Simulation of depolarizing channel exploring maximally non separable spin-orbit mode

1. Problem Statement

Quantum information processes are highly susceptible to decoherence, with the Depolarizing Channel being one of the most critical noise models to study. It represents a scenario where a qubit has a probability $\lambda$ of becoming a completely mixed state and a probability $(1-\lambda)$ of remaining unchanged.

• The Challenge: While theoretical models exist, experimentally emulating this channel requires robust platforms. Previous attempts using light degrees of freedom (DoF) often relied on the Solovay-Kitaev (SK) decomposition, which involves complex optical circuits with ancillary systems, multiple CNOT gates, and numerous optical elements (e.g., Dove prisms), leading to alignment difficulties and accumulated experimental errors.
• The Goal: To develop a simpler, more robust, and compact experimental method to simulate the dynamics of a single qubit under a depolarizing channel using classical light (intense laser beams) that mimics quantum behavior.

2. Methodology

The authors propose and experimentally implement two strategies using Spin-Orbit (SO) modes of light, where the polarization DoF acts as the system qubit and the transverse spatial mode (Hermite-Gaussian modes) acts as an ancillary system.

A. Theoretical Framework

• Depolarizing Channel: Defined by the map $\mathcal{E}(\rho) = \lambda \frac{I}{2} + (1-\lambda)\rho$. In the Bloch sphere representation, this corresponds to a uniform contraction of the Bloch vector.
• Spin-Orbit Modes: The authors utilize the mathematical equivalence between the tensor product of polarization and transverse modes and a bipartite entangled state. A maximally non-separable mode (analogous to a Bell state) is used to generate a maximally mixed state when the transverse mode is traced out.
• Quantum Coherence Measures: To quantify the effects of the channel, the authors employ:
  • $l_1$-norm Coherence ($C_{l1}$): Measures basis-dependent superposition.
  • Maximal Coherence ($C_{max}$): A basis-independent measure related to state purity.

B. Experimental Approaches

1. Baseline: Solovay-Kitaev (SK) Decomposition:

   • The authors first replicate the standard SK-decomposition method (previously implemented by their group).
   • Setup: Uses polarization as the system and HG modes as the ancilla.
   • Components: Requires four Dove prisms (for rotation in the ancilla space), wave plates for unitary transformations, and a complex arrangement of CNOT-like operations.
   • Control: The depolarization parameter $\lambda$ is controlled via convex combination weights ($p$) and rotation angles.

2. Proposal: Compact Linear Optical Circuit:

   • Core Concept: Instead of decomposing the channel into unitary operations, the authors exploit the properties of a maximally non-separable SO mode.
   • Mechanism:
     • An initial vertically polarized Gaussian beam is split by a Polarizing Beam Splitter (PBS).
     • One arm prepares an arbitrary initial qubit state.
     • The other arm passes through an S-plate (a specific optical element) that converts a Gaussian beam into a maximally non-separable state: $\frac{1}{\sqrt{2}}(|Hh\rangle - |Vv\rangle)$.
     • These two paths are recombined at a beam splitter.
   • Key Innovation: By tracing out the transverse mode (spatial mode) after the S-plate interaction, the polarization state naturally evolves into a mixture of the initial state and the maximally mixed state. The depolarization parameter $\lambda$ is directly and simply controlled by the angle $\theta$ of a Half-Wave Plate (HWP) at the input.
   • Advantage: This eliminates the need for multiple Dove prisms and complex ancillary logic, relying instead on the intrinsic properties of the non-separable mode.

3. Key Contributions

• Novel Simulation Method: The paper presents the first experimental demonstration of a depolarizing channel using a compact linear optical circuit based on maximally non-separable spin-orbit modes, bypassing the need for the complex Solovay-Kitaev decomposition.
• Direct Parameter Control: The proposed circuit allows for direct, continuous control of the depolarization parameter $\lambda$ via a single wave plate angle, whereas the SK method requires adjusting multiple parameters ($p, \alpha, \beta, \gamma$) simultaneously.
• First SK-Decomposition Results for Depolarizing Channel: The authors provide the first experimental results of the Depolarizing Channel implemented via SK-decomposition, serving as a rigorous benchmark.
• Robustness Analysis: A comparative analysis demonstrating that the compact circuit yields significantly higher fidelity and robustness against experimental noise compared to the SK-decomposition approach.

4. Experimental Results

The authors tested both methods using two input states:

1. Vertical Polarization ($|V\rangle$): A diagonal state in the density matrix.
2. Diagonal Superposition ($|+\rangle$): A state with maximum coherence.

Findings:

• Density Matrix Reconstruction:
  • SK-Decomposition: Showed good agreement with theory but suffered from lower fidelity at the initial state ($\lambda=0$) due to experimental errors introduced by the four Dove prisms (e.g., $F \approx 71.86\%$ for $|V\rangle$).
  • Compact Circuit: Achieved near-perfect fidelity ($F \approx 100\%$ for $|V\rangle$ at $\lambda=0$) and maintained high fidelity ($>99.8\%$) even at the maximally mixed state ($\lambda=1$).
• Coherence Evolution:
  • Both methods correctly reproduced the theoretical decay of $l_1$-norm coherence and maximal coherence ($C_{max}$) as $\lambda$ increased.
  • The compact circuit results showed less deviation from theoretical predictions, particularly for the $C_{max}$ of the $|V\rangle$ state, confirming the superior stability of the new method.
• Bloch Sphere Contraction: The experimental data confirmed the uniform contraction of the Bloch vector as predicted by the depolarizing channel theory.

5. Significance and Conclusion

• Simplicity and Efficiency: The proposed compact circuit drastically reduces the number of optical elements and alignment complexity compared to the SK-decomposition, making it a more practical tool for quantum information simulations.
• Resource Scaling: The method demonstrates that maximally non-separable classical modes can effectively emulate quantum noise channels without the overhead of complex ancillary logic.
• Applications: This setup serves as a powerful testbed for studying:
  • Noise-induced effects and robustness in quantum protocols.
  • The transition from quantum to classical behavior.
  • Protocols involving mixed states, such as quantum key distribution without shared reference frames or communication through entanglement-breaking channels.
• Future Outlook: The authors suggest this approach can be extended to investigate more complex channel architectures, such as superpositions of depolarizing channels with indefinite causal orders.

In summary, the paper successfully validates a streamlined, high-fidelity optical simulation of the depolarizing channel, offering a superior alternative to existing decomposition-based methods for studying quantum decoherence.