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The Universe as a Detector: A Quantum Filtering Formulation of the Diósi-Penrose Model

John Gough, Dylon Rees

TL;DR

The paper recasts the Diósi-Penrose gravity-induced decoherence as a quantum filtering problem driven by space-time measurements. It develops a quantum Kushner-Stratonovich type equation within the Hudson-Parthasarathy open quantum systems framework, linking the DP Lindbladian to a conditioned state evolution. A spectral diagonalization of the Newtonian Green's function and a space-time input-output formalism are used to derive both Gaussian and Poisson counting variants of the quantum filter, including a nonlocal kernel $1/|x-y|$. The result offers a conceptual link between gravity-induced decoherence and quantum filtering, framing the Universe as a continuous observer that shapes wave function collapse without specifying a microscopic mechanism.

Abstract

We consider the Diósi-Penrose problem but rather than postulating background gravitational fluctuations, we instead consider the quantum filter that arises from space-time homodyning the continuum of output quadrature described in the open quantum stochastic model presented here. This is described by a quantum Kushner-Stratonovich equation, typical of the form appearing in continuous-time collapse of the wave-function models in Quantum Decoherence Theory

The Universe as a Detector: A Quantum Filtering Formulation of the Diósi-Penrose Model

TL;DR

The paper recasts the Diósi-Penrose gravity-induced decoherence as a quantum filtering problem driven by space-time measurements. It develops a quantum Kushner-Stratonovich type equation within the Hudson-Parthasarathy open quantum systems framework, linking the DP Lindbladian to a conditioned state evolution. A spectral diagonalization of the Newtonian Green's function and a space-time input-output formalism are used to derive both Gaussian and Poisson counting variants of the quantum filter, including a nonlocal kernel . The result offers a conceptual link between gravity-induced decoherence and quantum filtering, framing the Universe as a continuous observer that shapes wave function collapse without specifying a microscopic mechanism.

Abstract

We consider the Diósi-Penrose problem but rather than postulating background gravitational fluctuations, we instead consider the quantum filter that arises from space-time homodyning the continuum of output quadrature described in the open quantum stochastic model presented here. This is described by a quantum Kushner-Stratonovich equation, typical of the form appearing in continuous-time collapse of the wave-function models in Quantum Decoherence Theory
Paper Structure (10 sections, 5 theorems, 63 equations)

This paper contains 10 sections, 5 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Diagonalizing the Poisson Equation Green's Function
  4. Quantum Open System Model
  5. Quantum Stochastic Calculus
  6. Unitary Dynamics
  7. Derivation of the Quantum Filter
  8. Comparison with Diósi's Model
  9. Number Counting
  10. Discussion and Conclusion

Key Result

Proposition 1

The Green's function $g$ admits a convolutional '' square root'' $\gamma$ such that This is explicitly given by

Theorems & Definitions (7)

  • Proposition 1
  • Remark 2
  • Remark 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Theorem 7: The Quantum Filtering Equation