barandes-theorem

STOCHASTIC-QUANTUM THEOREM.

I was listening to the following YouTube video the other day, which led me down an academic rabbit-hole: Jacob Barandes' Stochastic-Quantum Theorem. To appreciate his idea, we have to understand what stochastic systems are and how they differ from quantum systems (well, i had to). Using Wikipedia for the "mathy" stuff + GPT 4.5 Preview Research for a quick intuition regarding the math, I was able to integrate the paper quite nicely in my brain.

A stochastic system is simply a system where the outcome is not certain. In other words, there are many possible paths, each with its own probability. For example, tossing a coin (50/50 chance) or stock market changes. These systems use probability to describe how things evolve over time; often modeled using transition probabilities (i.e. Markov Chains) to describe the chances of moving from one state to another. Simply put: *you cannot predict _what_ will happen, but you _can_ predict the odds!*

So, how do stochastic systems differ from quantum ones?

Stochastic Systems:
	- they are "messy"
	- they can be irreversible at the level of their observed states (i.e. you can’t generally run them backward)
	- they include noise, unpredictability, or memory
	- information about prior states is typically lost unless the system is enlarged

Quantum Systems:
	- closed systems are smooth and reversible
	- governed by wave functions & interference
	- they conserve information (unitary evolution)

Which led me to ask: how could something as "structured" as quantum mechanics possibly explain something as chaotic as a random system (stochastic)? I'll let Barandes answer.

Barandes' theorem states:

	| Every stochastic system can be embedded inside a
	| quantum system.

In other words, what may look like randomness may just be the visible part of a deeper, information-preserving process happening in a larger, hidden space. Here, “hidden” does not mean classically deterministic in a hidden-variable sense, but rather unobserved degrees of freedom evolving unitarily. Think of it a bit like Plato's Cave:

	- you're watching shadows on a wall
	- the shadows dance about randomly
	- but you don't see the _actual_ 3D objects casting the shadows, which are rotating smoothly and predictably

What I think Barandes is saying is this: the shadows (the stochastic system) are not false, but incomplete. They are consistent projections of a higher-dimensional evolution (the quantum system). Plato’s Cave is useful here not because the shadows deceive us, but because they are lawful consequences of what we cannot see. The paper formalizes this intuition through an embedding.

Avoiding all of the math gumbo, here’s how embedding works (at least to me):

	- you define your stochastic system (i.e. a list of states & the probabilities of jumping between them)
	- then you construct a quantum system with additional degrees of freedom and structure
	- the stochastic behavior is what you observe when you restrict attention to only part of the larger system

The rotation is the key: closed quantum systems evolve by _unitary transformations_, which can be understood as smooth rotations in Hilbert space. When a quantum system is open—interacting with an environment—its evolution can appear non-unitary and stochastic, even though the combined system remains unitary. Thus, no information is lost at the global level. Everything is preserved. The randomness we observe arises because we are only viewing a subsystem and ignoring the rest. This is a powerful intuition.

Quantum systems are stable precisely because of how they evolve: by unitary transformations. These transformations do not create or destroy information; they redistribute it across the system. Therefore, Barandes' claim means:

	- systems with noise, uncertainty, or even memory can be represented as arising from an underlying unitary (rotational) evolution, provided the environment is allowed to carry the missing information
	- randomness may not be the deepest layer. *Rotation might be*

This realization reminded me of my earlier endeavors while studying Physics. To me, the fundamental conclusion of physics is structured around transformation. This paper highlights that even randomness *might* be another manifestation of transformation that we do not fully see.

This opens the following possibility (philosophical, not proven):

	| Quantum gravity might not be about gluing randomness to
	| geometry. It might be about recognizing that all systems
	| are already geometric. They just appear random when we
	| look at only one part of the picture.

Barandes' Stochastic-Quantum Theorem doesn’t just offer a new mathematical representation of stochastic systems. It also suggests that the noisy world may be a filtered view of a quieter, information-preserving structure underneath. By starting with familiar ideas like randomness and working toward the quantum description, we may begin to see that the chaotic surface is a shadow cast by a deeper organization.

In that structure, *rotation isn’t merely motion; it is how reality keeps its accounts balanced*.

— RIGOR NOTE —

The result discussed here is a mathematical representation theorem. It shows that a classical stochastic evolution can be realized as the reduced dynamics of a larger quantum system evolving unitarily. This guarantees the *existence* of such an embedding, not that nature itself must be described by it. The theorem concerns what is possible in principle within quantum theory, not a claim about the ultimate ontology of the universe.

— REPRESENTATION VS. ONTOLOGY —

It is important to distinguish between how a system can be represented and what the system fundamentally is. Saying that a stochastic system can be embedded in a quantum one does not mean that stochasticity is “illusory” or that randomness is eliminated. Rather, it means that randomness at one level of description can arise from information-preserving evolution at a higher level when degrees of freedom are ignored. The embedding explains *how* stochastic behavior can emerge, not *why* the universe must be quantum at bottom.

— TECHNICAL ANCHOR (WITHOUT MATH) —

Formally, Barandes’ result is closely related to well-known dilation theorems in quantum theory (such as Stinespring dilation). These theorems show that non-unitary, irreversible, or probabilistic dynamics can be understood as the restriction of a unitary evolution on a larger Hilbert space that includes an environment. Memory effects correspond to information stored in that environment, and apparent irreversibility arises from tracing it out. The “rotation” intuition used throughout this essay is a geometric way of understanding this unitary dilation.

Mata ne!