Divisible and indivisible Stochastic-Quantum dynamics

In memory of Joseana S. Soares (1989 - 2024)
for her friendship and her work making physics and astronomy accessible to the general public.

Predicting the future is not an easy task. Buy a telescope to watch an eclipse and I bet it will be cloudy. But I can't be sure. Can you?

In uncountable circumstances, the best one can do is to attribute probabilities to future events, regardless of their dependence on human choices, as one would normally do when predicting the next solar eclipse or, a much harder problem, if it is going to be cloudy that day. In any case, we can describe both situations (and many more) with probabilities that change as time goes by.

Screenshot from ESOcast 170 showing a clear-weather simulation of the 2019 total solar eclipse at La Silla. More information: www.eso.org/public/images/ann18054a/ Credit: M. Druckmüller, P. Aniol, K. Delcourte, P. Horálek, L. Calçada/ESO

On this page, we are going to explore a very general and yet poorly understood type of probabilistic time-evolution: indivisible stochastic dynamics.

Don't be fooled! Indivisible stochastic dynamics are rich enough to allow for quantum dynamics and even provide a derivation of the Hilbert space quantum formalism! This surprising result is called the Stochastic-Quantum Correspondence.

Here is what you are going to learn:

• the meaning of stochastic dynamics, divisibility and indivisibility;
• a geometrical interpretation of these concepts in the simplest non-trivial case, as presented in my paper, arXiv:2505.08785;
• some connections between geometry, divisibility and time;
• explain how these concepts are translated into the interactive diagrams below (drag the yellow, red and green circles below);
• get a glimpse into higher dimensional cases, through a final interactive diagram.

I will guide you on a step-by-step journey to understand each and every part of these interactive diagrams and in the end you will understand what indivisibility means and acquire geometric intuition that can be applied to qubits or any binary system undergoing divisible or indivisible stochastic dynamics.

A qubit, at the core of quantum computing and quantum information, or any 2-configurations physical system is our object of study. In fact, the present framework is more general, as it applies to any yes-no question which can be described by initial probabilities and conditional probabilities. Higher dimensional cases are still an open problem but, as promised, you'll get a glimpse of them.

What you are NOT going to obtain HERE but you can learn by reading the sections of my paper (and references therein) include:

• a deeper understanding of the connections between geometry, information decrease, divisibility and time, and
• the connection between coarse-graining and divisibility.

Of course, there is much more but let's establish some solid foundations to understand a very general type of probabilistic evolution, indivisible stochastic dynamics, which embraces both classical and quantum dynamics.

In this section you will understand the simplest geometric setting of probabilistic time evolution where we will encode probabilities as points in a line and or stochastic evolution via points and curves inside a square.

Call A the event corresponding to a YES answer to the question under consideration and B its complement, the answer NO. Let $p_A(t)$ be the probability of a YES at time t and $p_B(t)$ the probability of a NO at the same instant. Here is a graphical depiction of their sum to 100%.

Drag the white circles below.

Did you drag the white circles? If not, try it now.

In this interactive plot, the horizontal axis represents the probability of a YES and the vertical axis the probability of a NO. Since these depend on each other, as their sum is one, we can pick either axis to represent the probability distribution at a given time. In a moment, when we consider time-evolution, we will use both representations simultaneously.

These are related, by assumption, to the initial probabilities $p_A(0)$ and $p_B(0)$ via conditional probabilities.

\begin{align} p(t) &\equiv \begin{pmatrix} p_{A}\left(t\right)\\ p_{B}\left(t\right) \end{pmatrix} \nonumber\\ &=\begin{pmatrix} p\left(A, t| A, 0\right) & p\left(A, t| B, 0\right)\\ p\left(B, t | A, 0\right) & p\left(B, t| B, 0\right) \end{pmatrix} \begin{pmatrix} p_{A}\left(0\right)\\ p_{B}\left(0\right) \end{pmatrix} \nonumber\\ &\equiv \Gamma(t\leftarrow0)p(0) \end{align}

Conditional probabilities are, obviously, probabilities and, as such, need to sum to one when conditioned on the same event. For this reason we can write any $2\times2$ stochastic matrix in terms of its non-negative diagonal entries.

\begin{equation} \Gamma(t) \equiv \Gamma(t\leftarrow0) = \begin{pmatrix} p(t) & 1-q(t)\\ 1-p(t) & q(t) \end{pmatrix} \end{equation}

Now consider $(p,q)$ as the Cartesian coordinates of a point. This choice gives the following representation of matrices in a unit square.

