Introduction
The standard formulation of quantum theory rests on the Dirac–von Neumann axioms: states are vectors (or density matrices) in a Hilbert space, observables are self-adjoint operators, isolated systems evolve by a unitary operator (equivalently, by the Schrödinger equation), and measurement outcomes are governed by the Born rule (Dirac 1930; Neumann 1932). These axioms are empirically magnificent but conceptually divided: they treat “measurement” as an unexplained primitive and offer no account of what is happening in an experiment before a measurement is complete.
In a series of recent papers, Jacob Barandes has argued that a quantum system can instead be understood as a perfectly ordinary stochastic process unfolding in an old-fashioned configuration space, governed by probabilistic laws that are non-Markovian in a specific and previously unstudied sense. The central object is the indivisible stochastic process, and the central result is the stochastic–quantum correspondence (Barandes 2025b, 2025a):
Indivisible stochastic processes generically exhibit all the hallmark empirical features of quantum systems, including interference, decoherence, entanglement, and noncommutative observables, and a quantum system evolving unitarily can be understood as an indivisible stochastic process in disguise.
The payoff is a reformulation in which the Hilbert space and its ingredients—wave functions, state vectors, and Hamiltonians—are derived mathematical appurtenances rather than axioms, and in which the “measurement problem” is dissolved rather than solved.
The purpose of this post is to give a compact but self-contained account of the correspondence. I assume familiarity with probability and statistics, linear algebra, and the basics of quantum mechanics. Part I collects only the definitions the argument uses explicitly — stochastic processes, transition probabilities, Markovianity, divisibility, and stochastic matrices on the one hand; Hilbert spaces, states, observables, quantum channels, and unistochastic matrices on the other — and everything else is treated as known. I then give the formal definition of an indivisible stochastic process, the precise sense in which it fails to be Markovian, the dictionary that connects it to a Hilbert space, the Kraus and Stinespring machinery that yields unitary evolution, and finally the stochastic–quantum theorem itself. Throughout, definitions, lemmas, propositions, and theorems are stated formally and, where instructive, proved.
A reader who wants the conceptual through-line rather than the full machinery may read the introduction to each part and the statements of the main results, and treat the proofs and technical definitions as a reference. (Statement numbers follow a longer draft of this material, so they contain gaps.)
This post is purely expository. Every definition, result, interpretation, and argument below is due to Jacob Barandes and the other authors cited; I am reporting, organizing, and explaining their work, not proposing any new ideas of my own. The choice of material, the exposition, and any errors of presentation are mine, but the substance is theirs. Readers interested in the original development should consult the cited primary sources.
Roadmap
- Part I collects only the definitions the rest of the post uses: stochastic processes, transition probabilities, Markovianity, divisibility, and stochastic matrices; and Hilbert spaces, states, observables, quantum channels, and unistochastic matrices. All other prerequisites — probability theory, linear algebra, topology, measure theory — are assumed.
- Part II defines indivisible stochastic processes, explains divisibility and its failure, and states the axioms of indivisible quantum theory.
- Part III develops the stochastic–quantum correspondence: the time-evolution operator, Schur–Hadamard gauge freedom, the dictionary, the Hilbert-space representation, Kraus decompositions, dilations, and the stochastic–quantum theorem.
- Part IV discusses gauge invariance, dynamical symmetries, dilations, the complex numbers, and the interpretational consequences for measurement, hidden variables, Bell’s theorem, and causation.
Part I: Preliminaries
1. Preliminary Definitions
Definition 8 (Configuration space). The configuration space of a stochastic process is a fixed set \(\mathcal{C}\) whose elements \(i\in\mathcal{C}\) are the possible configurations of the system.
Relevance. the configuration space is the classical analogue of a Hilbert-space basis label; every construction in Parts II–III is indexed by it. (Kolmogorov 1933; Doob 1953).
Definition 8a (Standalone probability distribution). A standalone probability distribution is a time-dependent function \(p(i,t)\) giving the probability (or probability density) that the system is in configuration \(i\) at time \(t\), satisfying \[0 \le p(i,t) \le 1, \qquad \sum_i p(i,t) = 1. \tag{1}\]
Relevance. the standalone distribution is the analogue of the diagonal of the quantum state, \(p_i(t)\), and its normalization is imposed again in Proposition 45. (Kolmogorov 1933).
Definition 8b (Algebra of random variables). The algebra of random variables of a stochastic process with configuration space \(\mathcal{C}\) is the collection of time-dependent maps \(A(t):\mathcal{C}\to\mathbb{R}\), with values \(a(i,t)\), whose expectation is \[\langle A(t) \rangle \equiv \sum_i a(i,t)\, p(i,t). \tag{2}\]
Relevance. the algebra of random variables is the classical counterpart of the algebra of observables (Definition 25c). (Kolmogorov 1933; Doob 1953).
Definition 8c (Stochastic process). A stochastic process is the data of a configuration space \(\mathcal{C}\) (Definition 8), a standalone probability distribution \(p\) (Definition 8a), and an algebra of random variables \(A\) (Definition 8b).
Relevance. this triple \((\mathcal{C},p,A)\) is the object the whole post is about: indivisibility is a property of its transition probabilities. (Kolmogorov 1933; Barandes 2023).
Definition 9 (Transition probability). A transition probability of a stochastic process is a conditional probability of the form \(p(i,t \mid j_1,t_1;\dots;j_n,t_n)\), the probability that the system is in configuration \(i\) at time \(t\) given that it was in configurations \(j_1,\dots,j_n\) at the earlier times \(t_1>\cdots>t_n\).
Relevance. a transition probability is the conditional probability whose composition law is the entire content of divisibility (Definition 14). (Kolmogorov 1933; Doob 1953).
Definition 9a (Kolmogorov tower). The Kolmogorov tower of a general stochastic process is the infinite hierarchy of transition probabilities (Definition 9) \[ \begin{aligned} p(i,t \mid j_1, t_1) &\quad \text{[first order]}, \\ p(i,t \mid j_1, t_1; j_2, t_2) &\quad \text{[second order]}, \\ p(i,t \mid j_1, t_1; j_2, t_2; j_3, t_3) &\quad \text{[third order]}, \\ &\vdots \end{aligned} \tag{3} \] where \(t > t_1 > t_2 > t_3 > \cdots\).
Relevance. the Kolmogorov tower is the full specification of a general, possibly non-Markovian process; Markovianity is the collapse of this tower to first order. (Kolmogorov 1931, 1933).
Definition 10b (Chapman–Kolmogorov equations). The transition probabilities must satisfy the marginalization consistency conditions \[\sum_{j_k} p(\dots; j_k, t_k; \dots) = p(\dots \text{[no } j_k] \dots), \tag{5}\] for every time index \(k\): summing over the intermediate configuration reproduces the joint probability with that time omitted.
Relevance. the Chapman–Kolmogorov equations are the consistency conditions any valid transition probability satisfies, and that the dilated process of Proposition 61 is constructed to obey. (Kolmogorov 1933; Doob 1953; Ross 1995).
Definition 11 (Markov property). A stochastic process is Markovian if every second- and higher-order transition probability equals the corresponding first-order transition probability for the latest conditioning time, i.e. \[p(i,t \mid j_1,t_1; j_2,t_2; \dots) = p(i,t \mid j_1,t_1). \tag{6}\] The latest conditioning time \(t_1\) then “screens off” all earlier times.
Relevance. the Markov property is the property that indivisible processes generically fail to have (Proposition 36). (Kolmogorov 1933; Doob 1953).
Definition 12 (Transition matrix). For a stochastic process, the transition matrix \(\Gamma(t \leftarrow t')\) is the matrix whose \(i\)-th row, \(j\)-th column entry is the first-order transition probability \[\Gamma_{ij}(t \leftarrow t') \equiv p(i,t \mid j,t'). \tag{7}\]
Relevance. the transition matrix is the formalism in which the whole of Parts II–III is conducted. (Kolmogorov 1933; Doob 1953).
Proposition 13 (Column stochasticity). Every transition matrix is a (column) stochastic matrix: \[\Gamma_{ij}(t \leftarrow t') \ge 0, \qquad \sum_i \Gamma_{ij}(t \leftarrow t') = 1. \tag{8}\]
Relevance. column stochasticity is the normalizability constraint that the quantum dictionary must reproduce, and that becomes the summation condition of Proposition 45. (Kolmogorov 1933).
