toqito.matrix_props.is_ldoi

Check if a quantum state is LDOI (Local Diagonal Orthogonal Invariant).

Module Contents

toqito.matrix_props.is_ldoi.is_ldoi(mat, rtol=1e-05, atol=1e-08)[source]

Check if a quantum state is LDOI (Local Diagonal Orthogonal Invariant).

A matrix (A in mathcal{L}(mathcal{X} otimes mathcal{Y})) is called local diagonal orthogonal invariant (LDOI) if (Phi_O(A) = A), where (Phi_O) is the local diagonal orthogonal twirl map defined as:

[

Phi_O(A) = frac{1}{2^n} sum_{O in text{DO}(mathcal{X})} (O otimes O) A (O otimes O)

]

where (text{DO}(mathcal{X})) is the set of diagonal matrices with entries (pm 1).

LDOI states include many important families such as Werner states, isotropic states, X-states, and mixtures of Dicke states. This function efficiently checks the LDOI property using the standard basis representation.

Examples

X-states are examples of 2-qubit LDOI states:

```python exec=”1” source=”above” from toqito.matrix_props import is_ldoi import numpy as np

# Example X-state x_state = np.array([[1, 0, 0, 2],

[0, 3, 4, 0], [0, 5, 6, 0], [7, 0, 0, 8]])

print(is_ldoi(x_state)) ```

All diagonal states are LDOI:

```python exec=”1” source=”above” from toqito.matrix_props import is_ldoi import numpy as np

diagonal_state = np.diag([1, 2, 3, 4]) print(is_ldoi(diagonal_state)) ```

Non-LDOI states return False:

```python exec=”1” source=”above” from toqito.matrix_props import is_ldoi import numpy as np

# Random non-LDOI state non_ldoi = np.array([[1, 2, 3, 4],

[5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]])

print(is_ldoi(non_ldoi)) ```

Parameters:

  • mat (numpy.ndarray) – A matrix representing a quantum state.

  • rtol (float) – Relative tolerance parameter (default: 1e-05).

  • atol (float) – Absolute tolerance parameter (default: 1e-08).

Returns:

True if the matrix is LDOI, False otherwise.

Return type:

bool