toqito.state_props.is_unextendible_product_basis

Check if a set of states form an unextendible product basis.

Module Contents

toqito.state_props.is_unextendible_product_basis.is_unextendible_product_basis(vecs, dims)[source]

Check if a set of vectors form an unextendible product basis (UPB) [@Bennett_1999_UPB].

Consider a multipartite quantum system (mathcal{H} = bigotimes_{i=1}^{m} mathcal{H}_{i}) with (m) parties with respective dimensions (d_i, i = 1, 2, …, m). An (incomplete orthogonal) product basis (PB) is a set (S) of pure orthogonal product states spanning a proper subspace (mathcal{H}_S) of (mathcal{H}). An unextendible product basis (UPB) is a PB whose complementary subspace (mathcal{H}_S-mathcal{H}) contains no product state. This function is inspired from IsUPB in [@QETLAB_link].

Examples

See [tile()][toqito.states.tile.tile]. All the states together form a UPB:

`python exec="1" source="above" import numpy as np from toqito.states import tile from toqito.state_props import is_unextendible_product_basis upb_tiles = np.array([tile(i) for i in range(5)]) dims = np.array([3, 3]) print(is_unextendible_product_basis(upb_tiles, dims)) `

However, the first 4 do not:

`python exec="1" source="above" import numpy as np from toqito.states import tile from toqito.state_props import is_unextendible_product_basis non_upb_tiles = np.array([tile(i) for i in range(4)]) dims = np.array([3, 3]) print(is_unextendible_product_basis(non_upb_tiles, dims)) `

The orthogonal state is given by

[

frac{1}{sqrt{2}} |2rangle left( |1rangle + |2rangle right)

]

Raises:

  • ValueError – If product of dimensions does not match the size of a vector.

  • ValueError – If at least one vector is not a product state.

Parameters:

  • vecs (list[numpy.ndarray]) – The list of states.

  • dims (list[int]) – The list of dimensions.

Returns:

Returns a tuple. The first element is True if input is a UPB and False otherwise. The second element is a witness (a product state orthogonal to all the input vectors) if the input is a PB and None otherwise.

Return type:

tuple[bool, numpy.ndarray | None]