toqito.state_props.is_abs_ppt

Checks if a quantum state is absolutely PPT.

Module Contents

toqito.state_props.is_abs_ppt.is_abs_ppt(mat, dim=None, rtol=1e-05, atol=1e-08)[source]

Determine whether or not a matrix is absolutely PPT [@Hildebrand_2007_AbsPPT].

A Hermitian positive semidefinite matrix is absolutely PPT iff it is PPT under all unitary transformations. Equivalently, if the matrix operates on a Hilbert space (H_{nm}) of dimension (nm), then it is PPT under all possible decompositions of (H_{nm}) as (H_{n} otimes H_{m}). Being absolutely PPT is a spectral condition (i.e. it is a condition on the eigenvalues of the matrix).

The function allows passing a numpy ndarray or a cvxpy Variable for mat:

  • If mat is a numpy ndarray, the function first checks if mat is Hermitian positive

    semidefinite. Then, it checks if its eigenvalues satisfy the Gerschgorin circle property (see Theorem 7.2 of [@Jivulescu_2015_Reduction]). Then it checks if the matrix belongs to the separable ball by calling in_separable_ball. Finally, if all the above checks fail to return a definite result, it determines if the matrix is absolutely PPT by checking if all the constraint matrices returned by abs_ppt_constraints are positive semidefinite.

  • If mat is a cvxpy Variable, mat must be a 1D vector representing the eigenvalues of

    a matrix. The function then returns the list of cvxpy Constraints required for optimizing over the space of absolutely PPT matrices. This includes the positive semidefinite constraint on each constraint matrix as well as mat[0] ≥ mat[1] ≥ … ≥ mat[-1] ≥ 0.

This function is adapted from QETLAB [@QETLAB_link].

!!! Note

If min(dim) (leq 6), this function checks all constraints and therefore returns True or False in all cases. However, if min(dim) (geq 7), only the first (33592) constraints are checked, since there are over (23178480) constraints in this case [@Johnston_2014_Orderings]. Therefore the function returns either False if at least one constraint was not satisfied, or None if all checked constraints were satisfied.

Examples

A random density matrix will likely not be absolutely PPT:

`python exec="1" source="above" import numpy as np from toqito.rand import random_density_matrix from toqito.state_props import is_abs_ppt rho = random_density_matrix(9) # assumed to act on a 3 x 3 bipartite system print(f"ρ is absolutely PPT: {is_abs_ppt(rho, 3)}") `

The maximally-mixed state is an example of an absolutely PPT state:

`python exec="1" source="above" import numpy as np from toqito.states import max_mixed from toqito.state_props import is_abs_ppt rho = max_mixed(9) # assumed to act on a 3 x 3 bipartite system print(f"ρ is absolutely PPT: {is_abs_ppt(rho, 3)}") `

Raises:

  • TypeError – If mat is not a numpy ndarray or a cvxpy Variable.

  • ValueError – If mat is a numpy ndarray but is not square.

  • ValueError – If mat is a cvxpy Variable but is not 1D.

  • ValueError – If dim does not divide the dimensions of mat.

Parameters:

  • mat (numpy.ndarray | cvxpy.Variable) – A square matrix.

  • dim (int | None) – The dimension of any one subsystem on which mat acts. If None, dim is selected such that

  • `min –

  • [@Hildebrand_2007_AbsPPT]). ((see Theorem 2 of)

  • rtol (float) – The relative tolerance parameter (default 1e-05).

  • atol (float) – The absolute tolerance parameter (default 1e-08).

Returns:

If mat is a 1D cvxpy Variable, return a list of cvxpy Constraints required for optimizing over the space of absolutely PPT matrices.

Return type:

bool | None | list[cvxpy.Constraint]