Source
Barandes (2023; v3 2025), LIT-086, read in full in NOTE-080, which checked the algebra.
What was actually shown
Indivisible stochastic process (§2). A configuration space C = {1,…,N} and transition matrices Γ(t←t0), stochastic and with Γ(t0←t0) = 1, specified only from conditioning times. Divisibility, Γ(t←0) = Γ̃(t←t′)Γ(t′←0) with Γ̃ stochastic, is not assumed, and higher-order conditionals are left unspecified.
Dictionary (Eqs. 12–15). Γ_ij(t←0) = |Θ_ij(t←0)|² = tr(Θ†P_iΘP_j), with Σ_i |Θ_ij|² = 1. This is an identity: the entrywise square root always works (NOTE-080, C1). Then ρ(t) = Θ diag(p(0)) Θ† gives p_i(t) = tr(P_iρ(t)) (Eqs. 16–17), Kraus operators K_β = ΘP_β (Eqs. 25–27), and, after a cited Stinespring dilation, a unitary U with Γ_ij = |U_ij|² (Eqs. 30–32).
Interference (Eq. 43). Γ_ij(t←0) − [|U(t←t′)|²|U(t′←0)|²]ij = Σ{k≠l} U_ik(t←t′)Ψ_k(t′) conj(U_il(t←t′)Ψ_l(t′)). The cross-term is the gap between the actual transition probabilities and a pretend Markov factorization at t′. NOTE-080 confirms the algebra (C5).
Division events (Eqs. 45–58). If at t′ the system is perfectly correlated with an environment, p^{SE}{i′e′}(t′) = p^S{i′}(t′)δ_{e′e(i′)} with the e(i) distinct, and afterwards U^{SE} = U^S ⊗ U^E, the cross-terms vanish and Γ^S(t←0) = Γ^S(t←t′)Γ^S(t′←0). Regular division events with a time-homogeneous U^S give a Markov chain, Γ^S(nδt←0) = (Γ^S)ⁿ. The density matrix after a division event is diagonal (Eqs. 59–62).
Measurement (§4). Given a unistochastic evolution that correlates a device with the eigenbasis of a non-diagonal Ã^S (Eq. 73), the device configurations follow the Born rule for Ã^S (Eq. 78), and conditioning on the device reproduces collapse (Eqs. 89–94).
What could have come out otherwise. A stochastic matrix with no complex “square root” would have blocked the representation; there is none. The cross-terms could have survived a perfectly correlating environment; under Eqs. 45–46 they cannot.
What this does not say
- That quantum dynamics are derived from stochastic processes. The Schrödinger, von Neumann and Ehrenfest equations follow from the definition H ≡ iħ(∂U/∂t)U† (Eq. 33), and carry no content beyond the choice of U (NOTE-080, C4). The paper’s “new result” labels mark identities and definitions.
- That unitarity is inevitable. A unitary Θ on the original N configurations exists only for doubly stochastic Γ (Eq. 32); general Γ needs the dilated space, which is cited, not constructed. The claimed gauge freedom Θ_ij ↦ e^{iθ_ij}Θ_ij does not preserve unitarity (NOTE-080’s check on the Hadamard), so the choice of unitary representative is not gauge-invariant (C3, C13).
- That interference needs indivisible dynamics on any ontology. The decomposition is relative to the chosen configuration space. NOTE-080 notes that an odd-d stabilizer circuit shows Eq. 43’s interference on the computational configurations yet has a divisible deterministic model on LIT-007’s larger ontic space.
- That every quantum system is such a process in a physical sense. The quantum-to-stochastic direction, Γ = |U|², needs a chosen configuration basis, which the paper does not fix (C12). Multi-time statistics beyond the measurement model, locality and Bell-type questions are deferred.
- That Markovianity is explained in practice. Only the ideal division event is shown; the approximate case is asserted (fn. 18).
Connections
- Auffèves & Grangier (LIT-077) reach the same unistochastic matrices as a claimed consequence of postulates about contexts, where here they are a chosen representation between times (NOTE-080). Their contextual objectivity is THEORY-010.
- The paper places itself outside the ontological-models framework, and so outside PBR (THEORY-605), with a nomic wave function (§5.3).
- LIT-001 (Rejected) reaches a Schrödinger equation from stochastic mechanics but leaves no real stochastic process; this paper keeps the process and gives the equation no dynamical content (NOTE-080).