matrix_ops.tensor_comb

Compute tensor combination of list of vectors.

Functions

tensor_comb(states, k)

Generate all possible tensor product combinations of quantum states (vectors).

Module Contents

matrix_ops.tensor_comb.tensor_comb(states, k)

Generate all possible tensor product combinations of quantum states (vectors).

This function creates a tensor product of quantum state vectors by generating all possible sequences of length k from a given list of quantum states, and computing the tensor product for each sequence.

Given n quantum states, this function generates n^k combinations of sequences of length k, computes the tensor product for each sequence, and converts each tensor product to its corresponding density matrix.

For one definition and usage of a quantum sequence, refer to [1].

Examples

Consider the following basis vectors for a 2-dimensional quantum system.

\[e_0 = \left[1, 0 \right]^{\text{T}}, e_1 = \left[0, 1 \right]^{\text{T}}.\]

We can generate all possible tensor products for sequences of length 2.

>>> from toqito.matrix_ops import tensor_comb
>>> import numpy as np
>>> e_0 = np.array([1, 0])
>>> e_1 = np.array([0, 1])
>>> tensor_comb([e_0, e_1], 2)
{(0, 0): array([[1, 0, 0, 0],
       [0, 0, 0, 0],
       [0, 0, 0, 0],
       [0, 0, 0, 0]]), (0, 1): array([[0, 0, 0, 0],
       [0, 1, 0, 0],
       [0, 0, 0, 0],
       [0, 0, 0, 0]]), (1, 0): array([[0, 0, 0, 0],
       [0, 0, 0, 0],
       [0, 0, 1, 0],
       [0, 0, 0, 0]]), (1, 1): array([[0, 0, 0, 0],
       [0, 0, 0, 0],
       [0, 0, 0, 0],
       [0, 0, 0, 1]])}

References

[1]

Tathagata Gupta, Shayeef Murshid, Vincent Russo, and Somshubhro Bandyopadhyay. Optimal discrimination of quantum sequences. arXiv preprint arXiv:2409.08705, 2024.

Raises:

ValueError – If the input list of states is empty.

Parameters:

  • states (list[numpy.ndarray]) – A list of quantum state vectors represented as numpy arrays.

  • k (int) – The length of the sequence for generating tensor products.

Returns:

A dictionary where: - Keys represent sequences (as tuples) of quantum state indices, - Values are density matrices corresponding to the tensor product of

the state vectors for the sequence.

Return type:

dict