Definition 14 (Divisibility). A stochastic process is divisible over an interval if, for any intermediate time \(t'\) between \(t_0\) and \(t\), its transition matrices factorize as \[\Gamma(t \leftarrow t_0) = \Gamma(t \leftarrow t')\,\Gamma(t' \leftarrow t_0). \tag{13}\]
Relevance. divisibility is the property whose failure characterizes indivisible processes; it is the classical shadow of complete positivity of a quantum channel. (Wolf and Cirac 2008; Barandes 2023).
Proposition 15 (Markovianity implies divisibility). A Markovian process is divisible.
Relevance. this proposition supplies the classical direction of the correspondence: Markovian processes are divisible. (Kolmogorov 1933; Doob 1953).
Definition 18a (Column stochastic matrix). An \(N\times N\) matrix with non-negative real entries is column stochastic if for every column \(j\), \(\sum_{i=1}^{N} X_{ij}=1\).
Relevance. column stochastic matrices are exactly the admissible transition matrices of Part II. (Kolmogorov 1933).
Definition 18b (Doubly stochastic/bistochastic matrix). A matrix is doubly stochastic (or bistochastic) if it is column stochastic and row stochastic in the sense that for every row \(i\), \(\sum_{j=1}^{N} X_{ij}=1\) (equivalently both row and column sums are 1).
Relevance. doubly stochastic matrices are the bistochastic ones that arise as the squared moduli of unitaries (Proposition 32). (Horn 1954).
Definition 19 (Permutation matrix). A permutation matrix is a matrix of \(0\)s and \(1\)s with exactly one \(1\) in each row and in each column. Every permutation matrix is doubly stochastic (Definition 18b) and orthogonal, and therefore unitary (Definition 30b).
Relevance. permutation matrices are the transition matrices of permutation dynamics, and are the matrices fixed by the permutation lemma. (Horn 1954).
Lemma 20 (Permutation lemma). Let \(X\) and \(Y\) be \(N \times N\) matrices with non-negative entries and \(XY = \mathbb{1}\). Then \(X\) (and hence \(Y\)) is a permutation matrix.
Relevance. this lemma is what forces the modulus-squaring of a unitary to be doubly stochastic, i.e. what makes the first claim of Proposition 32 work. (Horn 1954; Barandes 2023).
Definition 21 (Pseudo-stochastic matrix). A matrix is pseudo-stochastic if each of its columns sums to \(1\) but its entries are real (or complex in generalizations) and need not be non-negative (Chruściński et al. 2015). Pseudo-stochastic matrices form a semigroup and, unlike stochastic matrices, allow affine combinations beyond non-negative convex combinations.
Relevance. pseudo-stochastic matrices are the matrices \(\Theta\) whose squared moduli are valid transition probabilities; they are the quantum amplitudes of Part III. (Chruściński et al. 2015; Barandes 2023).
Definition 23c (Sesquilinear form). Let \(V\) be a vector space over \(\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}\). A map \(b:V\times V\to\mathbb{K}\) is sesquilinear if for all \(u,v,w\in V\) and \(\lambda\in\mathbb{K}\): \[b(\lambda u+v,w)=\lambda\, b(u,w)+b(v,w),\qquad b(u,\lambda v+w)=\overline{\lambda}\,b(u,v)+b(u,w).\] (When \(\mathbb{K}=\mathbb{R}\) this reduces to bilinearity.)
Relevance. sesquilinearity is bilinearity with conjugation, the structure that makes quantum inner products complex and that Remark 71 argues is necessary. (Halmos 1958; Neumann 1955).
Definition 23c1 (Positive-definite form). A sesquilinear \(b\) is positive-definite if for all \(v\neq 0\), \(b(v,v)>0\) (in particular \(b(v,v)\in\mathbb{R}_{\ge0}\) for every \(v\)).
Relevance. positive definiteness is what makes \(\langle\psi|\psi\rangle\) a squared norm, hence non-negative, which the Born rule requires. (Halmos 1958).
Definition 23c2 (Inner product). An inner product on a \(\mathbb{K}\)-vector space \(V\) is a map \(\langle\cdot,\cdot\rangle:V\times V\to\mathbb{K}\) that is sesquilinear (Definition 23c), conjugate-symmetric (\(\langle w,v\rangle=\overline{\langle v,w\rangle}\) for all \(v,w\)), and positive-definite (Definition 23c1).
Relevance. the inner product defines lengths and angles of state vectors, and hence transition probabilities between states. (Halmos 1958; Neumann 1955).
Definition 23c3 (Norm induced by an inner product). An inner product \(\langle\cdot,\cdot\rangle\) induces the norm \(\|v\|\equiv\sqrt{\langle v,v\rangle}\) and the metric \(d(v,w)=\|v-w\|\).
Relevance. the induced norm is what the Cauchy–Schwarz bound and the unitarity constraint \(|\Theta_{ij}|\le 1\) rest on. (Halmos 1958).
Definition 23d (Hilbert space). A Hilbert space is a complete inner product space (every Cauchy sequence of vectors converges), here over \(\mathbb{R}\) or \(\mathbb{C}\) (Halmos 1958; Rudin 1991).
Relevance. the Hilbert space is the kinematical stage on which the Dirac–von Neumann axioms place the quantum state. (Dirac 1930; Neumann 1932; Halmos 1958).
Definition 23e (Complex Hilbert space). A complex Hilbert space is a Hilbert space (Definition 23d) whose scalar field is \(\mathbb{K}=\mathbb{C}\). In the finite-dimensional case \(\mathcal{H}\cong\mathbb{C}^N\) with the standard inner product \(\langle v\mid w\rangle=v^\dagger w=\sum_{k}\overline{v_k}\,w_k\).
Relevance. the complex Hilbert space is the specific kinematics adopted; Remark 71 discusses why \(\mathbb{R}\) and \(\mathbb{H}\) do not work. (Dirac 1930; Neumann 1932).
Notation 23h (Dirac bra-ket notation). A vector \(\psi\in\mathcal{H}\) is written as a ket \(\lvert\psi\rangle\) and its dual as a bra \(\langle\psi\rvert\); the inner product is the bracket \(\langle\phi\mid\psi\rangle=\langle\phi,\psi\rangle\), and the rank-one operator \(\lvert\psi\rangle\langle\phi\rvert\) acts by \(v\mapsto\langle\phi,v\rangle\,\psi\).
Relevance. Dirac notation is used throughout Parts III–IV to write \(\Theta_{ij}=\langle i|\Theta|j\rangle\) and the dictionary of Theorem 50. (Dirac 1942, 1930).
Definition 24a (Adjoint). On a complex Hilbert space \(\mathcal{H}\), the adjoint \(T^{\dagger}\) of an operator \(T:\mathcal{H}\to\mathcal{H}\) is the unique operator (where it exists) satisfying \(\langle\phi\mid T\psi\rangle=\langle T^{\dagger}\phi\mid\psi\rangle\) for all \(\phi,\psi\in\mathcal{H}\). For a matrix, \(T^{\dagger}=\overline{T^{T}}\) is the conjugate transpose.
Relevance. the adjoint of \(\Theta\) is what makes \(\Theta^\dagger P_i\Theta\) appear in the dictionary, and self-adjointness is the quantum shadow of real-valuedness of probabilities. (Halmos 1958; Rudin 1991).
Definition 24a1 (Self-adjoint operator). An operator \(T\) is self-adjoint (Hermitian) if \(T^{\dagger}=T\).
Relevance. self-adjoint operators are the observables (Definition 25c) and the configuration operators of Definition 52. (Neumann 1932; Halmos 1958).
Definition 24a2 (Positive-semidefinite operator). An operator \(T\) is positive-semidefinite if \(\langle\phi\mid T\phi\rangle\ge 0\) for every \(\phi\in\mathcal{H}\). It is positive-definite if \(\langle\phi\mid T\phi\rangle>0\) for every \(\phi\neq 0\).
Relevance. positive semidefiniteness is what makes \(\rho\) a legitimate density matrix and \(|\Theta_{ij}|^2\) a non-negative probability. (Halmos 1958).
Definition 24b (Trace). The trace \(\operatorname{tr}T=\sum_i\langle e_i\mid T e_i\rangle\) of an operator on a Hilbert space (when defined) is independent of the chosen orthonormal basis \(\{e_i\}\).
Relevance. the trace converts operators into the scalars that probabilities and expectation values are; it is the quantum analogue of summing over configurations. (Neumann 1932; Rudin 1991).
Definition 24c (Density matrix). A density matrix (density operator) on a complex Hilbert space \(\mathcal{H}\) is an operator \(\rho\) that is self-adjoint (Definition 24a1), positive-semidefinite (Definition 24a2), and satisfies \(\operatorname{tr}\rho=1\).
