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RDCompositeBasis

The RDCompositeBasis class represents a composite basis formed from multiple RDBasis objects. It creates a complete basis for a composite quantum system by taking the Kronecker product of the individual basis matrices of the constituent RDBasis objects. In other words, it builds symbolic composite basis elements as products of the basis elements from each subsystem and computes their corresponding matrix representations. This composite basis is essential for projecting operators or matrices onto a more manageable symbolic representation of the full system.

Parameters

Attributes

  • bases (list[RDBasis]):

    • The original list of RDBasis objects forming the composite basis.

  • dim (int):

    • The total dimension of the composite basis, calculated as the product of the dimensions of all provided RDBasis objects.

  • basis (ndarray):

    • An array containing the symbolic composite basis elements.

  • basis_matrices (ndarray):

    • An array of matrix representations corresponding to the composite basis elements.
    • Each matrix is computed as the Kronecker product of the matrices of the constituent basis elements.

  • basis_ling_alg_norm (int or float):

    • A normalization factor for the composite basis.
    • When there is only one composite basis element, this factor equals the composite dimension.
    • Otherwise, it is computed from the trace of the product of the conjugate transpose of each basis matrix with itself.

Methods

project(self, to_be_projected)

Projects a given matrix onto the composite basis.

  • Parameters:

    • to_be_projected (Matrix or compatible ndarray):
      • The matrix (or operator) that is to be projected onto the composite basis.
        Ensure that the dimensions of to_be_projected are compatible with the composite dimension.

  • Returns:

    • A symbolic expression representing the projection of the input matrix in the composite basis.

  • Raises:

    • ValueError:
      • If the dimensions of to_be_projected do not match the expected composite dimensions.

Usage Example

Below is an example demonstrating how to create a RDCompositeBasis instance from multiple RDBasis objects, and how to project a matrix onto the composite basis.

from sympt import RDCompositeBasis, RDBasis
import numpy as np

spin = RDBasis('sigma', dim=2)
charge = RDBasis('tau', dim=2)

# Initialize the composite basis using the two RDBasis objects.
composite_basis = RDCompositeBasis(bases=[spin, charge])

# Print composite basis information.
print("Composite Dimension:", composite_basis.dim)

# Suppose we have a matrix (operator) defined in the full composite space.
# This matrix should be of dimensions (dim x dim).
some_matrix = np.array([
    [0, 1, 0, 0],
    [1, 0, 0, 0],
    [0, 0, 0, -1],
    [0, 0, -1, 0]
])

# Project the matrix onto the composite basis.
projected_expression = composite_basis.project(some_matrix)
print("Projected Expression onto the Composite Basis:\n")
display(projected_expression)

Composite Dimension: 4
Projected Expression onto the Composite Basis:

\(\displaystyle \sigma_{3} \tau_{1}\)

License

SymPT is licensed under the MIT License. See the LICENSE file for details.

Citation

If you use SymPT in your research, please cite the following paper:

BibTeX Entry:

@misc{diotallevi2024symptcomprehensivetoolautomating,
      title={SymPT: a comprehensive tool for automating effective Hamiltonian derivations}, 
      author={Giovanni Francesco Diotallevi and Leander Reascos and Mónica Benito},
      year={2024},
      eprint={2412.10240},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2412.10240}, 
}

APA Citation:

Diotallevi, G. F., Reascos, L., & Benito, M. (2024). SymPT: a comprehensive tool for automating effective Hamiltonian derivations. arXiv preprint arXiv:2412.10240.

IEEE Citation:

G. F. Diotallevi, L. Reascos, and M. Benito, "SymPT: a comprehensive tool for automating effective Hamiltonian derivations," arXiv preprint arXiv:2412.10240, 2024.

References