Drag the yellow circle.

Note that the probabilities conditioned on a YES, the first column of $\Gamma$, is represented by the horizontal axis, while those conditioned on a NO, the second column of $\Gamma$, are represented by the vertical axis. This provides a representation of a stochastic matrix as a point in the unit square.

A few things remain unexplained: the checkboxes and the dynamical lines. Each checkbox constraints the possible matrices, the first forces it to be bistochastic, meaning that both its rows and columns have unit sum, while the second forces it to be degenerate, meaning that the stochastic matrix erases all information about the initial probabilities. Do you see why?

The bistochastic $2\times2$ matrices have $p=q$ and lie along the main diagonal of the square.

\begin{equation} \Gamma_{bistochastic} = \begin{pmatrix} p & 1-p\\ 1-p & p \end{pmatrix} \end{equation}

The degenerate $2\times2$ matrices have $p=1-q$ and lie along the secondary diagonal.

\begin{equation} \Gamma_{degenerate} = \begin{pmatrix} p & p\\ 1-p & 1-p \end{pmatrix} \end{equation}

For these reasons, when both constraints are imposed, the only possible matrix is located where these two lines intersect: the centre of the square. The centre corresponds to a perfect mixing: no matter the initial input, the output probabilities are 50% YES and 50% NO. We will gradually understand the dynamical lines. For now, keep in mind that two of them intersect at $(p,q)$ and the other two at $(1-p, 1-q)$ while they all emanate from the deterministic irreversible matrices.

Do you remember the concept of a prime number? That old notion of a natural number that is only divisible by 1 and itself? In this section you will see that some stochastic matrices are the matrix-analog of prime numbers. We shall proceed in 3 steps by:

1. providing a naive notion of divisibility which captures the essence of the concept,
2. defining rigorously a non-trivial notion of divisibility and
3. capturing the geometry of the divisors in an interactive diagram.

Step 1: A naive notion of divisibility. Let $\Gamma$ be a stochastic matrix. $\Gamma$ is naïvely divisible if there exist stochastic matrices $\hat\Gamma$ and $\tilde\Gamma$ such that

$$ \Gamma = \hat \Gamma \tilde \Gamma $$

In the case of 2 configurations, with the parameters below all in the interval $[0,1]$, we can write this condition as the existence of $r$ and $s$ defining $\hat \Gamma$, $u$ and $v$ defining $\tilde \Gamma$ such that, given $p$ and $q$ defining $\Gamma$, the following expression holds.

$$ \begin{pmatrix} p & 1-q\\ 1-p & q \end{pmatrix} = \begin{pmatrix} u & 1-v \\ 1-u & v \end{pmatrix} \begin{pmatrix} r & 1-s\\ 1-r & s \end{pmatrix} $$

This definition, as it is right now, is trivial because it can always be satisfied. The reason is simple: consider a system with N configurations and a stochastic matrix that flips the first 2 configurations and leaves the remaining ones unchanged. Flipping only the first 2 configurations again restores the original setup. Mathematically, $\sigma_{12}^2=\mathbb{1}_N^{}$. Consequently, $\Gamma = \sigma_{12} \left(\sigma_{12}\Gamma\right)$, meaning that it is always possible to write $\Gamma$ as a product of two stochastic matrices. More generally, given any permutation $\sigma$ defined on the $N$ configurations will provide such a rewriting of a stochastic matrix $\Gamma$ as a product of stochastic matrices.