Relevance. the density matrix encodes the standalone distribution \(p_i(t)\) in the quantum language and evolves according to Proposition 51; Gleason’s theorem is what makes such operators the only consistent state space. (Neumann 1932, 1927; Gleason 1957).
Definition 24c1 (Spectral decomposition). Let \(A\) be a self-adjoint operator on a finite-dimensional complex Hilbert space. Its spectral decomposition is a representation \(A=\sum_i a_i P_i\) with \(\{P_i\}\) a projection-valued measure (Definition 25b), \(a_i\in\mathbb{R}\) the eigenvalues of \(A\), and \(P_i\) the corresponding orthogonal projections (Definition 25a).
Relevance. a spectral decomposition writes a self-adjoint operator in an orthonormal eigenbasis, which is what lets observables be diagonalized. (Hellinger and Toeplitz 1910; Horn and Johnson 2013).
Definition 24c2 (Pure state). A pure state is a density matrix of the form \(\rho=\Psi\Psi^{\dagger}\) with \(\|\Psi\|=1\); equivalently \(\rho\) has rank one and \(\rho^2=\rho\).
Relevance. pure states are the rank-one density matrices produced by a definite configuration; they are the extreme points of the state space. (Neumann 1932).
Definition 24d (Antilinear map). A map \(T:V\to W\) between complex vector spaces is antilinear (conjugate-linear) if \(T(\alpha u+\beta v)=\overline{\alpha}\,T(u)+\overline{\beta}\,T(v)\) for all \(u,v\) and \(\alpha,\beta\in\mathbb{C}\).
Relevance. antilinear maps are needed for antiunitary symmetries, which Wigner’s theorem (Remark 69a) permits as time-reversal-type transformations. (Wigner 1959).
Definition 24e (Anti-unitary operator). A bijective antilinear map (Definition 24d) \(U:\mathcal{H}\to\mathcal{H}\) is anti-unitary if \(\langle Ux,Uy\rangle=\overline{\langle x,y\rangle}\) for all \(x,y\in\mathcal{H}\).
Relevance. anti-unitary operators are the anti-symmetries Wigner’s theorem allows, relevant to time reversal in Part IV. (Wigner 1959).
Definition 25a (Projection). A projection (idempotent operator) satisfies \(P^2=P\). An orthogonal projection on a complex Hilbert space also satisfies \(P^\dagger=P\).
Relevance. projections are the operators that assign a yes/no outcome to a configuration, and are the building blocks of a PVM. (Halmos 1958).
Definition 25b (Projection-valued measure). A projection-valued measure (PVM) is a family \(\{P_i\}\) of orthogonal projections with \(P_i P_j=\delta_{ij}P_i\) and \(\sum_i P_i=\mathbb{1}\) (in the strong operator topology, Definition 25d, when infinite) (Mackey 1952, 1957).
Relevance. a PVM is the measurement structure that translates a partition of the configuration space into quantum operators, as Proposition 49 does. (Mackey 1952, 1957).
Definition 25d (Strong operator topology). A sequence of operators \(A_n\) converges strongly to \(A\) if \(A_n v\to A v\) in norm for every vector \(v\) (Definition 23c3); this is the strong operator topology. Infinite sums such as \(\sum_i P_i=\mathbb{1}\) in Definition 25b are understood in this sense.
Relevance. the strong operator topology is the topology in which infinite Kraus and spectral sums converge. (Rudin 1991).
Definition 25c (Observable). An observable of a quantum system is a self-adjoint operator \(A\) on its complex Hilbert space (Definition 24a1); its possible measurement outcomes are the eigenvalues of \(A\), and it admits a spectral decomposition as in Definition 24c1.
Relevance. an observable is a self-adjoint operator whose spectral projectors are the measurement outcomes. (Neumann 1932; Hellinger and Toeplitz 1910).
Definition 27b (Positive map). A linear map \(\mathcal{E}\) on operators is positive if \(X\ge0 \implies \mathcal{E}(X)\ge0\).
Relevance. positivity is the minimal condition ensuring that states are mapped to states. (Stinespring 1955; Choi 1975).
Definition 27c (Completely positive). A map \(\mathcal{E}\) is completely positive (CP) if for every \(k\ge1\), \(\mathrm{id}_k \otimes \mathcal{E}\) is positive on operators of \(\mathbb{C}^k\otimes\mathcal{H}\).
Relevance. complete positivity, rather than mere positivity, is the condition that survives tensoring with an ancilla — exactly the operation used in the dilation of Theorem 59. (Choi 1975; Stinespring 1955; Paulsen 2003).
Definition 27d (Trace-preserving). A linear map \(\mathcal{E}\) is trace-preserving (TP) if \(\operatorname{tr}\mathcal{E}(X)=\operatorname{tr}X\) for all trace-class \(X\).
Relevance. trace preservation is the quantum form of the normalization \(\sum_i p_i = 1\) and of column stochasticity (Proposition 13). (Kraus 1971; Nielsen and Chuang 2000).
Definition 27e (Quantum channel). A completely positive trace-preserving (CPTP) map is called a quantum channel.
Relevance. a quantum channel is the completely positive trace-preserving map that a stochastic process becomes in the dictionary of Theorem 50. (Nielsen and Chuang 2000; Paulsen 2003).
Theorem 29 (Stinespring dilation). Every CPTP map \(\mathcal{E}\) on \(\mathcal{H}\) can be purified: there exist an ancillary Hilbert space \(\mathcal{H}'\) and a unitary \(U\) on \(\mathcal{H} \otimes \mathcal{H}'\) such that \[\mathcal{E}(X) = \operatorname{tr}'\big(U (X \otimes \mathbb{1}') U^\dagger\big),\] where \(\operatorname{tr}'\) is the partial trace over \(\mathcal{H}'\) (Stinespring 1955; Keyl 2002).
Relevance. the Stinespring dilation is the operator-algebraic theorem that Theorem 59 specializes to the stochastic setting. (Stinespring 1955).
Definition 30b (Unitary matrix). An \(N\times N\) complex matrix \(U\) is unitary if its columns (equivalently rows) form an orthonormal basis of \(\mathbb{C}^N\), i.e. \(U^\dagger U = U U^\dagger = \mathbb{1}\) where \(U^\dagger\) is the conjugate transpose.
Relevance. unitary matrices are the finite-dimensional amplitudes \(\Theta\) whose squared moduli give unistochastic transition probabilities. (Horn 1954; Horn and Johnson 2013).
Definition 31 (Unistochastic matrix). A stochastic matrix \(X\) is unistochastic if there exists a unitary matrix \(U\) (Definition 30b) with \[X_{ij} = |U_{ij}|^2 \quad \text{for all } i,j. \tag{68}\]
Relevance. unistochastic matrices are precisely the \(\Gamma\) arising as \(|U_{ij}|^2\) from a unitary \(U\) — the class in which Part III shows every indivisible process can be embedded. (Barandes 2023; Milz and Modi 2021).
Definition 31b (Orthostochastic matrix). A stochastic matrix \(X\) is orthostochastic if there exists a real orthogonal matrix \(U\) (a real matrix with \(U^{T}U=UU^{T}=\mathbb{1}\)) with \(X_{ij}=|U_{ij}|^2\) for all \(i,j\); i.e. \(X\) is unistochastic (Definition 31) with a real \(U\). The study of these matrices was initiated by Horn (Horn 1954), who called them “ortho-stochastic.”
Relevance. orthostochastic matrices are the real subclass arising from orthogonal matrices; the distinction from the unistochastic class is what Proposition 32 records. (Horn 1954).
Proposition 32. Every unistochastic matrix is doubly stochastic. Every \(2 \times 2\) doubly stochastic matrix is unistochastic. Beyond \(2 \times 2\), neither implication reverses: there exist doubly stochastic matrices that are not unistochastic, and unistochastic matrices that are not orthostochastic.
Relevance. this proposition establishes the hierarchy unistochastic \(\subset\) doubly stochastic and orthostochastic \(\subset\) unistochastic, bounding how classical the dictionary can be. (Barandes 2023; Horn 1954).
Part II: Indivisible Stochastic Processes
2. Definition and Immediate Properties
The idea of an indivisible stochastic process first appeared in passing in a 2021 review of quantum non-Markovianity (Milz and Modi 2021, fig. 6), with rudimentary forms traceable to work on open quantum systems (Wolf and Cirac 2008). Barandes builds them into a complete framework (Barandes 2025b).
Definition 33 (Indivisible stochastic process). An indivisible stochastic process is a tuple \[(\mathcal{C}, \mathcal{T}, \mathcal{T}_0, \Gamma, p, \mathcal{A})\] consisting of:
Relevance. this is the central object of the post: the class of processes that are generically non-divisible, hence non-Markovian, and that Part III embeds in quantum theory. (Barandes 2023; Wolf and Cirac 2008).
