Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the emergence of quantum mechanics from stochastic processes

Jason Doukas

Abstract

The stochastic--quantum correspondence reinterprets quantum dynamics as arising from an underlying stochastic process on a configuration space. We generalize the correspondence by lifting an arbitrary stochastic kernel $Γ$ in finite dimension to a map $φ$ on $B(\mathcal H)$, formulating the associated lift-compatibility relation, and giving an explicit dictionary between $Γ$ and CPTP (Kraus) maps. We isolate Chapman--Kolmogorov divisibility of the lifted family as the decisive additional constraint: when a CK-consistent CPTP family exists, the lift admits a Lindblad master equation form. In this picture, off-diagonal (phase) degrees of freedom act as a compressed carrier of history dependence not fixed by transition kernels alone; conversely, the apparent emergence of quantum phase information from a phase-blind stochastic description is explained as a memory effect. Finally, we state and prove a divisibility criterion for the underlying stochastic kernels, expressed as a condition involving divisibility of the lifted map together with a diagonality requirement on the density operator.

On the emergence of quantum mechanics from stochastic processes

Abstract

The stochastic--quantum correspondence reinterprets quantum dynamics as arising from an underlying stochastic process on a configuration space. We generalize the correspondence by lifting an arbitrary stochastic kernel in finite dimension to a map on , formulating the associated lift-compatibility relation, and giving an explicit dictionary between and CPTP (Kraus) maps. We isolate Chapman--Kolmogorov divisibility of the lifted family as the decisive additional constraint: when a CK-consistent CPTP family exists, the lift admits a Lindblad master equation form. In this picture, off-diagonal (phase) degrees of freedom act as a compressed carrier of history dependence not fixed by transition kernels alone; conversely, the apparent emergence of quantum phase information from a phase-blind stochastic description is explained as a memory effect. Finally, we state and prove a divisibility criterion for the underlying stochastic kernels, expressed as a condition involving divisibility of the lifted map together with a diagonality requirement on the density operator.
Paper Structure (20 sections, 1 theorem, 114 equations, 1 figure)

This paper contains 20 sections, 1 theorem, 114 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Stochastic processes
  3. Chapman--Kolmogorov (CK) property
  4. Short-time structure of the stochastic--quantum correspondence
  5. Generalizing the correspondence
  6. Operator lifts and the compatibility condition
  7. Explicit constructions of compatible lifts
  8. CK divisibility, composition, and Lindblad generators
  9. Example: CTMC
  10. Phases from phase-blind dynamics
  11. Updates, interventions and division events
  12. Divisibility criterion
  13. Operational division event from an initially uncorrelated environment.
  14. Conclusion
  15. The only Markovian $\theta$-process is the trivial process.
  16. ...and 5 more sections

Key Result

Theorem 1

If a stochastic process is Q-divisible at time $t_1$ (with $t_0<t_1<t_2$) and the lifted state $\rho(t_1)$ is diagonal at time $t_1$ for every initially diagonal state $\rho(t_0)$, then the process is C-divisible at time $t_1$.

Figures (1)

  • Figure 1: Commuting diagram for the lift compatibility condition. The map $D$ denotes the readout map from a diagonal operator to the corresponding probability vector, with $D\circ J=\mathrm{Id}$. The compatibility condition is $D\,\Pi\,\phi_{t,s}\,J=\Gamma(t\leftarrow s)$.

Theorems & Definitions (2)

  • Theorem 1: Divisibility criterion
  • proof