matrix_ops.tensor_comb¶

Compute tensor combination of list of vectors.

Functions¶

tensor_comb(states, k[, mode, density_matrix])

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

Module Contents¶

matrix_ops.tensor_comb.tensor_comb(states, k, mode='injective', density_matrix=True)¶

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])

result = tensor_comb([e_0, e_1], 2, mode="injective", density_matrix=True)

for key, mat in result.items():
    print(f"tensor_comb{key} =\n{mat}\n")

tensor_comb(0, 1) =
[[0 0 0 0]
 [0 1 0 0]
 [0 0 0 0]
 [0 0 0 0]]

tensor_comb(1, 0) =
[[0 0 0 0]
 [0 0 0 0]
 [0 0 1 0]
 [0 0 0 0]]

References

Raises:

ValueError – If the input list of states is empty.

Parameters:

states (list[numpy.ndarray]) – A list of state vectors.
k (int) – The length of the sequence.
mode (str) – Determines the type of sequences. Default is "injective". non-injective will allow repetitions in sequences, injective will ensures sequences are injective (no repetitions) and diagonal will allow sequences with repeated indices (diagonal elements).
density_matrix (bool) – Determines whether the return is a density matrix or a ket. Default is True.

Returns:

A dictionary where keys are tuples representing sequences of state indices, and values are density matrices of the tensor products of the corresponding state vectors or tensor products of the corresponding state vectors based on input density_matrix being either True or False.

Return type:

dict[tuple[int, Ellipsis], numpy.ndarray]