- a configuration space \(\mathcal{C}\), a fixed set of possible configurations;
- a set of target times \(\mathcal{T}\);
- a set of conditioning times (or division events) \(\mathcal{T}_0 \subseteq \mathcal{T}\), containing an initial time \(0\);
- a transition map \(\Gamma : \mathcal{C}^2 \times \mathcal{T} \times \mathcal{T}_0 \to [0,1]\) with entries \[\Gamma_{ij}(t \leftarrow t_0) \equiv p(i,t \mid j,t_0),\] defined only for conditioning times \(t_0 \in \mathcal{T}_0\), satisfying the normalization condition \[\sum_i \Gamma_{ij}(t \leftarrow t_0) = 1 \quad \text{for all } j \in \mathcal{C},\ t \in \mathcal{T},\ t_0 \in \mathcal{T}_0;\]
- a probability distribution \(p : \mathcal{C} \times \mathcal{T} \to [0,1]\), satisfying the normalization condition \(\sum_i p(i,t) = 1\) and the law of total probability \[p_i(t) = \sum_j \Gamma_{ij}(t \leftarrow t_0)\, p_j(t_0) \quad \text{for all } t \in \mathcal{T},\ t_0 \in \mathcal{T}_0; \tag{19}\]
- an algebra of random variables \(\mathcal{A}\): the maps \(A : \mathcal{C} \times \mathcal{T} \to \mathbb{R}\).
The transition map is also required to satisfy the trivialization condition \(\Gamma(t_0 \leftarrow t_0) = \mathbb{1}\) for every \(t_0 \in \mathcal{T}_0\), and the continuity condition \(\lim_{t \to t_0} \Gamma(t \leftarrow t_0) = \mathbb{1}\).
Remark 34 (What is and is not assumed). Three features distinguish this definition from a textbook stochastic process.
Relevance. the assumptions listed delimit what the correspondence does and does not claim, which is essential for reading Propositions 35 and 36 correctly. (Barandes 2023).
- Only first-order laws. The nomological content is a single first-order transition map, not an infinite Kolmogorov tower. The law of total probability (19) is the only Chapman–Kolmogorov equation available.
- Division events are sparse and system-centric. Transition probabilities are not assumed to exist for every conditioning time, only for division events. Division events are not global properties of the universe; they are features of a given subsystem and may be generated to excellent approximation through interactions (Barandes 2025a).
- No assumption that \(t > t_0\). The target time may lie before the conditioning time, so logical time-reversal invariance is not violated fundamentally.
3. Divisibility and Its Failure
Proposition 35 (Linearity). The law of total probability (19) is linear in the standalone distribution: \(p(t) = \Gamma(t \leftarrow t_0)\, p(t_0)\).
Proof. Immediate from (19) written in matrix-vector form. \(\blacksquare\)
Relevance. linearity of time evolution in the standalone distribution is what lets \(\Gamma\) act as a matrix and what the quantum dictionary preserves. (Kolmogorov 1933; Barandes 2023).
This linearity is the seed of the linearity of quantum time evolution (Barandes 2025a).
Proposition 36 (Indivisibility entails non-Markovianity). An indivisible stochastic process is generically not divisible, and therefore generically not Markovian.
Proof. Suppose one attempts to define an intermediate transition matrix between \(t_0\) and \(t\) through an intermediate time \(t'\): \[\tilde\Gamma(t \leftarrow t') \equiv \Gamma(t \leftarrow t_0)\,\Gamma^{-1}(t' \leftarrow t_0).\] By construction \(\Gamma(t \leftarrow t_0) = \tilde\Gamma(t \leftarrow t')\Gamma(t' \leftarrow t_0)\). But by Lemma 20, \(\tilde\Gamma(t \leftarrow t')\) is a stochastic matrix only if both factors are permutation matrices. In general it is pseudo-stochastic (Definition 21), with negative entries, and is not a legitimate transition matrix. Hence the dynamics do not divide at \(t'\). By the contrapositive of Proposition 15, the process is non-Markovian. \(\blacksquare\)
Relevance. this is the first main result: divisibility, and with it Markovianity, fails generically for indivisible processes. (Barandes 2023; Wolf and Cirac 2008).
The failure of divisibility is therefore not the familiar failure of Markovianity as memory in the sense of needing higher-order conditionals. An indivisible process has vastly less nomological structure than a textbook non-Markovian process, yet is still non-Markovian, because it does not divide at all.
Remark 37 (Non-Markovian realizers). Given only the sparse ingredients of Definition 33, there generically exists a large or infinite family of complete Kolmogorov towers consistent with them. Any such tower that matches the system’s statistics is called a non-Markovian realizer; an indivisible stochastic process encompasses an equivalence class of them. Only the minimalist ingredients are connected to empirical predictions, so the particular realizer is contingent, potentially unknowable, and arguably meaningless (Barandes 2025a).
Relevance. the realizers indicate how non-Markovianity is to be detected operationally, which is what makes Proposition 36 physically meaningful. (Barandes 2023; Milz and Modi 2021).
4. Markovian Dynamical Systems as a Special Case
Definition 38 (Markovian-homogeneous dynamical system). A discrete Markovian-homogeneous dynamical system is a triple \((\mathcal{X}, \mathcal{T}, f)\) where \(\mathcal{X}\) is a finite state space, \(\mathcal{T} \cong \mathbb{Z}\) under addition, and the dynamical map \(f : \mathcal{X} \times \mathcal{T} \to \mathcal{X}\) satisfies the homogeneity property (Definition 38a) \[f_{t+t'} = f_t \circ f_{t'} \quad \text{for all } t,t' \in \mathcal{T}, \tag{70}\] together with the trivialization condition \(f_0 = \mathrm{id}_{\mathcal{X}}\).
Relevance. Markovian-homogeneous dynamical systems are the reversible, divisible processes against which indivisibility is contrasted. (Kolmogorov 1933; Doob 1953).
Definition 38a (Homogeneity). A family of maps \(\{f_t\}\) indexed by \(\mathcal{T}\) is homogeneous if \(f_{t+t'}=f_t\circ f_{t'}\) for all \(t,t'\in\mathcal{T}\) (Definition 38).
Relevance. homogeneity in time is what makes the evolution a one-parameter group, which Proposition 39 then inverts. (Norris 1997).
Proposition 39. In a Markovian-homogeneous dynamical system, \(f_t\) is invertible for every \(t\), with \(f_t^{-1} = f_{-t}\).
Proof. Immediate from homogeneity and trivialization. \(\blacksquare\)
Relevance. invertibility of the evolution is the classical counterpart of unitarity, and is what a divisible but non-permutation process lacks. (Doob 1953).
Example 40 (Permutation dynamics is unistochastic). If the state \(i\) at time \(t = n\,\delta t\) is identified with the standard basis vector and \(f_{n\,\delta t} = \Sigma^n\) for a fixed permutation matrix \(\Sigma\), then \(\Sigma\) is both unitary and unistochastic (its entries are \(0\)s and \(1\)s, whose modulus-squares are themselves). Such a system is already a unistochastic process. There is even an analytic interpolation to continuous time via \(U(t \leftarrow 0) = \Sigma^{t/\delta t}\) (diagonalize \(\Sigma = V^\dagger D V\) and take eigenvalues to the power \(t/\delta t\)), generating a Hamiltonian as the infinitesimal generator (Barandes 2023).
Relevance. permutation dynamics is the explicit example showing that not every doubly stochastic matrix is unistochastic, and that unistochastic dynamics need not be Markovian. (Barandes 2023; Horn 1954).
5. Axioms of Indivisible Quantum Theory
The reformulation replaces the Dirac–von Neumann axioms with a smaller, more transparent set (Barandes 2025a).
Axiom 41 (Kinematical axiom). For each model, one selects an appropriate configuration space, whose members are the possible configurations of the system, treated as elementary. The configuration space is a fixed feature of the model.
Relevance. the kinematical axiom fixes the state space, the quantum analogue of the standalone distribution of Definition 8a. (Dirac 1930; Neumann 1932).
Axiom 42 (Dynamical axiom). For arbitrary target times and for conditioning times that are division events, the dynamical laws consist of transition probabilities \(p(i,t \mid j,t_0)\). Division events may arise intrinsically or through interactions; the transition probabilities are fixed features of the model.
Relevance. the dynamical axiom constrains the time evolution, corresponding to the transition probabilities of Definition 9. (Dirac 1930; Neumann 1932).
