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Symbolic Quantum State Representation and its Simulation

Simon Sekavcnik, Janis Noetzel

Abstract

We introduce a symbolic operator framework for simulating quantum photonic systems that works directly with the canonical commutation relations and the Weyl algebra. Unlike existing Fock-space or Gaussian simulators, our method treats temporal wave packets and polarization modes in a continuous setting and does not rely on discretization or Hilbert-space truncation. Device operations are expressed as algebraic rewrite rules acting on creation and annihilation operators, allowing exact evolution of finite-photon states through linear optical networks. As an illustration, we reproduce Hong-Ou-Mandel interference for Gaussian pulses with controlled temporal and spectral mismatch.

Symbolic Quantum State Representation and its Simulation

Abstract

We introduce a symbolic operator framework for simulating quantum photonic systems that works directly with the canonical commutation relations and the Weyl algebra. Unlike existing Fock-space or Gaussian simulators, our method treats temporal wave packets and polarization modes in a continuous setting and does not rely on discretization or Hilbert-space truncation. Device operations are expressed as algebraic rewrite rules acting on creation and annihilation operators, allowing exact evolution of finite-photon states through linear optical networks. As an illustration, we reproduce Hong-Ou-Mandel interference for Gaussian pulses with controlled temporal and spectral mismatch.
Paper Structure (36 sections, 69 equations, 1 figure)

This paper contains 36 sections, 69 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mathematical framework
  3. Single photon space
  4. Gaussian mode family
  5. Fock space
  6. Operator Algebra
  7. Creation and Annihilation operators
  8. Operator Algebraic Structures
  9. Operator Monomials
  10. Normal Ordering
  11. Polynomial Operators
  12. Quantum States
  13. Contraction rules
  14. Trace and Partial Trace
  15. Observables and Expectation Values
  16. ...and 21 more sections

Figures (1)

  • Figure 1: Simulated coincidence probability in relation to the time delay and spectral detuning $P(\Delta t, \Delta \omega)$. The dip occurs, as predicted, when the two incoming photons are indistinguishable in all modeled degrees of freedom.