$$\Gamma = \sigma \left(\sigma^{-1}\Gamma\right) = \left(\Gamma\sigma^{-1}\right)\sigma$$

This condition captures an essential point: we do not require any of the 3 matrices to be invertible nor to have a stochastic inverse. Divisibility amounts to the existence of two stochastic matrices that, once multiplied in the correct order, produce the matrix that we want to divide. this amounts to saying that one can always write

$$ \begin{pmatrix} p & 1-q\\ 1-p & q \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1-p & q\\ p & 1-q \end{pmatrix} $$

and

$$ \begin{pmatrix} p & 1-q\\ 1-p & q \end{pmatrix} = \begin{pmatrix} 1-q & p\\ q & 1-p \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}. $$

Note that the left-multiplication by $\sigma_x = \begin{pmatrix}0&1\\1&0\end{pmatrix}$ maps $(p,q)$ to $(1-p, 1-q)$ while the right-multiplication by $\sigma_x$ maps $(p,q)$ to $(1-q, 1-p)$.

These trivial decompositions motivate the following step.

Step 2: A rigorous notion of divisibility. A stochastic matrix $\Gamma$ is divisible if and only if it can be written as a product of stochastic matrices, none of which is a permutation. Conversely, $\Gamma$ is indivisible if any decomposition of $\Gamma = \hat \Gamma \tilde \Gamma$ as a product of stochastic matrices implies that either $\hat \Gamma$ or $\tilde \Gamma$ is a permutation, but not both.

This is the actual definition of divisibility as presented by vom Ende and Shahbeigi in a work where they also present all indivisible $3\times3$ stochastic matrices. These are, up to permutations, of the form

\begin{equation} \begin{pmatrix} 0 & 1-b & c \\ a & 0 & 1-c \\ 1-a & b & 0 \end{pmatrix} \end{equation}

where $a,b,c\in(0,1)$. For $2$ configurations the situation is simpler: all $2\times2$ stochastic matrices are divisible.

Step 3: The geometry of divisors. This is such a rich topic that it deserves a section for itself. Right now, this is only understood for $2\times2$ stochastic matrices, so let's dive in.

Given that a $2\times2$ stochastic matrix $\Gamma$ is always divisible, we will determine where in the square are its possible divisors $\hat \Gamma$ such that $\Gamma = \hat \Gamma \tilde \Gamma$.

We shall call $\hat \Gamma$ the left divisor and $\tilde \Gamma$ the right divisor. In the diagram below, the glowing circle represents $\Gamma$ and the grey areas represent the possible right divisors of $\Gamma$. We will see in Section V that the grey areas contain the possible pasts of $\Gamma$ given a division event. They are bounded by rays that emanate from the deterministic and degenerate matrices. The golden central parallelogram represents all possible matrices that have $\Gamma$ as their right divisor. In Section V we shall see that they represent the possible futures of $\Gamma$ given a division event.

Drag the yellow circle.

This means that, in the following equation,

$$ \begin{pmatrix} p & 1-q\\ 1-p & q \end{pmatrix} = \begin{pmatrix} u & 1-v \\ 1-u & v \end{pmatrix} \begin{pmatrix} r & 1-s\\ 1-r & s \end{pmatrix} $$

the values of $(r,s)$ correspond to points inside the closed grey regions. For each $(r,s)$ there is a unique pair $(u, v)$ such that the equation above is satisfied. The diagram below adds two cyan areas which represent the left divisors of $\Gamma$, the possible values of $(u, v)$ in the previous equation, given a matrix located at the point $(p, q)$.

The gray and golden areas are mapped to each other by a left-action of the permutation of the two configurations, represented by the Pauli matrix $\sigma_x$. The cyan areas are related by a right-multiplication by $\sigma_x$.

We add now a final layer of complexity before starting to explore the dynamics. In the following diagram, the transparent magenta layer corresponds to the action of $\Gamma$, represented by the yellow glowing circle, on the stochastic square.