Axiom 43 (Epistemic axiom). The system has a time-dependent standalone probability distribution, connected across times by the transition probabilities. It is contingent, varying between runs.
Relevance. the epistemic axiom concerns what is knowable about the state, matching the contingent character of the standalone distribution. (Barandes 2023).
The kinematical and epistemic axioms are essentially classical; the distinctively non-classical ingredient is the dynamical axiom, which replaces Markovian differential equations with a sparse set of indivisible transition probabilities. The configurations play the role historically assigned to “hidden variables,” but they are not hidden: they are exactly what one sees in experiments, and they replace wave functions as the ontological content (Barandes 2025a).
Part III: The Stochastic–Quantum Correspondence
6. The Time-Evolution Operator
Consider an indivisible stochastic process with finite configuration space of size \(N\), and fix an initial division event \(0\). Write the transition matrix as \[\Gamma_{ij}(t \leftarrow 0) \equiv p(i,t \mid j,0). \tag{24}\]
Observation 44 (Modulus-squaring). Since \(\Gamma_{ij}(t \leftarrow 0) \ge 0\), each entry can be written as the modulus-square of a complex number: \[\Gamma_{ij}(t \leftarrow 0) = |\Theta_{ij}(t \leftarrow 0)|^2. \tag{25}\] This is an identity, not a postulate: every non-negative real number is the modulus-square of some complex number. The matrix \(\Theta(t \leftarrow 0)\) is not unique, and is called the time-evolution operator.
Relevance. the modulus-squaring map \(\Gamma_{ij}=|\Theta_{ij}|^2\) is the dictionary’s first half, converting amplitudes into classical probabilities. (Barandes 2023; Milz and Modi 2021).
Proposition 45 (Summation condition). The column normalization of \(\Gamma\) implies \[\sum_{i=1}^N |\Theta_{ij}(t \leftarrow 0)|^2 = 1 \quad \text{for all } j, \tag{26}\] a generalization of the unitarity constraint. The trivialization condition gives \(\Theta(0 \leftarrow 0) = \mathbb{1}\) (the identity matrix).
Proof. Substitute (25) into \(\sum_i \Gamma_{ij} = 1\). \(\blacksquare\)
Relevance. the summation condition is the quantum form of column stochasticity (Proposition 13), and is what makes \(\Gamma\) a legitimate transition matrix. (Kolmogorov 1933; Horn and Johnson 2013).
Definition 46a (Schur–Hadamard product). For matrices \(X,Y\) of the same size, the Schur–Hadamard product is entry-wise multiplication (Schur 1911; Halmos 1958): \((X \odot Y)_{ij} \equiv X_{ij} Y_{ij}\).
Relevance. the Schur–Hadamard product is the entrywise product used to square amplitudes and to build the second gauge freedom of Proposition 67. (Schur 1911; Halmos 1958).
Definition 46b (Schur–Hadamard square). For a matrix \(\Theta\), define \(|\Theta|^{\odot 2}\equiv\overline{\Theta}\odot\Theta\), i.e. entrywise \(\big(|\Theta|^{\odot2}\big)_{ij}=\overline{\Theta_{ij}}\,\Theta_{ij}=|\Theta_{ij}|^2\), where \(\overline{\Theta}\) is the entrywise complex conjugate.
Relevance. the Schur square of \(\Theta\) is the non-negative matrix whose entries are the transition probabilities. (Schur 1911; Horn 1954).
Definition 46b1 (Schur–Hadamard gauge freedom). The Schur–Hadamard gauge freedom of a matrix \(\Theta\) is the freedom to replace \(\Theta_{ij}\mapsto\Theta_{ij}e^{i\theta_{ij}}\) for arbitrary real phases \(\theta_{ij}\), which leaves \(|\Theta|^{\odot2}\) unchanged.
Relevance. the gauge freedom records that many amplitudes give the same transition probabilities, which is why the dictionary is not invertible. (Barandes 2023).
Definition 46b2 (Gauge invariance). A gauge invariance (gauge symmetry) is a transformation of a mathematical representation that leaves all physical (empirical) quantities unchanged — here the transition matrix \(\Gamma\) (Definition 12), the probabilities, and the expectation values.
Relevance. gauge invariance is the statement that physically equivalent descriptions differ only by a phase redefinition; it is the theme of Part IV. (Barandes 2023).
Proposition 47 (Schur–Hadamard gauge freedom). The transformation \[\Theta_{ij}(t \leftarrow 0) \mapsto \Theta_{ij}(t \leftarrow 0)\, e^{i\theta_{ij}(t)} \tag{30}\] for arbitrary real phases \(\theta_{ij}(t)\) leaves every entry of \(\Gamma\) unchanged, and hence has no physical effect.
Proof. \(|\Theta_{ij} e^{i\theta_{ij}}|^2 = |\Theta_{ij}|^2\). \(\blacksquare\)
Relevance. the second gauge freedom shows that the choice of phases in \(\Theta\) is unphysical, which is what makes the dictionary a correspondence rather than an identification. (Barandes 2023).
This is a genuine gauge invariance, first described in (Barandes 2025a); it is not merely a change of relative phases of state vectors. It has downstream effects on the Hilbert-space ingredients to be defined, arranged so that all empirical predictions are preserved.
7. The Configuration Basis and the Dictionary
Definition 48 (Configuration basis). The configuration basis of \(\mathcal{H}=\mathbb{C}^N\) is the standard orthonormal basis \(e_1,\ldots,e_N\), with the \(i\)-th configuration identified with \(e_i\).
Relevance. the configuration basis is the isomorphism between configurations and Hilbert-space vectors that makes the dictionary possible. (Dirac 1930; Halmos 1958).
Definition 48b (Configuration projector). The configuration projector \(P_i\) is the rank-one orthogonal projection onto \(\operatorname{span}(e_i)\): \[P_i \equiv e_i e_i^\dagger = \operatorname{diag}(0,\dots,1,\dots,0), \tag{78}\] with the \(1\) in the \(i\)-th diagonal position.
Relevance. configuration projectors are the quantum encoding of “the system is in configuration \(i\)”. (Neumann 1932; Mackey 1957).
Proposition 49 (Configuration projectors form a PVM). The \(P_i\) satisfy \(P_i P_j = \delta_{ij} P_i\) and \(\sum_i P_i = \mathbb{1}\).
Proof. Direct computation using \(e_i^\dagger e_j = \delta_{ij}\). \(\blacksquare\)
Relevance. this proposition is what lets the classical partition of the configuration space be represented as a PVM, a prerequisite for Theorem 50. (Mackey 1952, 1957).
Theorem 50 (The dictionary). The modulus-squaring identity (25) is equivalent to the matrix-trace identity \[\boxed{\ \Gamma_{ij}(t \leftarrow 0) = \operatorname{tr}\!\left(\Theta^\dagger(t \leftarrow 0)\, P_i\, \Theta(t \leftarrow 0)\, P_j\right).\ } \tag{39}\]
Proof. The \(j\)-th diagonal entry of \(\Theta^\dagger P_i \Theta\) is \((\Theta^\dagger P_i \Theta)_{jj} = \sum_k \overline{\Theta_{kj}} (P_i \Theta)_{kj} = \overline{\Theta_{ij}}\Theta_{ij} = |\Theta_{ij}|^2\). Multiplying by \(P_j\) and taking the trace selects this \(j\)-th diagonal entry, giving \(|\Theta_{ij}|^2 = \Gamma_{ij}\). \(\blacksquare\)
Relevance. the dictionary is the central translation between the stochastic and quantum descriptions; Propositions 51, 52b and 61 and Theorem 62 all quote it. (Barandes 2023; Milz and Modi 2021).
The dictionary is the foundational equation of the stochastic–quantum correspondence: it translates the stochastic process on the left into Hilbert-space data on the right. It is the direct analogue of the correspondence between Newtonian and Hamiltonian mechanics.
8. The Hilbert-Space Representation
Given an initial standalone distribution \(p_j(0)\), define the initial density matrix \[\rho(0) \equiv \operatorname{diag}(p_1(0), \dots, p_N(0)) = \sum_{j=1}^N p_j(0)\, P_j. \tag{41}\]
Proposition 51 (Density-matrix evolution). The law of total probability implies \[\rho(t) = \Theta(t \leftarrow 0)\, \rho(0)\, \Theta^\dagger(t \leftarrow 0), \tag{42}\] which is self-adjoint, positive-semidefinite, and unit-trace. The probability formula becomes the Born rule, \[p_i(t) = \operatorname{tr}\!\left(P_i\, \rho(t)\right). \tag{46}\]
Proof. Insert the dictionary into the law of total probability and use \(\sum_j P_j = \mathbb{1}\) and \(\Theta \rho(0) \Theta^\dagger\). Self-adjointness and positivity follow from those of \(\rho(0)\); unit-trace follows from the summation condition (26). \(\blacksquare\)
Relevance. density-matrix evolution is the quantum restatement of \(p(t)=\Gamma(t\leftarrow t_0)p(t_0)\) in Proposition 35, and is the Schrödinger picture of the dynamics. (Schrödinger 1926; Neumann 1932; Barandes 2023).
