Spatial and particle-particle entanglement in 1D quantum walks of two distinguishable or indistinguishable bosonic particles

Abstract: We present entanglement measures between spatially separated regions and between two distinguishable or indistinguishable particles in one-dimensional two-particle continuous-time quantum walks governed by the Hubbard Hamiltonian. The left-right entanglement checks the entropy of coarse-grained states counting the numbers of particles on the left and right halves of the lattice while the particle-particle entanglement is based on the entropy of the singular values of the time-evolved Fock state. With separable, entangled, and doubly occupied initial states, we examine initial entanglement and the following growth in different entanglement measures. While the entanglement measures of the indistinguishable cases resemble those of the distinguishable cases when the initial states are comparable, the long-time limits of the entanglement measures are typically non-monotonic as the onsite repulsion increases. We also discuss possible implications for future research of entanglement in multi-particle quantum dynamics.

• The paper demonstrates that spatial and particle-particle entanglement in 1D quantum walks depends non-monotonically on the interaction strength (U/J) across various initial states.
• It employs Hubbard and Bose-Hubbard Hamiltonians with optimized Fock basis construction to model and measure entanglement for both distinguishable and indistinguishable bosonic particles.
• The numerical analysis supports experimental realizability in quantum optical lattices and photonic chips, offering insights into many-body dynamics and entanglement scaling.

Spatial and Particle-Particle Entanglement in 1D Quantum Walks of Bosonic Particles

Model Framework and Entanglement Measures

The study presents a thorough analysis of entanglement in one-dimensional continuous-time quantum walks (QWs) of two bosonic particles, focusing on both distinguishable and indistinguishable cases. The dynamics are governed by the Hubbard and Bose-Hubbard Hamiltonians for distinguishable and indistinguishable particles, respectively, on a lattice of size $L$, explicitly incorporating onsite repulsion ($U$) and nearest-neighbor hopping ($J$). The authors systematically construct the Fock basis for both particle types, exploiting number conservation in the indistinguishable case for optimized computational scaling.

For quantification, two central entanglement measures are implemented:

• Spatial (left-right) entanglement: Defined as the von Neumann entropy of coarse-grained sector probabilities $p_j$ for configurations with $n_L$ and $n_R$ particles on the left and right halves of the lattice, respectively, yielding $ES_{LR} = -\sum_j p_j \ln p_j$.
• Particle-particle entanglement: For distinguishable particles, the entropy derives from singular values of the time-evolved coefficient matrix in the Schmidt basis; for indistinguishable particles, the measure generalizes using the SVD of the symmetrized Fock-state coefficients to obtain $S_i = -\sum_k 2|y_k^S|^2 \ln(2|y_k^S|^2)$.

Numerical Analysis: Distinguishable Particles

For two distinguishable walkers, the authors analyze three canonical initial states:

• Separable (adjacent, singly occupied sites): Shows rapid growth in spatial entanglement, with maximal values achieved around intermediate interaction strengths, while the particle-particle entropy remains zero for $U=0$ and exhibits a non-monotonic dependence on $U/J$ for $U$0.
• Entangled (adjacent, spin-entangled): Maintains elevated particle-particle entanglement throughout, due to initial spin correlation, while spatial entanglement mirrors behavior seen in the separable case.
• Doubly occupied (both particles at same site): Initial spatial entanglement is suppressed, but evolves non-trivially depending on $U$1. Energy conservation maintains a preference for doubly occupied states at large $U$2, and the particle-particle entropy demonstrates a peak at intermediate $U$3 but ultimately diminishes for very strong interactions.

  Figure 1: Left-right and up/down (spin) entanglement measures for two distinguishable particles, demonstrating distinct time evolution and interaction dependence for a separable initial state.

  

  Figure 2: Quantitative evaluation of left-right and spin entanglement for the entangled initial state, highlighting the effect of initial spin correlations.

  

  Figure 3: Entanglement profiles for the doubly occupied initial state, showing interplay of spatial distribution and interaction-induced correlation.

  

  Figure 4: Time-evolved occupation numbers for each spin component, visualizing dynamics for all three initial states.

Numerical Analysis: Indistinguishable Particles

The indistinguishable case, realized as two bosons with symmetrized Fock states, exhibits key qualitative similarities to the distinguishable scenario. For adjacent singly occupied and doubly occupied initial states:

• Singly occupied (adjacent sites): Spatial entanglement grows rapidly for low and moderate $U$4, with maximal values near $U$5. Particle-particle entropy starts at $U$6 due to intrinsic bosonic correlation and is robust under time evolution.
• Doubly occupied: The left-right entropy attains a maximum at intermediate $U$7 but is substantially suppressed for strong interactions, reflecting the dominance of doubly occupied outcomes. Notably, the particle-particle entanglement remains zero for $U$8, as non-interacting dynamics preserve the structure of the initial Fock state.

  Figure 5: Left-right and particle-particle entanglement for two indistinguishable particles with an adjacent, singly occupied initial state, revealing similarities to the distinguishable entangled case.

  

  Figure 6: Results for the doubly occupied initial state, showing reduced spatial entanglement at large $U$9 and time-evolution profiles for particle-particle correlations.

  

  Figure 7: Occupation number dynamics for two indistinguishable particles, contrasting the spread for adjacent and doubly occupied initial conditions.

Comparative Insights and Theoretical Implications

A central result is the non-monotonic dependence of long-time entanglement on the interaction parameter $J$0, robust across both particle types and various initial states. The study demonstrates that indistinguishable bosons with adjacent, singly occupied initial configuration mimic the entangled distinguishable case in their spatial and particle-particle entanglement characteristics, supporting the generalizability of the employed measures.

The analysis confirms that quantum dynamics, governed by energy conservation and the interplay of kinetic and interaction energies, crucially shape entanglement generation and saturation. For both bosonic species, the entanglement measures capture intricate time-dependent correlations that can now be directly probed in experimental setups, including quantum optical lattices and photonic chips.

The techniques are extensible to other statistics, including fermionic systems, with adaptations to account for antisymmetrization and the Pauli exclusion principle. Additionally, the spatial coarse-graining approach circumvents issues inherent to partial trace-based measures for indistinguishable particles, offering a unified framework for entanglement quantification.

Experimental Realizability and Prospects

Advances in quantum state tomography and reconstruction methods facilitate practical extraction of these entanglement measures in real-world quantum walk experiments. The measures are relevant for foundational studies in quantum computation, quantum simulation, and probing many-body quantum dynamics.

Future work may address:

• Scaling behavior with larger particle numbers and longer-range interactions,
• Real-time control protocols to maximize entanglement for quantum technologies,
• Extensions to network structures and higher-dimensional lattices.

Conclusion

This study provides a technically rigorous and numerically robust characterization of entanglement in 1D quantum walks of two bosonic particles, emphasizing the significance of spatial and particle-particle correlations as functions of interaction strength and initial state. The findings elucidate general principles applicable to a broad class of quantum dynamical systems and establish methodological foundations for further exploration of multi-particle entanglement in both theoretical and experimental contexts.