_quantum.pdf

Here is the basis points, although you’d need to look at the book to see more technical details, since I will just explain them in English here, whereas the actual book is mostly math on every page.

1. Einstein pointed out that quantum mechanics is much less “weird” if we recognize that it is a kind of ensemble theory. Ensemble means, for example, if I say there is a 25% chance of X happening, and a 75% chance of Y happening, what I am really claiming is that if you repeat the same experiment thousands of times, and distribute your results, you will find a physical distribution of ~25% of the distribution being X and ~75% being Y. Probabilities become physical proportions over an ensemble (ensemble = large sample size). Similarly, this means that the “waves” in quantum mechanics are actually only physical properties of an ensemble: statistics over large numbers of particles form wave-like patterns, but there exists no wave for individual particles. Believing that the wavefunction describes an individual system, rather than ensemble, as Einstein argued, leads to mental confusion, like believing a cat is both alive and dead at the same time until you look, when the superposition of states is describing that, over a large sample size, the cat is sometimes alive and sometimes dead, but in the individual run, it is always one or the other.

2. The Soviet physicist Dmitry Blokhintsev mostly agreed with Einstein but also criticized him, because Einstein was strongly an adherent to a “principle of locality,” which quantum mechanics seemed to violate. Blokhintsev was a dialectical materialist and argued, from that perspective, that the universe is holistic, and if it is holistic, there is no good philosophical reason to be such a strong adherent to locality. Blokhintsev thus argued we should adopt an understanding of quantum mechanics that is indeed an ensemble description, like Einstein advocated for, but is holistic, so it does not need to be local, although he never formalized what that would look like.

3. A few year after Blokhintsev published one of main books on this subject, the physicist John Bell wrote a proof showing that the empirical predictions of quantum mechanics, if they are confirmed, are not actually compatible with both an ensemble interpretation and locality at the same time. These empirical predictions were confirmed and even awarded the Nobel prize in 2022. Many people take this to mean that the ensemble view was debunked, that particles are in multiple places at once when you’re not looking, that a cat can be both dead and alive at the same time until you open the box. But that’s not true: Blokhintsev’s ensemble interpretation is still on the table, because Bell’s theorem is perfectly compatible with it as long as it is explicitly non-local.

4. Just last year (2025), the Harvard physicist Jacob Barandes published a paper that shows you can formulate quantum mechanics entirely in terms of statistics without referencing the wavefunction at all, and called this unistochastic dynamics. What is interesting about unistochastic dynamics is that it is quite explicitly holistic. In order to predict how any physical interaction will alter the system, you have to consider the arrangement of the whole experiment. For example, in the case of the Mach-Zehnder interferometer, you have two beam splitters, which are identical pieces of glass. However, unistochastic dynamics would describe the second piece of glass as having different behavior from the first, because it is situated after the first. (This holism is represented by taking all physical interactions, described by unitary operators, up to a given time interval you want to predict, and composing them together, which then yields the unistochastic matrix needed to predict the statistical distribution of the system at that time interval.)

5. One issue with unistochastic dynamics, as pointed out by the physicist David Albert, is that it represents quantum systems as a stochastic process, but does not actually allow you to simulate a quantum mechanics as a stochastic process, because it leaves the step-by-step transition law ambiguous. It puts constraints on what it should be, but many plausible laws fit those constraints. However, there is actually an answer to this as well.

6. Schrödinger also believed quantum systems could be understood as a kind of stochastic process and proposed a method for computing the step-by-step transitions, as he proved that, in one of his lesser known papers, under some reasonable assumptions, the solution is merely a solution to a scaling problem. However, he did not know how to actually solve the scaling problem. The thing is, in modern times, the solution to that scaling problem is a well-known algorithm called iterative proportional fitting, and so, using modern knowledge Schrödinger was not aware of because they weren’t invented yet, it is trivial to use his method to compute the step-by-step transition matrices.

7. You can then combine Barandes’ unistochastic dynamics and Schrödinger’s step-by-step transitions to derive a transition law, entirely terms of statistics without reference to the wavefunction, that gives any arbitrary quantum system an ensemble description entirely in terms of statistics. This law is explicitly holistic because it requires you to factor in the whole arrangement of your experiment up to the given time interval you are asking about. This then ultimately allows us to formalize what Blokhintsev saw as the correct way to think about it.

In the book I use this framework to then analyze all the major foundational papers on this topic, such as the PBR theorem, the famous Frauchiger-Renner paradox, the double-slit experiment, etc, and show directly how it gives a simple intuition for all of them.

The formalism is mathematically equivalent to the standard wavefunction formalism by construction, as it reproduces the Born rule probabilities at every time interval by construction. It also is only defined, in the book, over abstract quantum information, meaning, no specific physics goes into its assumption. That latter point is intentional, because it means the application of this interpretation does not depend upon any specific physical criteria in the model. As long as the model is informationally structured as a quantum mechanical model, whatever the physics are, you can apply this approach to interpret it.

The interpretation even works for something as abstract as a quantum computer, which, under this interpretation, just becomes a normal computer where a “qubit” doesn’t even exist. I intentionally never use the word “qubit” in the book, because, in my view, there is no such thing as a “qubit” as a distinct physical object (only as an abstract mathematical construct). A quantum computer just contains bits, but the logic gates are (1) stochastic, and (2) have holistic behavior, so computing the stochastic effect of the logic gate requires composing together the full arrangement of the circuit prior to the gate.

I put together a universal quantum computing simulator to demonstrate this. Unlike most simulators which just give you the statistical distribution at the end for the bits when you make a “measurement,” this simulator gives you the values of the bits at every time interval after every gate, and show you how the bit values change over time, step-by-step, through the circuit:

https://qansel.foleosoft.com/

(The random sampling on the version on my website is actually being done using a real quantum random number generator.)

You can change it from “single” to “shots” mode to then run the same program many times over (by default 1024 times), and the overall statistical distribution you get for the bits at the end is the Born rule.

I’ve also been trying to produce a video series explaining it since I fear most people don’t read books these days, but it is not complete yet, but the first two are viewable at v01.mp4 and v02.mp4.

What exactly don’t you understand? Is there something in particular that you would like to know?

Even as a Computer Engineering Bsc, I only heard about it, but usually in any talk of quantum mechanics I stay like that meme of the dog in front of a computer.