Hence the linearity of the law of total probability underwrites the linearity of the density-matrix evolution. For a rank-one \(\rho(t) = \Psi(t)\Psi^\dagger(t)\), one obtains a state vector \(\Psi(t) = \Theta(t \leftarrow 0)\Psi(0)\) of unit norm, and the Born rule takes the textbook form \(p_i(t) = |\Psi_i(t)|^2\).
Definition 52 (Configuration operator). For a random variable \(A(t)\) with values \(a(i,t)\) on the configuration basis, the associated configuration operator (diagonal operator) is \[A(t) \equiv \sum_{i=1}^N a(i,t)\, P_i = \operatorname{diag}(a_1(t), \dots, a_N(t)). \tag{48}\] Its values \(a(i,t)\) are the eigenvalues of \(A(t)\), and the projectors \(P_i\) are the configuration projectors (Definition 48b).
Relevance. the configuration operator is the observable whose spectral outcome \(i\) reports that the system is in configuration \(i\). (Neumann 1932).
Proposition 52b (Expectation via the trace). For a configuration operator \(A(t)\) (Definition 52) and a state \(\rho(t)\), the expectation equals \[\langle A(t) \rangle = \operatorname{tr}\!\left(A(t)\, \rho(t)\right). \tag{49}\]
Relevance. the trace formula is the quantum version of the expectation in (2), and is what makes classical statistics read off from \(\rho\). (Neumann 1932; Born 1926).
Definition 52c (Beable). A random variable \(A(t)\) whose values are the eigenvalues of a configuration operator (Definition 52), i.e. a diagonal self-adjoint operator, is called a beable (following Bell) (Bell 1971, 1976): when the system is in configuration \(i\), each beable has the definite value \(a(i,t)\).
Relevance. beables are the quantities assumed to have definite values independent of measurement, which the Kochen–Specker theorem constrains. (Bell 1971, 1976).
Definition 52d (Emergeable). An observable (Definition 25c) that is not a beable — i.e. represented by a non-diagonal self-adjoint operator — is called an emergeable; by the Kochen–Specker theorem (Remark 52e), not all observables can be beables (Bell 1966; Kochen and Specker 1967).
Relevance. emergeables are the quantum quantities that behave like classical beables, the notion Part IV needs in order to state what is and is not classical. (Barandes 2023).
Remark 52e (Kochen–Specker theorem). The Kochen–Specker theorem states that in a Hilbert space of dimension at least three there is no assignment of definite values to all observables (self-adjoint operators) that is consistent with the functional relations among them. Hence not every observable can be a beable (Definition 52c) (Bell 1966; Kochen and Specker 1967).
Relevance. the Kochen–Specker theorem is the no-go result forcing beables to be context-dependent, and hence what motivates the emergeable notion. (Bell 1966; Kochen and Specker 1967).
Definition 53 (Heisenberg picture). Define \[\rho^H \equiv \rho(0), \qquad \Psi^H \equiv \Psi(0), \qquad A^H(t) \equiv \Theta^\dagger(t \leftarrow 0)\, A(t)\, \Theta(t \leftarrow 0). \tag{51}\] The probability and expectation formulas become \(p_i(t) = \operatorname{tr}(P_i^H(t)\rho^H)\) and \(\langle A(t)\rangle = \operatorname{tr}(A^H(t)\rho^H)\).
Relevance. the Heisenberg picture is the description in which observables evolve and states are fixed, the mirror image of the Schrödinger picture of Proposition 51. (Heisenberg 1925; Neumann 1932).
Example 54 (Velocity as an emergeable). For a stochastic particle, the derivative \(\dot A \equiv \lim_{t \to 0} \frac{d}{dt} A^H(t)\) is self-adjoint but generically non-diagonal, and does not commute with \(A\). It has a transparent physical meaning through \(\operatorname{tr}(\dot A \rho(0)) = \lim_{t\to 0} \frac{d}{dt}\langle A(t)\rangle\), yet no direct interpretation on the configuration space: velocity is an emergeable, an amalgam of kinematical and dynamical ingredients (Barandes 2025b).
Relevance. the velocity example exhibits a concrete emergeable, showing that the notion is non-vacuous. (Barandes 2023).
9. Kraus Decomposition and Quantum Channels
Definition 55 (Kraus operators). For each \(\beta = 1, \dots, N\), define \[K_\beta(t \leftarrow 0) \equiv \Theta(t \leftarrow 0)\, P_\beta, \tag{58}\] i.e. the matrix whose \(\beta\)-th column agrees with that of \(\Theta\) and which is zero elsewhere: \(K_{\beta,ij} = \delta_{\beta j}\Theta_{ij}\).
Relevance. Kraus operators are the matrices \(K_\beta\) whose squared moduli give the transition probabilities; they are the object Proposition 57 constructs. (Kraus 1971).
Proposition 56 (Kraus condition). The summation condition (26) implies \[\sum_{\beta=1}^N K_\beta^\dagger(t \leftarrow 0)\, K_\beta(t \leftarrow 0) = \mathbb{1}. \tag{59}\]
Proof. \([K_\beta^\dagger K_\beta]_{ij} = \sum_k \overline{K_{\beta,ki}} K_{\beta,kj} = \delta_{\beta i}\delta_{\beta j}\sum_k |\Theta_{k\beta}|^2 = \delta_{\beta i}\delta_{\beta j}\). Summing over \(\beta\) gives \(\delta_{ij}\). \(\blacksquare\)
Relevance. the Kraus condition is the normalization that guarantees the resulting map is trace preserving, i.e. that probabilities still sum to one. (Kraus 1971; Choi 1975).
Proposition 57 (Kraus decomposition of the dynamics). The dictionary becomes \[\Gamma_{ij}(t \leftarrow 0) = \sum_{\beta=1}^N \operatorname{tr}\!\left(K_\beta^\dagger(t \leftarrow 0)\, P_i\, K_\beta(t \leftarrow 0)\, P_j\right), \tag{60}\] and the density-matrix rule becomes \[\rho(t) = \sum_{\beta=1}^N K_\beta(t \leftarrow 0)\, \rho(0)\, K_\beta^\dagger(t \leftarrow 0). \tag{61}\]
Relevance. this decomposition writes the process dynamics as a quantum channel, which is what Theorem 59 then dilates. (Kraus 1971; Choi 1975).
Definition 58 (Quantum channel). Abstracting, define the map \[\mathcal{E}_{t \leftarrow 0}(X) \equiv \sum_{\beta=1}^N K_\beta(t \leftarrow 0)\, X\, K_\beta^\dagger(t \leftarrow 0), \tag{63}\] which is a CPTP map, or quantum channel, on the algebra \(\mathbb{C}^{N \times N}\).
Relevance. the quantum channel is the structural form of the dynamics that Theorem 50 and Proposition 57 establish. (Nielsen and Chuang 2000; Paulsen 2003).
The Kraus operators are not unique: any set satisfying the Kraus condition and reproducing the dictionary yields a valid transition matrix. The existence (not uniqueness) of these operators is what permits the final step.
10. Dilation and Unitary Evolution
Theorem 59 (Dilation). By the Stinespring dilation theorem (Theorem 29), the quantum channel \(\mathcal{E}_{t \leftarrow 0}\) can be purified. There exists an integer \(\tilde N\) with \[N \le \tilde N \le N^3 \tag{94}\] (\(\tilde N\) a multiple of \(N\)), an ancillary space \(\mathcal{H}' = \mathbb{C}^{N'}\) with \(\tilde N = N N'\), and an \(\tilde N \times \tilde N\) unitary \(\tilde U(t \leftarrow 0)\) on \(\tilde{\mathcal{H}} = \mathcal{H} \otimes \mathcal{H}'\) such that the dictionary becomes \[\Gamma_{ij}(t \leftarrow 0) = \operatorname{tr}\!\left(\operatorname{tr}'\!\left(\tilde U^\dagger(t \leftarrow 0)\,[P_i \otimes \mathbb{1}']\, \tilde U(t \leftarrow 0)\,[P_j \otimes P'_{j'}]\right)\right). \tag{96}\]
Proof sketch. Given Kraus matrices \(K_\beta\), define an \(N^3 \times N^2\) matrix \(\tilde V\) by \(\tilde V_{(i\beta m)(jl)} = K_{\beta,ij}\delta_{lm}\). One shows \(\tilde V^\dagger \tilde V = \mathbb{1}_{N^2 \times N^2}\), so \(\tilde V\) is a partial isometry; it extends to a unitary \(\tilde U\) by adjoining \(N^3 - N^2\) mutually orthogonal columns, chosen so that \(\tilde U(0\leftarrow 0) = \mathbb{1}\). Substituting \(\tilde U\) into the right-hand side of (96) reproduces the dilute dictionary. \(\blacksquare\)
Relevance. the dilation theorem is the statement that the process can be embedded in a unitary dynamics on a larger Hilbert space. (Stinespring 1955; Barandes 2023).
