Symbolic Quantum State Representation and its Simulation

1. Problem Statement

Current simulation frameworks for quantum photonic systems face significant limitations when handling non-Gaussian states, continuous-mode fields, and mode distinguishability (temporal, spectral, or polarization mismatch).

• Fock-space approaches (e.g., QuTiP, Strawberry Fields) rely on truncated, orthogonal bases. They require discretization of continuous wave packets, which can introduce errors and struggle with non-orthogonal mode overlaps.
• Gaussian simulators (e.g., The Walrus) are efficient for Gaussian states but fail when non-Gaussian resources or complex spectral/temporal distinguishability are introduced.
• Quantum Input-Output (QIO) theory offers elegant continuous-time descriptions but is primarily a theoretical framework that does not generate explicit multi-photon output states for interferometric networks.

There is a lack of a unified simulation framework that can handle arbitrary finite-photon states in a continuous-mode setting without discretization or Hilbert-space truncation.

2. Methodology

The authors propose a Symbolic Operator Framework that operates directly on the Canonical Commutation Relations (CCR) and the underlying Weyl algebra. Instead of representing states as vectors in a truncated matrix, the method treats quantum states and device operations as algebraic expressions.

A. Mathematical Foundation

• Hilbert Space Construction: The single-photon space ($H_1$) is defined as the tensor product of a temporal degree of freedom ($L^2(\mathbb{R}, \mathbb{C})$) and a polarization degree of freedom ($\mathbb{C}^2$).
• Global Fock Space: The global state space is constructed as a tensor product of Fock spaces for distinct optical channels. Crucially, modes in different channels are orthogonal, while modes within the same channel can be non-orthogonal (continuous-mode wave packets).
• Operator Algebra:
  • Ladder Operators: Creation ($\hat{a}^\dagger$) and annihilation ($\hat{a}$) operators are defined for arbitrary temporal profiles, polarizations, and paths.
  • Commutation Relations: The framework utilizes the smeared ladder operators where $[\hat{a}(m_i), \hat{a}^\dagger(m_j)] = \langle m_i, m_j \rangle$, allowing for non-orthogonal mode overlaps.
  • Polynomial Representation: Quantum states are represented as polynomials of normally ordered monomials acting on the vacuum. For example, a two-photon state is $\hat{a}^\dagger(m_1)\hat{a}^\dagger(m_2)|0\rangle$.
  • Contraction Rules: Calculations of inner products, traces, and expectation values are performed by applying normal ordering rules. When an annihilation operator passes a creation operator, it generates a contraction term equal to the inner product of the modes ($\langle m_i, m_j \rangle$). Terms with uncontracted operators vanish when acting on the vacuum.

B. Device Maps (Heisenberg Picture)

Device operations are modeled as algebraic rewrite rules that transform the ladder operators:

• Sources: Prepend creation operators to the state (e.g., deterministic single-photon sources or entangled pair sources).
• Linear Passive Devices: Beam splitters, phase shifters, and polarization rotators are implemented as unitary transformations on the operator tuple, preserving CCRs.
• Lossy/Filtering Channels: Modeled via Bogoliubov transformations that mix system modes with environmental vacuum modes. This allows for the simulation of attenuation, spectral filtering, and polarization filtering without explicit discretization.
• Amplifiers: Modeled as phase-insensitive amplifiers that introduce necessary quantum noise to preserve CCRs.
• Detection: Implemented via Positive Operator-Valued Measures (POVMs) acting on the operator polynomials.

3. Key Contributions

1. Discretization-Free Simulation: The framework eliminates the need for time-bin discretization or Fock-space truncation, enabling exact evolution of continuous-mode wave packets.
2. Unified Treatment of Non-Orthogonal Modes: By working directly with the CCR algebra, the method naturally handles non-orthogonal temporal and spectral modes, a capability often missing in standard Fock-space simulators.
3. Algebraic Rewrite System: The introduction of device maps as symbolic rewrite rules acting on ladder operators provides a flexible and exact method for simulating linear optical networks.
4. Prototype Implementation: The authors developed a lightweight symbolic simulation prototype that implements the core algebraic machinery (operator formation, normal ordering, and contraction).

4. Results

The framework was validated by simulating the Hong-Ou-Mandel (HOM) interference effect with Gaussian wave packets.

• Setup: Two photons with controlled temporal ($\Delta t$) and spectral ($\Delta \omega$) mismatch were injected into a 50/50 beam splitter.
• Outcome: The simulation successfully reproduced the analytical HOM dip formula:
  $$P_{coinc} = \frac{1}{2}(1 - |\gamma|^2)$$
  where $\gamma$ is the overlap integral of the two photon modes.
• Visualization: The generated plots (Fig. 1) show the coincidence probability as a function of time delay and spectral detuning, perfectly matching theoretical predictions. The simulation correctly demonstrated that distinguishability (mismatch) reduces the interference visibility.

5. Significance

• Bridging Theory and Simulation: This work bridges the gap between rigorous continuous-mode theoretical frameworks (like QIO) and practical numerical simulation tools.
• Scalability for Complex Systems: It enables the study of quantum photonic systems involving non-Gaussian states, complex pulse shapes, and mode mismatch without the approximation errors inherent in discretized methods.
• Foundation for Future Tools: While the current prototype focuses on linear optics and finite-photon states, the algebraic structure is extensible. It lays the groundwork for simulating more complex quantum optical networks, including those with non-linear interactions and continuous-variable Gaussian states, in a unified manner.
• Faithfulness to Physics: The method remains faithful to the underlying optical physics by avoiding artificial basis truncations, making it particularly valuable for high-precision quantum metrology and communication protocols.