When the magenta region is superposed to the yellow region it is displayed with a salmon tone, the sum of the colours. Similarly, the cyan regions acquire a darker tone when superposed to the grey regions. The two cyan regions and the two halves of the magenta region, above and below the secondary diagonal (in orange) are related by a right-action of $\sigma_x$.

It is time to turn our attention to the time-evolution.

Now we introduce evolution. We say a stochastic dynamics evolving the probabilities from time $t=0$ to time $t$ is divisible at time $t^\prime\in(0,t)$ if there is a stochastic matrix $\Gamma(t \leftarrow t^\prime)$ such that

\begin{equation} \Gamma(t) = \Gamma(t \leftarrow t^\prime)\Gamma(t^\prime). \label{divisibility} \end{equation}

Geometrically, we can represent stochastic dynamics by a curve on a right prism that has the stochastic square as its base and the height corresponds to time. One can project this curve directly on the square and represent dynamics as curves over the stochastic square. The following interactive diagram implements the latter representation.

Drag the yellow circle and visually explore the divisibility constraint by checking and unchecking the box.

Divisible dynamics correspond to curves such that the track of the matrix is fully contained inside the grey regions and its future motion is bound to the interior of the golden parallelogram. Indivisible dynamics have at least part of the past curve outside the grey regions.

This final visualisation for the $2\times2$ case combines the previous diagram a more explicit time-dependence using a treadmill. We represent the current stochastic matrix by its diagonal entries in a square and by its second row entries on a horizontal slider.

$$ \Gamma= \begin{pmatrix} 1-Red & 1-Green\\ Red & Green \end{pmatrix} $$

The initial configuration corresponds to the identity matrix.

You can control both the visual representation (the treadmill animation) and the mathematical representation (the matrix in parameter space) simultaneously. You can drag the yellow, red and green circles below to adjust the parameters. The golden circle in the unit square represents the diagonal (p,q)=(1-Red, Green) of the stochastic matrix. The red represents 1-p and the green represents q. A path in the square corresponds to a stochastic dynamics given by $\Gamma(t)$ with p(t) and q(t) in the diagonal.

Visualising the probability space for more than 2 configurations is possible but it requires some tricks. In fact, the horizontal slider above is our entrance to the $3\times3$ dimensional case.

For a 3-configurations system, there is a one-to-one map between a stochastic matrix and a set of 3 points in an equilateral triangle.

Drag the red, green and blue circles below.

Can you find the indivisible matrices in the triangle?

Each edge of the triangle serves as a 1-dimensional slider for a given pair of circles as we saw in the previous section. The reason is that if only 2 out of 3 configurations evolve, the system has effectively 2 configurations from a dynamical point of view. The system is, however, allowed to explore the interior of the triangle.

The case of four configurations corresponds to 4 points on a tetrahedron and, more generally, a stochastic matrix acting on N configurations can be represented by N points inside an $(N-1)$-dimensional simplex. The extension of these resutls to higher dimensions may open up rich possibilities for understanding complex stochastic systems.

The author, Leandro Silva Pimenta, would like to thank Lucas Tavares Cardoso, Rafael Chaves Souto Araújo, Dmitry Melnikov, Rodrigo Pereira, Daniel Augusto Turolla Vanzella, Diogo de Oliveira Soares-Pinto for helpful conversations. Special thanks for D. O. Soares-Pinto, L. T. Cardoso for comments on different versions of the manuscript and to Jacob Barandes for his comments on the arxiv version. The author also thanks Analia Silva, Sebastião das Graças Pimenta and Laura Keil for support during important steps of this project.

References:

1. Divisible and indivisible Stochastic-Quantum dynamics, Leandro Silva Pimenta (May 13, 2025), arXiv:2505.08785 [quant-ph] (and references therein).
2. Generating Sets of Stochastic Matrices, Frederik vom Ende, Fereshte Shahbeigi (Nov 28, 2024), arXiv:2411.18946 [math.RA].

This page was created with the help of the following AI tools: Manus AI, Claude and ChatGPT.