The dilated dictionary can be written directly in terms of the unitary entries: \[\tilde\Gamma_{ii',jj'}(t \leftarrow 0) = |\tilde U_{ii',jj'}(t \leftarrow 0)|^2, \tag{105}\] so the \(\tilde N \times \tilde N\) transition matrix \(\tilde\Gamma\) is unistochastic.
Definition 60 (Dilated process). Define the dilated indivisible stochastic process \((\tilde{\mathcal{C}}, \tilde{\mathcal{T}}, \tilde{\mathcal{T}}_0, \tilde\Gamma, \tilde p, \tilde{\mathcal{A}})\) by
Relevance. the dilated process is the unistochastic process of which the original is a subsystem, as Theorem 62 asserts. (Barandes 2023; Milz and Modi 2021).
- \(\tilde{\mathcal{C}} \equiv \mathcal{C} \times \mathcal{C}'\) (the Cartesian product, with \(|\mathcal{C}'| = N'\));
- \(\tilde{\mathcal{T}} \equiv \mathcal{T}\) and \(\tilde{\mathcal{T}}_0 \equiv \{0\}\);
- \(\tilde\Gamma\) given by the dilated dictionary (105), which is unistochastic;
- \(\tilde p_{ii'}(t) \equiv \sum_j \tilde\Gamma_{ii',jj'}(t \leftarrow 0)\, p_j(0)\);
- \(\tilde{\mathcal{A}}\) the maps \(\tilde{\mathcal{C}} \times \tilde{\mathcal{T}} \to \mathbb{R}\).
Proposition 61 (Marginalization). The dilated process marginalizes back to the original: \[\sum_{i'=1}^{N'} \tilde\Gamma_{ii',jj'}(t \leftarrow 0) = \Gamma_{ij}(t \leftarrow 0), \qquad \sum_{i'=1}^{N'} \tilde p_{ii'}(t) = p_i(t). \tag{108,110}\]
Proof. Tracing over the ancillary index in (96) and using \(\sum_{i'} P'_{i'} = \mathbb{1}'\). \(\blacksquare\)
Relevance. marginalization is the consistency requirement that the dilated process reproduce the original statistics, the quantum form of (5). (Kolmogorov 1933; Barandes 2023).
Definition 61a (Marginal distribution). Given a joint distribution \(\tilde p_{ii'}(t)\), the marginal over the ancillary index is \(\sum_{i'}\tilde p_{ii'}(t)\); likewise, summing a transition matrix over one index gives the marginal transition matrix, as in Proposition 61.
Relevance. the marginal distribution is the classical standalone distribution obtained by tracing out the ancilla. (Kolmogorov 1933).
Theorem 62 (The Stochastic–Quantum Theorem). Every indivisible stochastic process can be regarded as a subsystem of a unistochastic process. (Barandes 2023)
Proof. By Observation 44, write \(\Gamma_{ij} = |\Theta_{ij}|^2\). By §7–§9, build the Hilbert-space representation, the density matrix, and the CPTP channel. By Theorem 59, dilate to a unitary \(\tilde U\) on \(\tilde{\mathcal{H}}\). By Proposition 61, the original process is a subsystem of the dilated, unistochastic process \((\tilde{\mathcal{C}}, \tilde{\mathcal{T}}, \tilde{\mathcal{T}}_0, \tilde\Gamma, \tilde p, \tilde{\mathcal{A}})\). \(\blacksquare\)
Relevance. the Stochastic–Quantum Theorem is the post’s main result: every indivisible stochastic process is a subsystem of a unistochastic process. (Barandes 2023).
Corollary 63. To study the class of indivisible stochastic processes, it suffices to study the subclass of unistochastic processes. Every indivisible stochastic process corresponds to a unitarily evolving quantum system, and unitarily evolving quantum systems represent the most general way to model a system with stochastic dynamical laws (Barandes 2023).
Relevance. this corollary packages the reverse direction, that unistochastic processes are the natural container for indivisible ones. (Barandes 2023).
Corollary 64 (The correspondence is bidirectional). Reading the dictionary in the other direction, any quantum system can be modeled as an indivisible stochastic process. One thereby obtains the stochastic–quantum correspondence between indivisible stochastic processes and quantum systems (Barandes 2025b).
Relevance. bidirectionality is what makes the correspondence usable in both directions, from stochastic processes to quantum dynamics and back. (Barandes 2023; Milz and Modi 2021).
Definition 65 (Unistochastic process). An indivisible stochastic process (Definition 33) whose transition matrix (Definition 12) is unistochastic (Definition 31) at every target and conditioning time is a unistochastic process.
Relevance. unistochastic processes are the class closed under dilation, which is why the correspondence singles them out. (Barandes 2023).
Proposition 65a (Unistochastic dictionary). For a unistochastic process (Definition 65) one may take \(\Theta=U\) unitary (Definition 30b), and the dictionary reads \[\Gamma_{ij}(t \leftarrow 0) = \operatorname{tr}\!\left(U^\dagger(t \leftarrow 0)\, P_i\, U(t \leftarrow 0)\, P_j\right), \tag{65}\] with the state evolving as \(\rho(t) = U(t\leftarrow 0)\rho(0)U^\dagger(t\leftarrow 0)\).
Relevance. the unistochastic dictionary is the dictionary specialised to the dilated process, in which the amplitude \(\Theta\) is unitary. (Barandes 2023; Horn 1954).
The correspondence is many-to-one in both directions, exactly as the correspondence between Newtonian systems and Hamiltonian mechanics: a given indivisible stochastic process admits infinitely many Hilbert-space representations, and a given Hilbert-space system represents infinitely many qualitatively different indivisible stochastic processes (one per choice of orthonormal basis serving as a configuration space) (Barandes 2025b).
Part IV: Further Implications
11. Hamiltonian Dynamics and the Emergence of the Schrödinger Equation
If the unitary time-evolution operator is differentiable in \(t\), define the Hamiltonian \[H(t) \equiv i\hbar\, \frac{\partial U(t \leftarrow 0)}{\partial t}\, U^\dagger(t \leftarrow 0) = H^\dagger(t). \tag{69}\]
Proposition 66. The density matrix satisfies the von Neumann equation, a rank-one state vector satisfies the Schrödinger equation, expectation values satisfy the Ehrenfest equation, and Heisenberg-picture operators satisfy the Heisenberg equation of motion: \[i\hbar\, \frac{\partial \rho(t)}{\partial t} = [H(t), \rho(t)], \tag{70}\] \[i\hbar\, \frac{\partial \Psi(t)}{\partial t} = H(t)\Psi(t), \tag{71}\] \[\frac{d\langle A(t)\rangle}{dt} = \left\langle \frac{i}{\hbar}[H(t), A(t)] \right\rangle + \left\langle \frac{\partial A(t)}{\partial t} \right\rangle. \tag{72}\]
Relevance. the von Neumann, Schrödinger, Ehrenfest, and Heisenberg equations are all derived here rather than postulated. (Barandes 2023, 2025a).
Thus the Schrödinger equation is derived, not postulated, from an indivisible stochastic process with a differentiable unistochastic transition matrix. The conditions of linearity and probability conservation alone are insufficient to pick out unitarity—CPTP maps such as the bit-flip channel are linear and probability-conserving without being unitary—so the derivation of unitarity is a substantive result (Barandes 2025a).
12. Gauge Invariance and Dynamical Symmetries
12.1 Foldy–Wouthuysen gauge transformations
Proposition 67 (Second gauge invariance). Let \(V(t)\) be a time-dependent unitary matrix. The transformation \[ \begin{aligned} \rho(t) &\mapsto V(t)\rho(t)V^\dagger(t), \\ \Psi(t) &\mapsto V(t)\Psi(t), \\ A(t) &\mapsto V(t)A(t)V^\dagger(t), \\ \Theta(t \leftarrow 0) &\mapsto V(t)\,\Theta(t \leftarrow 0)\, V^\dagger(0) \end{aligned} \tag{74} \] leaves the probabilities, expectation values, and the transition matrix unchanged (Barandes 2025b).
Relevance. the second gauge invariance is the Part IV statement that physical predictions depend only on the equivalence class of amplitudes. (Barandes 2023; Milz and Modi 2021).
This is a gauge invariance first written down for the Dirac equation by Foldy and Wouthuysen (Foldy and Wouthuysen 1950). If \(V\) is time-independent it is a change of orthonormal basis; if it depends on time it is a local-in-time unitary transformation of each Hilbert-space fiber. Under it, when \(\Theta = U\) is unitary, the Hamiltonian transforms as a non-Abelian gauge potential: \[H(t) \mapsto V(t)H(t)V^\dagger(t) - i\hbar\, V(t)\frac{\partial V^\dagger(t)}{\partial t}. \tag{75}\] Choosing \(V(t) = U^\dagger(t\leftarrow 0)\) makes \(H_V = 0\)—which is precisely the Heisenberg picture. Consequently the Hamiltonian is not a gauge-invariant observable, and the familiar Hilbert-space ingredients are highly gauge-dependent; one may reasonably be skeptical of assigning them direct ontological meaning (Barandes 2025b).
12.2 Dynamical symmetries and Noether’s theorem
Definition 68 (Dynamical symmetry). For an invertible transformation of configurations \(P_i \mapsto \tilde P_i = V^\dagger P_i V\) (with \(V\) unitary), the transformation is a dynamical symmetry if it leaves the transition matrix invariant: \[\operatorname{tr}\!\left(\Theta^\dagger \tilde P_i \Theta \tilde P_j\right) = \operatorname{tr}\!\left(\Theta^\dagger P_i \Theta P_j\right). \tag{83}\]
Relevance. a dynamical symmetry is a symmetry of the equations of motion; Part IV relates it to conserved quantities. (Noether 1918; Sakurai 1994).
Proposition 69 (Condition for a dynamical symmetry). A unitary \(V\) is a dynamical symmetry if and only if \[V\,\Theta(t\leftarrow 0)\,V^\dagger = \Theta(t\leftarrow 0) \odot \left(e^{i\theta_{ij}(t)}\right) \tag{87}\] for some phase matrix; that is, \(V\Theta V^\dagger\) differs from \(\Theta\) by at most a Schur–Hadamard gauge transformation (Definition 46b1). The unitary case \(V\Theta V^\dagger = \Theta\) and the anti-unitary case \(V\Theta V^\dagger = \overline{\Theta}\) are special cases (Barandes 2025b).
Relevance. this condition is the operator equation whose solutions generate the dynamical symmetries of the process. (Noether 1918; Sakurai 1994).
Remark 69a (Wigner’s theorem). Wigner’s theorem states that any bijection of the set of pure states (Definition 24c2) of a complex Hilbert space that preserves transition probabilities \(|\langle\phi,\psi\rangle|^2\) is induced by a unitary or anti-unitary operator (Definition 24e) (Wigner 1959).
Relevance. Wigner’s theorem is what guarantees that the symmetries of a quantum theory are implemented by unitary or anti-unitary operators. (Wigner 1959).
For a unitary symmetry generated by a self-adjoint \(G\) that commutes with \(U\), Noether’s theorem follows as the conservation statement \[\langle G(t) \rangle = \operatorname{tr}\!\left(G U(t\leftarrow 0)\rho(0)U^\dagger(t\leftarrow 0)\right) = \langle G(0)\rangle. \tag{93}\]
Remark 69b (Noether’s theorem, here). For a continuous symmetry generated by a self-adjoint \(G\) commuting with the evolution, Noether’s theorem here is the conservation statement \(\langle G(t)\rangle=\langle G(0)\rangle\) above (Barandes 2025b).
Relevance. Noether’s theorem is the classical statement that continuous symmetries yield conserved quantities, the pattern Part IV follows. (Noether 1918).
Restricting to transformations that must preserve the Born rule in every orthonormal basis yields the stronger condition \(|\tilde\Theta'_{\alpha\beta}|^2 = |\Theta'_{\alpha\beta}|^2\), whose solutions are precisely the unitary and anti-unitary operators of Wigner’s theorem (Wigner 1959).
13. Dilations, Spin, and the Complex Numbers
Proposition 70 (Dilation as a free enlargement). For any integer \(D \ge 2\), the Hilbert-space formulation can be enlarged without changing the process by \[\Theta \mapsto \Theta \otimes \mathbb{1}^{\mathcal{I}}, \quad P_i \mapsto P_i \otimes \mathbb{1}^{\mathcal{I}}, \quad P_j \mapsto P_j \otimes P^{\mathcal{I}}_\gamma, \tag{98}\] where \(\{P^\mathcal{I}_\gamma\}\) is a PVM on an internal space \(\mathcal{H}_\mathcal{I}\) of dimension \(D\). The dictionary is unchanged after a partial trace over \(\mathcal{H}_\mathcal{I}\).
Relevance. the freeness of the dilation is what makes the embedding of Theorem 62 a genuine extension rather than a reformulation. (Barandes 2023; Stinespring 1955).
Dilations provide a larger “canvas” for designing transition matrices and can make new dilation-emergeables—such as intrinsic spin—manifest. Representing spin this way requires dilating the Hilbert space to \(N D\) dimensions and introducing a \(D\)-dimensional representation of \(SO(3)\) for the internal space, so the underlying indivisible stochastic process involves no preferred direction (Barandes 2025b).
Remark 71 (Why the complex numbers appear). Unistochastic matrices are generally not orthostochastic, so if the transition matrix \(\Gamma\) is already unistochastic there is no guarantee that its unitary square root \(U\) can be taken real orthogonal. The correspondence therefore implies that, to represent the most general indivisible stochastic processes, the complex numbers \(\mathbb{C}\) are an important structural feature of quantum theory (Barandes 2023). One is always free to choose \(\Theta\) real, in which case the dilated \(\tilde U\) is real orthogonal; and one can even double the dimension to represent the imaginary unit by a real \(2\times 2\) matrix (Myrheim 1999) and the complex-conjugation operator \(K\) by \(\sigma_x\) (Stueckelberg 1960). But even so, the algebraic structure of the complex numbers—and with it the spectral theorem, eigenvectors, anti-unitary operators, self-adjoint Hamiltonians, and the Schrödinger equation—becomes hard to avoid. These ideas are developed further in the deflationary account of (Barandes 2026).
Relevance. the appearance of \(\mathbb{C}\) is the reason the quantum formalism is the natural language for indivisible stochastic processes. (Barandes 2025a; Dirac 1930).
15. Summary
- An indivisible stochastic process (Definition 33) is a stochastic process whose dynamical laws consist of a single first-order transition map, defined at division events, without an infinite Kolmogorov tower.
- The free positivity of transition probabilities lets one write \(\Gamma_{ij} = |\Theta_{ij}|^2\) (Observation 44), introducing the time-evolution operator with its Schur–Hadamard gauge freedom (Proposition 47).
- The dictionary \(\Gamma_{ij} = \operatorname{tr}(\Theta^\dagger P_i \Theta P_j)\) (Theorem 50) translates the stochastic process into Hilbert-space language, and the law of total probability’s linearity becomes the linearity of density-matrix evolution and the Born rule (Proposition 51).
- A Kraus decomposition (Propositions 56–57) and Stinespring dilation (Theorem 59) produce a unitary operator on a space of dimension at most \(N^3\), whose associated transition matrix is unistochastic.
- The stochastic–quantum theorem (Theorem 62) states that every indivisible stochastic process is a subsystem of a unistochastic process, and the stochastic–quantum correspondence (Corollary 64) runs in both directions.
- The framework derives the Schrödinger equation, accommodates interference, decoherence, entanglement, and noncommutative observables, and reframes the measurement problem. It comes with two forms of gauge invariance, a Noether theorem, and a natural account of why complex numbers appear.
The most remarkable feature is how little is assumed. From ordinary probability, an old-fashioned configuration space, and a sparse set of first-order transition probabilities—no Hilbert space, no wave function, no unitarity postulate—one recovers the empirical content of quantum theory. Whether the residual non-Markovianity can be reconciled with relativistic locality, and whether the framework yields genuinely new predictions, are the questions that will determine its ultimate significance (Barandes 2025a, 2026).